I CD
V
PROJECTIVE GEOMETRY
BY
OSWALD VEBLEN
PROFESSOR OF MATHEMATICS, PRINCETON UNIVERSITY AND
JOHN WESLEY YOUNG
PROFESSOR OF MATHEMATICS, DARTMOUTH COLLEOE
VOLUME II
BY OSWALD VEBLEN
GINN AND COMPANY
BOSTON • NKW YORK • CHICAGO • LONDON ATLANTA • i>.\i I \-( • COLUMBUS • SAN FRANCISCO
QK
V7I
COPYRIGHT, 1918, BT OSWALD VEBLEN
ALL RIGHTS RB8ERVBD PRINTED IN THE UNITED STATES OF AMERICA
130.7
GINS AM) COMPANY • PRO- PRIETORS • BOSTON • U.S.A.
PREFACE
The present volume is an attempt to carry out the program out- lined in the preface to Volume I. Unfortunately, Professor Young was obliged by the pressure of other duties to cease his collabora- tion at an early stage of the composition of this volume. Much of the work on the first chapters had already been done when this hap- pened, but the form of exposition has been changed so much since then that although Professor Young deserves credit for constructive work, he cannot fairly be held responsible for mistakes or oversights.
Professor Yroung has kindly read the proof sheets of this volume, as have also Professors A. B. Coble and A. A. Bennett Most of the drawings were made by Dr. J. W.Alexander. I offer my thanks to all of these gentlemen and also to Messrs. Ginu and Company, who have shown their usual courtesy and efficiency while converting the manuscript into a book.
The second volume has been arranged so that one may pass on a first reading from the end of Chapter VII, Volume I, to the beginning of Volume II. The later chapters of Volume I may well be read in connection with the part of Volume II from Chapter V onward.
I shall pass by the opportunity to discuss any of the pedagogical questions which have been raised in connection with the first vol- ume and which may easily be foreseen for the second. It is to be expected that there will continue to be a general agreement among those who have not made the experiment, that an abstract method of treatment of geometry is unsuited to beginning students.
In this book, however, we are committed to the abstract point of view. We have in mind two principles for the classification of any theorem of geometry : (a) the axiomatic basis, or bases, from which it can be derived, or, in other words, the class of spaces in which it can be valid ; and (b) the group to which it belongs in a given space.
iii
ix PBBFAI i-:
In tlu- first, volume we were always concerned with theorem* be- longing to the protective group, and these theorems were class! lift I According as they were consequences of the groups of Assumptions A, 1 II : A, I •:. 1'; or A, E, P, HO. Among the spaces satis-
i\ in:; A. K. I* (the properly projective spaces) may be mentioned the modular spaces, the rational nonmodular space, the real space, ;m<l the complex space. Any one of these may be specified categorically by adding the proper assumptions to A, E, P. The passage from the point of view of general projective geometry to that of the particular spaces is made in the first chapter of this volume.
Having fixed attention on any particular space, we have a set of groups of transformations to each of which belongs its geometry. For example, in the complex projective plane we find among others, (1) the group of all-continuous one-to-one reciprocal transformations (analysis situs), (2) the group of birational transformations (algebraic geometry), (3) the projective group, (4) the group of non-Euclidean geometry, (5) a sequence of groups connected with Euclidean geometry (cf. § 64). The groups (2), (3), (4), and (5) all have analogues in the other spaces mentioned in the paragraphs above, and consequently it is desirable to develop the theorems of the corresponding geometries in such a way that the assumptions required for their proofs are put in evidence in each case. This will be found illustrated in the chapters on affine and Euclidean geometry.
The two principles of classification, (a) and (J), give rise to a double sequence of geometries, most of which are of consequence in present-day mathematics. It is the purpose of this book to give an elementary account of the foundations and interrelations of the more important of these geometries (with the notable exception of (2)). May I venture to suggest the desirability of other books taking account of this logical structure, but dealing with particular types of geometric figures ?
The ideal of such books should be not merely to prove every theorem rigorously but to prove it in such a fashion as to show in which spaces it is true and to which geometries it belongs. Some idea of the form which would be assumed by a treatise on conic sections written in this fashion can be obtained from § 83 below. Other subjects for which this type of exposition would be feasible at the present time are quadric surfaces, cubic and quartic curves,
PREFACE v
rational curves, configurations, linear line geometry, collineation groups, vector analysis.
Books of this type could take for granted the foundational and coordinating work of such a book as this one, and thus be free to use all the different points of view right from the beginning. On the other hand, a general work like this one could be much abbreviated if there were corresponding treatises on particular geometric figures (for example, conic sections) to which cross references could be made.
OSWALD VEBLEX BROOKLIN, MAINE AUGUST, 1917
CONTENTS
CHAPTER I
FOUNDATIONS SECTION PAOK
1. Plan of the chapter 1
2. List of Assumptions A, E, P, and H0 1
3. Assumption K 3
4. Double points of proj activities 5
5. Complex geometry 6
6. Imaginary elements adjoined to a real space 7
7. Harmonic sequence 9
8. Assumption H 11
9. Order in a net of rationality 13
*10. Cuts in a net of rationality 14
*11. Assumption of continuity 16
*12. Chains in general ., 21
*13. Consistency, categoricalness, and independence of the assumptions ... 23
*14. Foundations of the complex geometry . 29
*15. Ordered projective spaces 32
*16. Modular projective spaces 33
17. Recapitulation 36
CHAPTER II ELEMENTARY THEOREMS ON ORDER
18. Direct and opposite projectivities on a line 37
19. The two sense-classes on a line 40
20. Sense in any one-dimensional form 43
21. Separation of point pairs 44
22. Segments and intervals 45
23. Linear regions 47
24. Algebraic criteria of sense 49
25. Pairs of lines and of planes 50
26. The triangle and the tetrahedron 52
27. Algebraic criteria of separation. Cross ratios of points in space .... 65
28. Euclidean spaces 58
29. Assumptions for a Euclidean space 69
30. Sense in a Euclidean plane 61
*31. Sense in Euclidean spaces 63
•32. Sense in a projective space 04
33. Intuitional description of the projective piaue 67
vii
CIIAITKK 111 mi u i IM SB01 I' IN TIII-: PLANE
PAGE
84. The geometry corresponding to a given group of transformations ... 70
85. Euclidean plane and the aftine group 71
80. 1'arallel linen 72
87. Ellipse, hyperbola, parabola 73
88. The group of translations 74
88. Self-conjugate subgroups. Congruence 78
40. Congruence of parallel point pairs 80
41. Metric properties of conies 81
«. Vector* 82
48. Ratios of collinear vectors 85
44. Theorems of Menelaus, Ceva, and Carnot 89
46. Point reflections 92
44). Extension of the definition of congruence 94
47. The homothetic group 95
48. Equivalence of ordered point triads 96
49. Measure of ordered point triads 99
50. The equiaffine group 105
•61. Algebraic formula for measure. Barycentric coordinates 106
•62. Line reflections 109
•68. Algebraic formulas for line reflections 116
64. Subgroups of the affine group 116
CHAPTER IV EUCLIDEAN PLANE GEOMETRY
66. Geometric* of the Euclidean type 119
66. Orthogonal lines 120
67. Displacements and symmetries. Congruence 123
68. Pairs of orthogonal line reflections 126
60. The group of displacements 129
60. Circles 131
61. Congruent and similar triangles 134
62. Algebraic formulas for certain parabolic metric groups 135
63. Introduction of order relations 138
•U. The real plane 140
66. Intenectional properties of circles 142
66. The Euclidean geometry. A set of assumptions 144
n '.< • n • 147
«. Area 149
60. The measure of angles 151
70. The complex plane 164
• '•HH of rircles 167
72. Measure of line pairs 168
78. Generalization by projection ,167
CONTENTS ix CHAPTER v
ORDINAL AND METRIC PROPERTIES OF CONIC8 SECTION PAGE
74. One-dimensional projectivities 170
75. Interior and exterior of a conic 174
70. Double points of projectivities 177
77. Ruler-and-compass constructions 180
78. Conjugate imaginary elements 182
79. Projective, affine, and Euclidean classification of conies 180
80. Foci of the ellipse and hyperbola * .... 189
81. Focus and axis of a parabola 193
82. Eccentricity of a conic 196
83. Synoptic remarks on conic sections 199
84. Focal properties of collineations 201
85. Homogeneous quadratic equations in three variables 202
86. Nonhomogeneous quadratic equations in two variables 208
87. Euclidean classification of point conies 210
88. Classification of line conies 212
*89. Polar systems 215
CHAPTER VI INVERSION GEOMETRY AND RELATED TOPICS
90. Vectors and complex numbers 219
91. Correspondence between the complex line and the real Euclidean plane . 222
92. The inversion group in the real Euclidean plane 226
93. Generalization by inversion 231
94. Inversions in the complex Euclidean plane 235
96. Correspondence between the real Euclidean plane and a complex pencil
of lines 238
96. The real inversion plane 241
97. Order relations in the real inversion plane 244
98. Types of circular transformations 246
99. Chains and antiprojectivities 250
100. Tetracyclic coordinates 253
101. Involutoric collineations 257
102. The projective group of a quadric 269
103. Real quadrics 262
104. The complex inversion plane 264
105. Function plane, inversion plane, and projective plane 268
106. Projectivities of one-dimensional forms in general 271
*107. Projectivities of a quadric 273
*108. Products of pairs of involutoric projectivities 277
109. Conjugate imaginary lines of the second kind 281
110. The principle of transference 284
x CONTENTS
CllAITKU VII
AK1 1I'1.\N ..I -Ml HIV <)K THREE DIMENSIONS
••TTlOX PAOK
111. Affin* geometry 287
11*. Vector*, equivalence of point trials, etc 288
113. The parabolic im-iric group. Orthogonal lines and planes 293
114. Orthogonal plum- n-tlt-. -timis 295
116. Displacements ami .-yiimirtni->. Congruence _".'7
116. Kiu-lii'.ciin yf •mi-try »i tlin-i- dimensions 301
•117. (ieneralizalion to n dimriisions 304
1 18. Equations of the affine and Euclidean groups 305
110. Distance, area, volume, angular measure 311
190. The sphere and other quadrics 315
1^1. Ui-.soliiti'in of a displacement into orthogonal line reflections .... 317
1*2. Rotation, translation, twist 321
123. Properties of displuri-ments 325
124. Correspondence between the rotations and the points of space .... 328 126. Algebra of matrices 333
126. Rotations of an imaginary sphere 336
127. Quaternions 337
128. Quaternions and the one-dimensional projective group 339
•129. Representation of rotations and one-dimensional projectivities by
points 342
130. Parameter representation of displacements 344
CHAPTER VIII NON-EUCLIDEAN GEOMETRIES
181. Hyperbolic metric geometry in the plane 350
132. Orthogonal lines, displacements, and congruence 352
188. Types of hyperbolic displacements 355
184. Interpretation of hyperbolic geometry in the inversion plane .... 357
186. Significance and history of non-Euclidean geometry 360
186. Angular measure 302
187. Distance 3^4
188. Algebraic formulas for distance and angle 305
•180. Diffi-n-iitial <.f arc 366
140. Hyperbolic geometry of three dimensions 369
141. Elliptic plant- geometry. iMinition 371
142. Elliptic geometry of three dimensions 373
148. Double elliptic geometry 375
144. Euclidean geometry as a limiting case of non-Euclidean :;:.,
!**• Parameter representation of elliptic displacements 377
1441. Parameter representation of hyperbolic displacements 380
COX TK NTS x'i
CHAPTER IX
THEOREMS ON SENSE AND SEPARATION SECTION PACK
147. Plan of the chapter 385
148. Convex regions 385
149. Further theorems on convex regions 388
160. Boundary of a convex region 392
151. Triangular regions 395
1 ."»!'. The tetrahedron . . . .' 397
153. Generalization to n dimensions 400
154. Curves 401
155. Connected sets, regions, etc 404
156. Continuous families of sets of points 405
157. Continuous families of transformations 406
158. Affine theorems on sense 407
l"i'.'. Elementary transformations on a Euclidean line 409
160. Elementary transformations in the Euclidean plane and space . ... 411
161. Sense in a convex region 413
162. Euclidean theorems on sense 414
163. Positive and negative displacements 416
164. Sense-classes in projective spaces 418
165. Elementary transformations on a projective line 419
166. Elementary transformations in a projective plane 421
167. Elementary transformations in a projective space 423
*168. Sense in overlapping convex regions 424
*169. Oriented points in a plane 425
*170. Pencils of rays 429
•171. Pencils of segments and directions 433
*172. Bundles of rays, segments, and directions 435
*173. One- and two-sided regions 436
174. Sense-classes on a sphere 437
175. Order relations on complex lines 437
176. Direct and opposite collineations in space 438
177. Right- and left-handed figures 441
178. Right- and left-handed reguli, congruences, and complexes 443
•179. Elementary transformations of triads of lines 446
•180. Doubly oriented lines 447
•181. More general theory of sense 451
182. Broken lines and polygons 454
183. A theorem on simple polygons 457
184. Polygons in a plane 458
185. Subdivision of a plane by lines 460
186. The modular equations and matrices 464
187. Regions determined by a polygon 467
188. Polygonal regions and polyhedra 473
189. Subdivision of space by planes 475
190. The matrices J/j, 7/2, and 7f8 477
191. The rank of II., .' 479
xu < "N TENTS
••mo* FAOK
IWt. Polygon, in space 480
11)3. Odd ami even polyhedra 482
1M. Region* bounded by a polyhedron 483
It*. The matrices A*, and Et for the projective plane 484
1IM. Odd and even polygons in the projective plane 489
i toe- and two-aided polygonal regions 490
106. One- and two-sided polyhedra 493
1W. Orientation of space 490
INDEX . 601
PROJECTIVE GEOMETRY
CHAPTER I
FOUNDATIONS
1. Plan of the chapter. In the first volume of this book we have been concerned with general projective geometry, that is to say, with those theorems which are consequences of Assumptions A, E, P. In many cases we also made use of Assumption HQ, but most of the theorems which we proved by the aid of this assumption remain true (though trivial) when this assumption is false. The class of spaces to which the geometry of Vol. I applies is very large, and the set of assumptions used is therefore far from categorical
The main purpose of geometry is, of course, to serve as a theory of that space in which we envisage ourselves and external nature. This purpose can be accomplished only partially by a geometry based on a set of assumptions which is not categorical. We therefore pro- ceed to add the assumptions which are necessary in order to limit attention to the geometry of reals, the geometry in which the number system is the real number system of analysis.
These assumptions are stated in two ways, the one (§ 3) dependent on the theory of the real number system and the other (§§ 7~13) independent of it. We also state the assumptions (§§ 5, 14, 15, 16) necessary for certain other geometries which are of importance because of their relations to the real geometry and to other branches of mathematics. At the end of the chapter we give, a summary of the assumptions for the various projective geometries which we are considering.
2. List of Assumptions A, E, P, and H8. For the sake of having all the assumptions before us in the present chapter, we reprint A, E, P, and HQ. The assumptions serve to determine a class S of elements called points, and a class of subclasses of S called lines. The phrase
l
KM NDATIONS
< II A f. I
E
44 a point b on a line " or " a line is on a point " means that the point belongs to tin- lim- («-f. p. 1 ft. Vol. I).
ASSUMPTIONS OF ALHSNMKM :
A 1 // .1 nut/ ll are distinct points, there is at least one line on both A nn.i /:.
A 2. If A and B are distinct points, there is not more than one
II l*>th .1 <//«/ /•'.
A3. If A, B,C are points not nil on the same line, and D and E (D * E) are points such that B, C, D are on a line <in>f (', A, E are on a line, there is a point F such that A, B, F are on a line and also D, E, F are on a line.
_, FIG. 1
ASSUMPTIONS OF KXTKNSION:
E 0. There are at least three points on every line.
E 1. There exists at least one line.
E2. All points are not on the same line.
E 3. All points are not on the same plane*
E 3'. If S8 is a three-space,^ every point is on Sg.
ASSUMPTION OF PROJECTIVITY :
P. If a projectivity leaves each of three distinct points of a line invariant, it leaves every point of the line invariant.!-
ASSUMPTION H0:
HO. The diagonal points of a complete quadrangle are noncollinear.§
i As was explained when Assumption P was first introduced, this
assumption does not appear in the complete list of assumptions for the geometry of reals, but is replaced by certain other assumptions from whirh it (as well as HQ) can be derived as a theorem. The list of assumptions for this geometry will consist of Assumptions A, K, ami the new assumptions.
•Cf. §7. V,,l. I. t Cf. i », Vol. I.
JCf. §35, Vol. I. { Cf . § 18, Vol. I.
§§•_>,»] ASSUMPTION K 3
3. Assumption K. The most summary way of completing the list of assumptions for the geometry of reals is to introduce the following :
K. A geometric number system (Chap. VI, Vol. I) is isomorphic* with the real number system of analysis.
Thus a complete list of assumptions for the geometry of reals is A, £, K.
The use of Assumption K implies a previous knowledge of the real number system. f Its apparent simplicity therefore masks certain real difficulties. What these difficulties are from a geometric point of view will be found on reading §§ 7-13, where K is analyzed into indej>endent statements II, C, R. These sections, however, may be omitted, if desired, on a first reading.
Since a geometric number system in one one-dimensional form is isomorphic with any geometric number system in any one-dimensional form in the same space, it is evident that the principle of duality is valid for all theorems deducible from Assumptions A, E, K.
In order that the results of Vol. I be applicable to the geometry of reals, it must be shown that Assumption P is a logical conse- quence of Assumptions A, E, K. Since multiplication is commuta- tive in the real number system, this result would follow directly from Theorem 7, Chap. VI, Vol. I. The proof there given is, how- ever, incomplete. It is shown (Theorem 6, loc. cit.) that if P holds, multiplication is commutative; but it is not there proved that if multiplication is commutative, P is satisfied. The needed proof may be made as follows :
THEOREM 1. Assumption P is valid in any space satisfying Assumptions A and E and such that multiplication is commutative in a geometric number system (Chap. VI, Vol. I).
Proof. It is obvious that the number systems determined by any two choices of the fundamental points HJi^H* are isomorphic (cf. Theorems 1 and 3, Chap. VI, Vol. I), so that we may base our argument on an arbitrary choice of these points. We are assuming that multi- plication is commutative, and are to prove that any projectivity II
* This term is defined in § 52, Vol. I.
t The real number system is to be thought of either as defined in terms which rt-st ultimately on the positive integers (cf. Pierpont, Theory of Functions of Keal Variables, pp. 1-94 ; or Fine, College Algebra, pp. 1-70) or by means of a set of postulates (cf. K. V. Iluiitington, Transactions of the American Mathematical Society, Vol. VI (1905), p. 17).
4 Kol' N PATH >NS [CHAP. I
which leaves three disiin.-t ^.ints of a line fixed is the identity. I'.y .teliniiioii. II i> ihf resultant of a sequence of perspectives
win-re [//I denotes tlu- points of the given line. By Theorem 5, Chap. III. Vol. I, this chain of perspectives may be replaced by three perspectivities
Moreover, by Theorem 4, Chap. Ill, Vol. I, the pencils [P] and [Q] may be chosen so that their respective axes pass through two of the given fixed points of II. Let us denote these points by Hx and H9
FIG. 2
respectively and let 7/» be the third fixed point. By another applica- tion of Theorem 4 the pencils [P] and [Q] may be chosen so that their common point R is on the line SHn (fig. 2).
Now, since II » is transformed into itself, S, H*, and U must be colliuear. Since Hf is fixed, T, Ifx, and U must be collinear. Since //r is fixed, .V, T, and H9 are collinear. If H is any point of the line //,//,. it is transformed by the perspectivity with S as center to a point /' of tlu- line //,#; the per.spe.e.tivily with T as center trans- forms P to a point Q of the line RHV; the perspectivity with U as
55 :«.-»] DOUBLE POINTS OF PROJECTIVITIES 5
center transforms Q back to a point //' of the line HXHV. We have to show that H' = H.
Let //0 be the trace on the line HxHf of PT; let HI be the trace of RT; and H' is the trace of UQ.
The complete quadrangle TRSP determines Q (ff0ffvHlt and hence (Theorem 3, Chap. VI, Vol. I) in the scale
The complete quadrangle TRQU determines Q (HQHxHlt and hence in the scale HHI^
Since multiplication is commutative, H= H', which proves the theorem.
The reader will find no difficulty in using the construction above to prove that the validity of the theorem of Pappus (§ 36, Vol. I) is necessary and sufficient for the commutative law of multiplication and for Assumption P.
4. Double points of projectivities. DEFINITION. A projective trans- formation of a real line into itself is said to be hyperbolic, parabolic, or elliptic* according as it has two, one, or no double points.
It was proved in § 58, Vol. I, that the determination of the double points of a projective transformation t
px'0^ax0 + bx1 px[ = cx0 + dxt
depends on the solution of the equation (2) p2-(a
where A = ad — be. This equation has two real roots if and only if
its discriminant 7,2
(a + df — 4 A
is positive. Hence we have
If A < 0, the transformation (1) is hyperbolic. For an elliptic or parabolic projectivity A is always positive.
* These terms are derived from the corresponding types of conic sections (see § :>7). Jn a complex one-dimensional form a somewhat different terminology is used (cf. § 98).
t In this volume we shall generally write homogeneous coordinates in the form (x0, Xj), whereas in Vol. I we used (x,, x.,).
,; FOl M>ATK>NS ' HAP. I
In i-as*' thi« projft-livity (1) is an involution, a= — d(§ 54, Vol. I), and hfiiiv 4 A is tin- discriminant of (2). Hence
AH inn>ltiti»n in illiptic or h t//>e rbolic according as A is ponifirr
The intimate connection of these theorems with the theory of linear
order is evident on comparison with the first sections of Chap. II.
A i led urt inn of the corresponding theorems from the intuitive concep-
tions of order is to be found in Chap. IV of the Geometria Projet-
i nriques.
EXERCISE
A projectivity for which A>0 is a product of two hyperbolic involutions. A prnjrrtivity f<>r which A<0 is a product of three hyperbolic involutions.
5. Complex geometry. Assumption K provides for the solution of many problems of construction which could not be solved in a net of rationality. But even in the real space the fundamental problem of finding the double points of an involution has no general solution.
To see this it is only necessary to set up an involution for which A > 0. Take any involution of which two pairs of conjugate points .1.1' and Bff form a harmonic set U(AA', SB'). If the scale /'. /.'. /; is chosen so that A = P0, A' = R, B = Plt then B' = P_l and the involution is represented by the bilinear equation (§ 54, Vol. I)
The double points of this involution, if existent, would satisfy the
«-«j nation -2_ 1
or — 1,
which has no real roots.
An effect of Assumption K is thus to deny the possibility of solving this problem. If, however, we negate Assumption K and replace it by properly chosen other assumptions, we are led to a geometry in which this problem is always soluble, namely, the geometry of the space in which the geometric number system is isomorphic with the complex number system of analysis. Although thia geometry does not have the same relation to the space of external nature as the real geometry, it is extremely important because of its relation to other branches of mathematics.
§§.-.,»•>] IMAGINARY ELEMENTS 7
( hie way of founding this geometry is to replace Assumption K by another assumption of an equally summary character, namely,
.1. A geometric n/uiil»'r .X//.S/VM in isomorphic with the complex num- ber ni/fttc/n of analysis.
Since this number system obeys the commutative law of multi- plication, the corresponding geometry satisfies Assumption P, and all the theorems of Vol. I apply. Thus, a set of postulates for the com- plex geometry is A, E, J.
The problem of finding the double points of a one-dimensional
* projectivity is completely solvable in the complex geometry ; for
any such projectivity may be represented by the bilinear equation
(S 54, Vol. I)
c#a/+ dx '— ax — b = 0,
and therefore its double points are given by the roots of
which exist in the complex number system.
The analogous result holds good for an 7t-dimensional projectivity. In this case the problem reduces to that of finding the roots of an algebraic equation of the nth degree.
6. Imaginary elements adjoined to a real space. In this connection it is desirable to think of another point of view which we may adopt toward the complex space. Suppose we are working in a real geometry on the basis of A, E, K (or of A, E, H, C, R ; see below). It is a theorem about the real number system* that it is contained in a number system (the complex number system) all of whose elements are of the form ai + b where a and b are real and i satisfies the equation t*+l=0.
Hence it is a theorem about the real space that it is contained in another space which contains the double points of any given involution. Tli is may be seen in detail as follows: By the theory of homo- geneous coordinates the points of a real projective space S are in a correspondence with the ordered tetrads of real numbers (XQ, xv x^ a-8), except (0, 0, 0, 0), such that to each tetrad corresponds one point, and to each point a set of tetrads, given by the expression (mxQ, mxlt
* This same question is discussed from the point of view of a general space and a general field in Chap. IX, Vol. I.
g I'M NDATIoNS [CHAP. I
nXf wucj where a?0, •£,, xt, xt are tixed and m takes on all mil num- .ilues except /.«•!••>. lly ihe i>roj*Tiy of the real number system mentioned al»o\v, the set of all ordered tetrads of real numbers is contained in the set of all ordered tetrads (z0, zlt 22, z^ where : rit zt, zt an- < mnjilex numbers.
Ix-i us define a c<>ni]}l<:r point as the class of all ordered tetrads of complex numU'rs of the form
(kzo, kzl} kzs, kz)
where for a given class ZQ, zlt 22, 2, are fixed and not all zero and k takes on all complex values different from zero. Let the set of these classes satisfying two independent linear equations
be called a complex line. With these conventions it is easy to see that the set of all complex points and complex lines satisfies the assumptions A, E, P, and thus the complex points constitute a proper projective space. Let us call this space S^
The space Sc contains the set of all complex points of the form
(kxQ, kxlt kxa, kxj
where x0, xlt a:f, xt are all real. Let us call this subset of complex points S,. If any set of complex points of Sr which satisfy two equa- tions of the form (3) with real coefficients be called a " real line," we have, by reference to the homogeneous coordinate system in S, that the complex points of Sr are in such a one-to-one correspondence with the points of S that to every line in S corresponds a " real line" in Sr, and conversely.
Thus, Sr is a real projective space and is contained in the complex projective space S,.. Obviously S may also be regarded as contained in a complex projective space S' where S' consists of the points of S together with the points of Se which are not in Sr, and where each line of S' consists of the complex points of S' which satisfy two equations of the form (3) together with the points of S whose coordi- nates satisfy the same two equations.
DKFJNITION. Points ()f the renl space S aje called real points, and points of the extended space S', complex points. Points in S' but not in .S' are called imaginary points.
§§6,7] IMAGINARY ELEMENTS 9
This discussion of imaginary elements does not require a detailed knowledge or study of the complex number system as such. It is, in fact, a special case of the more general theory in Chap. IX, Vol. I (cf. particularly § 92), which applies to a general protective space- It serves in a large variety of cases where it is sufficient to know merely the existence of the complex space S' containing S and satis- fying Assumptions A, E, P. It is a logically exact way of stating the point of view of the geometers who used imaginary points before the advent of the modern function theory.
There are problems, however, which require a detailed study of the complex space, and this implies, of course, a study of the complex number system and such geometrical subjects as the theory of chains (see §§H, 12, below, and later chapters).
There is a very elegant and historically important method of intro- ducing imaginaries in geometry without the use of coordinates, namely, that due to von Staudt.* It depends essentially on the properties of involutions which are developed in Chap. VIII, VoL I, and §§ 74-75 of this volume. The reader will find it an excellent exercise to generalize the Von Staudt theory so as to obtain the result stated in Proposition K2, Chap. IX, VoL I.
7. Harmonic sequence. We shall now take up a more searching study of the assumptions of the geometry of reals. In Chap. IV, Vol. I, it was proved that every space satisfying Assumptions A, E contains a net of rationality R8, and that this net is itself a three-space which satisfies not only Assumptions A and E but also Assumption P (Theorem 20). To this rational subspace, therefore, apply all the theorems in Vol. I which do not depend essentially on Assumption HQ. For example, every line of R3 is a linear net of rationality and may be regarded (with the exception of one point chosen as oc) as a com- mutative number system all of whose numbers are expressible as rational combinations of 0 and 1.
Throughout VoL I we left the character of this net indeterminate. It might contain only a finite number of points or it might contain an infinite number. We propose now to introduce a new assumption which will fix definitely the structure of a net of rationality.
* Cf . K. G. C. von Staudt, Beitrage zur Geometric der Lage, Niirnberg (1856 and 1857). ,1. Liiroth, Mathcinatische Annalen, Vol. VIII (1874), p. 145. Segre, Memorie della R. Accademia delle scienze di Torino (2), Vol. XXXVIII (1886).
10
ForXDATIONS
. i
DRFIMII»N. Ix-t 77 , //,. 77,, I*- any three distinct points of a line // ; Iri N an«l '/' !>«• i\v.» distinrt joints collinear with //. but not on // ; and let A*0 be a jMtint of intersection of *S7/o and 777^ Denote the
S
II, H, H, FIG. 3
points of the line h by [//] and those of the line K0ffn by [A'], and let II be a projectivity defined by perspectivities as follows :
The set of points
H H H • • • If 77
0' 1* 2' ' • ' i + l'
such that H(/fi) = ffi + l, together with the set
such that H(H_._l) = H_i, is called a harmonic sequence. The point 77. is not in the sequence but is called its limit point.
The projectivity II is evi- dently parabolic and carries
"
"r
TIIKOKKM 2. The middle one of any three consecutive * po in t x of a harmonic sequence is the 1< a nnonic conjugate of the limit I" nut of the sequence with re- to the other two.
11*
Proof. By construction we have
FIG. 4
• This tiTin refers to the Kulwripts in the imUtinn //,.
§§7,8] HARMONIC SEQUENCK 11
COROLLARY. All points of a harmonic sequence belong to the same net of rational it//.
THEOREM 3. Two harmonic sequences determined by 11 , //,, //„ and by M n, J/j. J/OB are protective in any projectivity II by which
Proof. By Theorem 3, Chap. IV, Vol. I, the projectivity II transforms harmonic sets of points into harmonic sets.
8. Assumption H. By reference to fig. 3 it is intuitively evident to most observers that in any picture which can be drawn representing points by dots, and lines by marks drawn with the aid of a straight- edge, no point Hi which can be accurately marked will ever coincide with Hj (i 3= j). On the other hand, there is nothing in Assumptions A and E to prove that Ht =£ Hp because (Introduction, § 2, Vol. I) these assumptions are all satisfied by the miniature spaces discussed in § 72, Chap. VII, Vol. I, and if the number of points on a line is finite, the sequence must surely repeat itself. Thus we are led to make a further assumption.
ASSUMPTION H.* If any harmonic sequence exists, not every one contains only a finite number of points.
The existence of a harmonic sequence determined by any three points follows directly from Assumptions A and E. That any two sequences are projective follows from Theorem 3. Hence Assumption H gives at once
THEOREM 4. Any three distinct collinear points Jf0, Hr H* deter- mine a harmonic sequence containing an infinite number of points and having JfQ and HI as consecutive points and If* as the limit point.
Tiii.(u;KM 5. The principle of duality is valid for all theorems deducible from Assumptions A, E, H.
Proof. This principle has been proved in Chap. I, Vol. I, for all theorems deducible from A and E. If T;(), i^, ?/„ are any three planes on a line /, let a line-/' meet them in //0, //t, -//«, respectively. The projection by / of. the harmonic sequence determined on I' by 7/o, 7/x, II ^ is the space dual of a harmonic sequence of points. Since the
* Cf. Gino Fano, Giornale di Matematiche, Vol. XXX (1892), p. 106. Obviously Assumption H0 (Vol. I, p. 45) is a cmisrqiicMce of H. Hence, after introducing Assumption H, we have that a net of rationality satisfies not only A, E, P but also H0, and thus every theorem in Vol. I can be applied to a net of rationality.
FOUNDATIONS [CHAP. I
. , ; . ,,.,,;, is intinitf, s<. is ili«- sri|ii.-niv (.I1 phim's. Hfii'-c the space dual of Assumption H is true. The principle of duality in a plane or a bundle follows as iu § 11, Chap. I, Vol. I.
By reference to the definition of addition in Chap. VI, Vol. I, it is evident on the basis of Assumptions A and E alone that the trans- formation x' = x + a is a parabolic projectivity. Denoting it by a, it is clear that if there is any integer n such that a" is the identity, tluMi a***" = am, k and in being any integers. Hence, if a has a finite {K-riod, there is only a finite number of points in a harmonic sequence, contran- to Assumption H. Hence
THEOREM 6. A parabolic projectivity iiever has a finite, period. In other word*, if of three points determining a harmonic sequence the Hin it point is taken as oo in a scale and two consecutive points as 0 and 1, then the sequence consists of
0
1 -1
1+1=2 _i_i=_2
2+1=3 _2-l=-3
3+1=4 -3-l=-4
that is, of zero and all positive and negative integers.
COROLLARY 1. The net of rationality determined by 0, 1, oo consists
fwi
of zero and all numbers of the form — where m and n are positive or
n negative integers.
Proof. By Theorem 14, Chap. VI, Vol. I, the net of rationality determined by 0, 1, oo consists of all numbers obtainable from 0 and 1 by the operations of addition, multiplication, subtraction, and division (excluding division by zero).
COROLLARY 2. The homogeneous coordinates of any point in a linear planar or spatial net of rationality may be taken as integers.
Proof. If *0, xj( xf, xt are the homogeneous coordinates of a point in the net, they are defined, according to Chap. VII, Vol. I, in terms of the coordinates in certain linear nets. Hence they may be taken
in the form 0 or — -1 where m^ and nl are integers. If m is the product
i of their denominators, mx0, mxv mx^ mxt are integers.
§§8,9] HAKMCXNIC SEQUENCE 13
The first of these corollaries enables us to obtain the following simple result with regard to the construction of any point in a net of rationality. Let //t be the harmonic conjugate of Jfn with regard to
N
/fj and H_r The sequence
• • »i H-±> H-±> H-v ^-' HV •"$» •"$> '**
is protective (fig. 5) with
Jf-a, H_z, H_l, Jf0, Hv H.lt Jft, • ••
and therefore must be harmonic. The points H0, Jflt If* determine a harmonic sequence
•••., H _a, H_2> H_i, HQ, HI, H-2, HS, ••••.
n n n n n n
By Cor. 1, any point of the net of rationality is contained in a sequence of the last variety for some value of n.
9. Order in a net of rationality. DEFINITION. If A and B are points of R(//0AT1^/,«)) different from Hx, A is said to precede B with respect to the scale 7/0, HI} Hn if and only if the nonhomogeneous coordi- nate (cf. § 53, -Vol. I) of A is less than the nonhomogeneous coordinate of B, If A precedes B, B is said to follow A.
From the corresponding properties of the rational numbers there follow at once the fundamental propositions : With respect to the scale HO, H^ Hn, (1) if A precedes B, B does not precede A\ (2) if A precedes B and B precedes C, then A precedes C ; (3) if A and B are distinct points of ^((HJIJI^), then either A precedes Bon B precedes^.
n
\I>ATI«>NS MAP. I
of the pro|H-r!i.-. of numbers in the argument above and in analoffouti cas. > doe> not imply that our treatment of geometry is dependent foundation*. K\ery theorem which we emplov here is a logical COnwqucnce "' ''"' a-Mimption-. A. K. II alone.
The argument which is involved in the present case may be Mated as follow*: The .-.M.rdinates relati\e to a scale //0, //„ //„ of the points
of a harmonic sequence, \\li.-n combined according to the rules for addition and multiplication ijiv.-ii in Cha|.. VI, Vol. I, satisfy the conditions which are known to characterize the system of positive and negative integers (including ,. From these conditions (the axioms of the system of positive and ne-a- tiv,. follow theorems which state the order relations amoni; these
integer-, and also theorems which state the order relations simony tin- rational niimhers, the latter Kein^ defined in terms of the integers. lint \>\ 'I'lieorem ti, Cor. 1, the rational iiiimoers are the coordinates of jioints in R( //,,7/j//^). Hence the jH.iuts of R (//„//]//„,) satisfy the conditions given alx.ve.
It \\oiihl of course !..• t-ntirely feasible to make the discussion of order in a net of rationality without the use of coordinates.
* 10. Cuts in a net of rationality. DEFIMTION. Two subsets, [A] ami [//", of a net of rationality R(//i//1//00) constitute a cut (A, B) ,i-,tk rfspect to the scale HQ, II ^ 7/« if and only if they satisfy the following conditions: (1) Every point of the net except //„ is in [A] or [It] ; (2) with respect to the scale 7/0, H^ //„ every point of [.4] pre-
M every point of [!>]. If there is a point O in [A] or in [Ji] such that every point of [A] distinct from 0 precedes it and every point of [ft] distinct from n follows it, the cut is said to be closed and to have 0 as its ntt-jHn'/tt ; otherwise, the cut is said to be open. The class [.•/] is said to be the Inn ,- x/Wr and [If] to be the upper side of the cut.
With respect to the scale 7/n, 11 ^ //«, any point 0(0 =^Jfx) of a. net R(//o//l//«) determines two sets of points [A] and [/•] such that every .1 precedes or is identical with O and 0 precedes every B. These of points are therefore a closed cut having O as cut-point. Not every cut, however, is closed, for consider- the set [A], including all points whose coordinates in a system of DOnhomogeneoas coordinates hav- ing //, as the point oo are negative or, if positive, such that their squares are le-s than '2 ; ami the set [II], including all points whose
• An asterisk :vi the left of a section number indicates that the section may be omitted on a lii-t reading. We have marked in this manner most of the sections which are nut .x-ential to an understanding of the discussion of metric geometry in rhaj>s. Ill and IV.
§10] CUTS IN A NKT 15
coordinates are positive and have their squares greater than 2. Since no rational number can satisfy the equation
this equation is not satisfied by the coordinates of any point in the net. The sets [A] and [#] constitute an open cut.
DEFINITION. With respect to the scale Jf0, Hlt //«,, an open cut precedes all the points of its upper side and is preceded by all points of its lower side. A closed cut precedes all the points which its cut- point precedes and is preceded by all points by which its cut-point is preceded. A cut (A, B) precedes a cut (C, D) if and only if there is a point H preceding a point C.
THEOREM 7. (I) If a cut (A, B} precedes a cut (C, D), then (C, D) does not precede (A, B}.
(2) If a cut (A, B} is not the same as the cut (C, D), then either (A, B) precedes (C, D} or (C, D) precedes (A, B), or both cuts are closed and have the same cut-point.
(3) If a cut (A, B) precedes a cut (C, D} and (C, D) precedes a cut (E, F), then (A, B) precedes (E, F).
Proof. These propositions are direct consequences of the definition above and of the corresponding properties of the relation of precedence between points.
DEFINITION. With respect to the scale HQ, Jf^ //„, a cut (Alt A2) is said to be between two cuts (B^ B^ and (Clt C^ in case (Blf J?o) pre- cedes (Alt A^ and (Av A^ precedes (Clt C^ or in case (Clt C2) precedes (Alt A2) and (Alt AJ precedes (B^ BZ). If any one of these cuts is closed, it may be replaced by its corresponding cut-point in this defi- nition. (Thus, for example, any open cut is between any point of its upper side and any point of its lower side.)
An open cut (/I, />) is said to be algebraic if there exists an equation, a^ + a,*"-1 + ... + aH = 0,
with integral coefficients, and two joints A0, B0, such that the coordinates of all jK>ints of [A~\ between A0 and B0 make the left-hand member of this equation greater than zero and all points of [#] between A0 and B0 make it lo> than /ero.* If it is assumed that this equation has a root between A0 and B0, this is equivalent to assuming that there exists a point corresponding to the cut (A, B) on the line A0B0 but not iu the given net.
* It is perhaps needless to remark that not every algebraic equation with integral coefficients can be associated in this way with a cut. For example, x2 + 1 = 0.
K>1 NDATIONS [CHAP. I
• the pur|«ose)* of geometric (-(instructions it would be sufficient to . the existence of cut-jHuiita for all algebraic oi>en cuts (see Chap. IX, Vol. I). many puri lows, indeed, it would l»e desirable to make the assumption , ,.u p. '.'7. Cliap. IV. Vol. I, and which we here put down for refer- ence M Awmniption (}.
AMI' MI- 1 1.. N (J. 'I'/" <•• /.« not more than one net of rntionaHtij on a line. Hut it is .list. unary in analysis to assume the existence of an irrational Hum: -Binding to every o]>en cut in the system of rationals, and it is
convenient iii jjei.metrv to have a one-to-one corresjxmdence between tin- points of a line and tin- system of real numbers. Hence we make the assumption which follows in the next section.
It must not be supposed that in the assumption which follows we are introducing new joints in any respect different from those already considered. What we are doing is to postulate that a space is a class of points having certain additional properties. The assumption limits the type of space which we consider ; it does not extend the class of points. In this respect our pro- cedure is not parallel to the genetic method of developing the theory of irrational numbers.
EXERCISE
The points of R (//0//1//co), together with the open cuts with respect to the M-ali- //„, //,, //„, constitute a set [A'] of things having the following property : If [.V] and [7*] are any two subclasses of [A] including all A"'s and such that every >' precedes every T, then there is either an X or a T which precedes all other 7**s and is preceded by all other .S"s.
* 11. Assumption of continuity. We shall denote the cut-point of a closed cut (Af, N) by I*M V). In the following assumption it is not stated whether the cuts (A^ A^, (/^, Z?2), and (/),, /)„) are open or closed. If one of them is closed, therefore, the corresponding one of the symbols A'*,. .«,)» ^!B,. «,)» ail(l A'/',. /',) must be understood in the sense just denned.
A>sr\ii'n<iN (J. If every net of rationality contains an infinity of points, then on one line I in one net R(//o/T1//.) there is associated with every open cut (A, B), with respect to the scale JfQ, H^ H,., a point I , , which is on I and such that the following conditions are satisfied :
(1) If two open cuts (A, B) and (C, D) are distinct, the points J'A.ID anfl %C,D) are distinct;
i If (Alt Af) and (7^, 7?a) are any two cuts and (Clt CJ any open cut betwetn two points A and B of R(^o/T1//ae), and if T is a projec-
•'I *tirh
T('<V,.rt)) M a point associated with some cut (D^ /)2) betiveen (A,. Jj
§ ii] ASSUMPTION C 17
DEFINITION. The set of all points of ^(H^H^H*), together with all points associated with cuts in R(/IQJflJ/atl), with respect to the scale HQ, Hlt //., is called the chain C(/fo^//.). The points of R(//o7/lJ//.) are called rational, and any other point of the chain is called irrational with respect to R(//0^1/T»). A point associated with a cut which fol- lows HQ is called positive, and one associated with a cut which precedes HQ is called negative.
THEOREM 8. The point JtA<R), associated, by Assumption C, with an open cut (A, B) of ^(H^H^HJ, is not a point of R(/T07/1^00).
Proof. The associated point could not be Jfn, because there are projectivities of R(7/o//rl^00) which leave Hm invariant and change the given cut into different cuts, and therefore, by Assumption C, change the associated point. Now suppose a point D, distinct from Hm but in R(^T0^1//a>), to be associated with some open cut. Since the given cut is open, there must be a point A between D and the cut. If B is a point on the opposite side of the cut from D, A and H both precede or both follow D with respect to the scale HQ, H^ II m. The transformation which changes every point of I into its harmonic conjugate with regard to Hx and D has, when regarded as a trans- formation of the points of R(7/0/^1//00) with respect to the scale JfQ, //j, //„,, the equation
x' = 2 d — x,
where d is the coordinate of D. It therefore transforms rational points which follow D into rational points which precede it, and vice versa. Hence A and B are transformed into two points, A' and B' , which precede D if A and B follow D, or which follow D if A and B pre- cede D. By Assumption C (2), the point D which is associated with an open cut between A and B is transformed into a point D' associated with a cut between A' and B'. By Assumption C(l), D' is distinct from D-, contrary to the hypothesis that D is a fixed point of the transformation.
THEOREM 9. The points of C(//0^1//x), excluding Ifx, form, with reference to the scale in which ff0= 0, -?/,= 1, ffx = oo, a number sys- ti in isomorphic with the real number system of analysis.
Proof. The definitions of Chap. VI, Vol. I, give a meaning to the operations of addition and multiplication for all points of the line /. In that place we derived all the fundamental laws of operation, except
Ig ml NhATIONS [CHAP. 1
the commutative law of multiplication, on the basis of Assumptions A and K. We have also seen in tin- present chapter (Theorem 6, Cor. 1) that the coordinates ,,f points in R (////,//,, arc t In- ordinary rational iiuinU'rs. Hence it remains to show that the geometric laws of coiu- liiiiaiinii as a]i]'li>-(l to the irrational points of C(/fQHl Hx) are the same as for the ordinary irrational numl>ers.
The analytic definition of addition of irrational numbers* may
be stated as follows: If a and b are two numbers defined by cuts
// ) and (#s, ya), then a + b is the number defined by the cut
<•',+ •'•,- //, + y,).
To show that our geometric number system satisfies this condition in C(//(// 7/x), suppose first that a is a rational point of C(H0HJI9) and l> an irrational point. The projective transformation
(4) x'=x + a
changes the set of points [#J into the set [#2+ a], which is the same as [xf+ x^. Similarly, it changes [yj into [y2+ yj. Hence, it changes the cut (xt, ya) into (xt + x^ 3^ + y^, and hence, by Assumption C (2), changes b into a point determined by a cut which lies between every pair xl + xt and y^ + yn. Therefore b is changed into the point asso- ciated with the cut (^+ #2, yt+ ^2). But the transform of b is a + b. Hence the geometric sum a + b is the number defined by the cut
(*,+ *,. y,+y2)-
Next, suppose both a and b irrational. The transformation (4) changes [j;J into the set of irrational points [#2-r-«], b into b+a, and [yj into [ya-l- «]. By the paragraph above, the cut which defines any x^+ a precedes the cut which defines any y2+a. Hence, by Assumption C(2), the cut which defines any point #2+ a precedes the cut which defines b + a, and this precedes the cut which defines yt+ a. Any point a^-f #2 of the lower side of the cut («t+ #2, y^\- y.z) precedes the cut defining one of the points s>'2+ a, by the paragraph above, and hence precedes the cut defining b + a. Similarly, any point of the upper side of this cut follows the cut defining b + a. Hence (xt+xt, y, + //.,) is the cut defining b + «. Thus we have identified geometric addition of points in C(//0//17/<C) with the addition of (•nlinary real numbers.
• Cf . Fine, College Algebra, p. 60 ; or Veblen and Lewies, Infinitesimal Analysis, Chap. I
§11] ASSUMPTION C 19
The analytic definition of multiplication of irrational numbers may be stated as follows : If a and b are positive numljers defined by the cuts (.' p //j) and (xn, y^y let [x[] be the set of positive values of xv Then ab is the number defined by the cut (x(x.2, y,y2). If a is nega- tive and b positive, ab = — (—a)b. If a is positive and b negative, ab = — (a (—b)). If both a and b are negative, ab = (—a)(—b). If a = 0 or b = 0, a& = 0.
Consider the transformation
x' = ax.
If a is positive and rational while b is positive and irrational, this transforms [.«J into [«#2], which is the same as [#{#2]. It also trans- forms b into ab and [y2] into [«y2], which is the same as [y,yj. Hence, by Assumption C(2), ab is the number associated with
If both a and 6 are irrational and positive, we again have [a; ], i, and [yj transformed into [a#J, ai, and [ayj, where, as in the analogous case of addition, the cut defining ax2 precedes the cut defining ab, which in turn precedes the cut defining ay2. Moreover, any x[x9 precedes some ax^, and any y1y2 follows some ayv Hence, by the same argument as in the case of addition, (x(x2, y^y^ is the cut with which ab is associated.
The transformation
ar = (— 1) x
changes the cut (x^ x^ defining the irrational number a into the open cut (— #2, — a^), which therefore defines an irrational «'. But since xl— x2 may be any negative rational and ,#g— x^ may be any positive rational, the sum of a and a', which has been proved to be determined Ity the cut (xl— #2, #2— ^j), must be zero. Hence we have that (— l)a is the irrational — a such that — a + a = 0.
The transformation
«'=./•(-!)
is the same as x'= (— 1 )x for all rational points. Hence, by Assumption C (2), these transformations are the same for all points of C(//o//i//x). Hence, for points of C (//////,), (- l)x= x(- 1).
I'.y the associative law of multiplication (which, it is to be remem- bered, depends only on Assumptions A and E) we have, if a is nega-
tive and b positive,
ab = — (— a) b,
•jo FolNDATIONS [THAI-. I
\v here (— a)b is determine.! l>y the analytic (cut) rule. If a is posit i vi- and 6 is negutive. it follows similarly, with the aid of the relation (_l)a = a(-l), that
«ft-«(-l)(- ft)— <•<->));
and if both " ami b are negative,
aft = (- 1)(- a)(- 1)(- ft) = (- a)(- ft).
COROLLARY. With respect to a scale in which /7X= oo, JfQ= 0, HI=\, we have ab = ba whenever a and b are in C(/7o/7,/7x).
THEOREM 10. Any projectivity which transforms 77Q, H^ and //„ into points of the chain C(7/0/7j/7x) transforms any point of the chain into (i {mint of the chain.
Proof. We have seen that d=ax and x' = x + a, for rational or irrational values of a, are projectivities which change H^ into itself and all other points of C(^T0/f1//eD) into points of the chain. The transformation x'= \/x is a projectivity which interchanges Hu and H o (see § 54, Chap. VI, Vol. I), and by Theorem 9 it changes every point of C (J/9HtHm)t except //. and HQ, into a point of 0(^^/7.).
As in the proof of Theorem 11, Chap. VI, Vol. I, it follows that /70, /7j, Hm can be transformed into any three points of the chain by a product of transformations of these three types. Moreover, any projectivity is fully determined as a transformation of 0(^^/7.) by the three points /?0, Blt Bn into which it transforms HQ, H^ H^. For, suppose there were two such projectivities, II and II', the prod- uct fl"1!!' would transform HO, Hv H^ into themselves. Hence, by Theorem 16, Chap. IV, Vol. I, it would leave invariant every p>int of R(/7o77]77x). Hence, by Assumption C (2), it would leave invariant every point of C(/70/7j77x). Hence II'1!!' would be the identity for all (X)ints of the chain, and II would be the same as II' for all points of the chain. Hence every projectivity changing HQ, Hlt Jfx into ]>oint8 of the chain is expressible as a product of projectivities of the forms a/= ax, xf= x+ a, x'= \/x. As.all these transform the chain into itself, the theorem follows.
COROLLARY 1. Any projectivity leaving invariant three points of the chain C(// //,//, ) leave* every point of the chain invariant.
Proof. I^et II be the given projectivity leaving the given points, say /• /• /.'.. invariant. Let P be the projectivity such that P(/.'o//l/.'r) = (77^77, 77,). Then PHP'1 leaves 7/0, 7/lf H9 invariant and hence
§§ ii, 12] CHAINS 21
leaves all points of the chain invariant, as shown in the proof of the theorem. Hence II leaves all points of the chain invariant.
COROLLARY 2. Any projectivity of the chain ^(H^H^H^) into itself is of the form
f\ !»' — ft /I* I /. ,•
a o
c d
px = c where the coefficients are real numbers.
* 12. Chains in general. DEFINITION. If (A, B) is an open cut in any net of rationality R (JT^J^JKi,) with respect to the scale KO, KJf Kx, let II be a projectivity transforming ^(K^K^K^ into R (H0H1Hai>') and K9 into Hn. This projectivity transforms (A, B) into a cut (C, D) in R^Zfj/T.) with respect to the scale ffQ, Hlt H^ If X is the point associated by Assumption C with (C, D), the point II~1(JT) = X' is called the irrational cut-point associated with (A, B).
The point X' is independent of the particular projectivity II. For let II' be any projectivity changing (A, B} into a cut (E, F) in R (ff^HJ with respect to the scale HQ, H^, Hv> and let Y be the point associated with (E,F) and Y'=H'-l(Y). Then II • U'~l changes (E, F) into (C, D) and hence, by Assumption C (2), must change Y into X. This can take place only if Y1 = X', that is, only if the cut-point X' asso- ciated with (A, B) is unique.
By projecting any net of rationality into R (H^H^^} it is shown that the cut-points associated with it satisfy the conditions stated for the points associated with the cuts of R (H^H^^) in Assumption C. Hence the theorems of the last section also apply to any chain what- ever, a chain being denned as follows :
DEFINITION. The totality of points of a net of rationality R(ABC), together with all the irrational cut-points defined by open cuts with respect to the scale A, B, C in R(ABC), is called the chain defined by A, B, C and is denoted by C(ABC). The irrational cut-points are said to be irrational with respect to R(ABC).
Thus we have
THEOREM 11. (1) The projective transform of a chain is a chain.
(2) Every open cut in any net of rationality defines a unique irrational cut-point collinear with, but not in, the net.
(3) If two such cuts with respect to the same scale and in the same net are distinct, their cut-points are distinct.
-2-2 FOUNDATIONS [CHAP. i
(4) Jftwoopfn cuts nre homoloytmx in n projectivity, their cut- tir( homologous in tin snntc jtroji'ctiriti/.
(5) Any projectivity which transforms three points A, />, C into three points of the chain C(ABC) transforms any point of the chain into a point of the chain.
TIIKORKM 12. There is one and only one chain containing three dis- tinct points of a line.
Proof. Let A, Bt C be the given points. They l>elong to the chain C(ABC) into which C(/ro/T1//(B) is transformed by a projectivity such that HJI^H^ABC. By Theorem 11 (5) any projectivity such that ABC-^BAC transforms all points of C(ABC) into points of C(ABC). But by definition such a projectivity transforms C(ABC) into C(BAC)', hence C(BAC) is contained in C(ABC). In like man- ner C(ABC) is contained in C(BAC). Hence C(ABC} = C(BAC) =
Now suppose A, B, C to be points of some other chain C (PQR). By Theorem 11(5) a projectivity such that* PQRA-^QPAR changes all points of C(PQR) into points of C(PQfi). But by definition it changes C(PQfi) into C(QPA). Hence C(QPA) is contained in C(PQR). But the same projectivity changes C(QPA) into C(PQJl). Hence C(PQR) = C(QPA). In like manner C(QPA) = C(PBA) =
COROLLARY. A chain contains the irrational cut-point of every open cut in any net of rationality in the chain.
THEOREM 13. THE FUNDAMENTAL THEOREM OF PROJECTIVITY FOR A CHAIN. If A, Bt C, D are distinct points of a chain and A', Ji', C' any three distinct points of a line, then for any projectivities (A, B, C, D) -£ (A1, B', C', D') and (A, Ji, C, D) j; (A1, B', C", D[)
Proof. Let II, Hl be the two projectivities mentioned in the theorem, H"1!! then leaves every point of C(ABC) fixed; for it leaves every point of R(ABC) fixed, and hence, by Theorem 11 (4), must leave every irrational cut-point of an open cut in R(ABC) fixed. But n,"1!! is then the identical transformation as far as the points of C(ABC) are concerned. Hence 7/ = 7>[. •
•Cf. Theorem 2, Chap. Ill, Vol. I.
§§ii',i3] ASSI.MITIOX R 23
This theorem may also be stated as follows :
Any jirojective correspondence between the points of two chains is uniquely determined by three pairs of homologous points.
( >ur list of assumptions for the geometry of reals may now be com- pleted by the following assumption of closure.
ASSUMPTION R On at least one line, if there is one there is not more than one chain.
It follows at once, by Theorem 12, that every line is a chain. It also follows, by an argument strictly analogous to the proof of Theorem 5, that the dual propositions of Assumptions C and R are true. Hence we have
THEOREM 14. The principle of duality is valid fqr all theorems deduciblc from Assumptions A, E, H, C, R.
* 13. Consistency, categoricalness, and independence of the assump- tions. Let us now apply the logical canons explained in the Intro- duction (Vol. I) to the foregoing set of assumptions.
THEOREM 15. Assumptions A, E, H, C, R are consistent if the real number system of analysis is existent.
Proof. Consider the class of all ordered tetrads of real numbers (#0, x^ xa> x8), with the exception of (0, 0, 0, 0). Any class of these ordered tetrads such that if one of its members is (aQ) a^ «2, a8) all its other members are given by the formula (maQ, malt w«2, mag), where m is any real number not zero, shall be called a point. Any class consisting of all points whose component tetrads satisfy two independent linear homogeneous equations
shall be called a line. The class of all points and lines so defined satisfy the assumptions A, E, H, C, R (cf. § 4, Vol. I).
THEOREM 16. Assumptions A, E, H, C, R/orra a categorical set.
Proof. In Chap. VII, Vol. I, it has been proved that the points of a space satisfying Assumptions A, E, P can be denoted by homogeneous coordinates which are numbers of the geometric number system of Chap. VI, VoL I. Since P is a logical consequence of A, E, H, C, R (cf. Theorem 13), this result 'applies here, and by Theorem 9 the
FOUNDATIONS [CHAP. I
number system in question is isomorphic with the real number system of analysis.
Now if two spaces Sj and Sf satisfy A, E, H, C, R, consider a homo- geneous coordinate system in each space and let each point of St
:•.-••.: ;: .• [M)in( < : S win. h hit- iln- MIMIC rnnnlmatrs. This correspondence is evidently such that if three points of Sj are collinear, tlu-ir correspondents in S, are collinear.
It if worthy of remark that the above correspondence may be set up in as many ways as there are collineations of Sl into itself.
THEOREM 17. Assumptions A 1, A 2, A 3, E 0, E 1, E 2, E 3, E :V, H, C, R are an independent set.
Proof. The method of proving that a given assumption is not a logical consequence of the other assumptions was explained in the Introduction, p. 6, VoL I. Suppose there is given a class of objects [x] and a class of subclasses of [x]. If we call each x a point and each element of the class of subclasses a line, then each of our assumptions, when thus interpreted, will be either true or false * with respect to this interpretation. If all the assumptions but one are true and the one is false, it cannot be a logical consequence of the others ; for a logical consequence of true statements must be true. In the sequel we shall call the objects, x, pseudo-points, and the subclasses of [x] which play the role of lines, pseudo-lines.
A 1. The pseudo-points shall be the points of a real projective plane TT together with one other point 0. The pseudo-lines shall be the lines of TT. A 1 is false because there is no pseudo-line contain- ing 0. A 2 is true because it is satisfied by the ordinary projective plane. A 3 is true because the only sets of points A, B, C, D, E which satisfy its hypothesis are in TT. The only pseudo-plane is TT, and there is no pseudo-space. Hence it is evident that E 0, E 1, E 2, E 3 are true and E 3' is vacuously true. Assumptions H, C, R are evidently true.
• If the hypothesis of a statement is not verified, we regard the statement as true. Following the terminology of E. H. Moore (Transactions of the American M:ittn-in:itic;il Society, Vol. Ill, p. 489), we shall describe statements which are inn- in this sense as "vacuously true" or "vacuous."
It is pomible to put any or all of the assumptions into a form such that they are vacuous for the ordinary real space. For example. I'rofessor Moore has pointed out that A 1 could be replaced by the following proposition, which is vacuous for ordinary space.
A 1 I.,-t .! be a point and B be a point. K there is no line which is on A »nd on B, then A = B.
§13] INDEPENDENCE PROOFS 25
A 2. The pseudo-points shall be the points of a real projective three-space Sg together with one other pseudo-point 0. The pseudo- lines shall be the lines of Sg, each pseudo-line, however, containing 0. Thus any two pseudo-points are collinear with 0 ; a pseudo-plane is an ordinary plane together with 0; a pseudo-space is S8 together with 0. Hence it is evident that A 2 is false and A 1, A3, E 0, El, K L', E 3, E 3' are true. There exist harmonic sequences of pseudo- points, some of which are ordinary harmonic sequences. Hence Assumption H is true. By reference to the definition of a quad- rangular set and harmonic conjugate it is clear (because every line contains 0) that any pseudo-point P is harmonically conjugate to 0 with regard to any two pseudo-points which are collinear with P. Hence a linear net of rationality contains all the pseudo-points of a pseudo-line. The operations of addition and multiplication are not unique, however, and hence the definition of order does not apply ; there are no open cuts, and Assumptions C and R are vacuously true.
A 3. The pseudo-points shall be the points of a real projective space S8, with the exception of a single point 0. The pseudo-lines shall be the lines of Sg, except that in case of those lines which pass through 0 the pseudo-lines do not contain 0. Clearly A 3 is false whenever the pseudo-points A, B, C, D, E are chosen so that the lines AB and DE meet in a A 1, A 2, E 0, E 1, E 2, E 3, E 3' are obviously true. A harmonic sequence and a net of rationality of pseudo-points can be found identical with an ordinary harmonic sequence and net of rationality on any line not passing through 0. Hence H, C, and R are also true.
E 0. The pseudo-points shall be the vertices of a tetrahedron, and the pseudo-lines the six pairs of pseudo-points. Thus the pseudo- planes are the trios of pseudo-points, and a pseudo-space consists of all four pseudo-points. A 1 and A 2 are obviously true. A 3 is true because we may have E=A and D=B. E 1, E 2, E 3, E 3' are true. H, C, R are vacuously true.
E 1. There shall be one pseudo-point and no pseudo-line. E 1 is false and all the other assumptions are vacuously true.
E 2. There shall be three pseudo-points and one pseudo-line con- taining all three pseudo-points. A 1, A 2, E 0, E 1 are true. A 3, E 3, E 3', H, C, R are vacuously true.
26 FOUNDATIONS [THAI-, i
E3. The pseudo-points and pseudo-lines shall be the point ;unl line* of a real projective plane. A 1, A 2, A 3, E 0, E 1, E 2, H, ( , It are true and K 3' is vacuous.
Tlif ps«Mido-points and pseudo-lines shall be the points and lines of a real four-dimensional projective space. E 3' is false and all the -.th.-r assumptions are true.
H. The pseudo-jwints and pseudo-lines shall l>e the points and lines df any modular projective three-space (cf. § 72, Vol. I, and § 16, U'low). All the assumptions A and E are true, H is false, and C and K are vacuously true.
C. The pseudo-points and pseudo-lines shall be the points and linear nets of rationality of a three-dimensional net of rationality in an ordinary real projective space. All the assumptions are true except C, which is false. R is vacuously true.
R The pseudo-points and pseudo-lines shall be defined as the points and lilies in Theorem 15, the coordinates, however, being ele- ments of the system of ordinary complex numbers. All the assuni[>- tious are true except R, which is false.
Assumption C, which is more complicated in its statement than the others, is, however, such that neither of the two statements into which it is separated may be omitted. This result is established in the following theorem :
THEOREM 18. Assumption C(l) is not a consequence of Assump- tion C (2) and all the other assumptions. Assumption C (2) is not a • i .,'.,••• of ( ' ' l \ and "/ tin otii, ,- nxxii motion* >'/•••// if u-e d<lil /<> C(l) the following ; If a projectivity transforms H9 into itself and 7/0 ami 7/[ into points of ^(H^H^H^), and transforms an open mf (.i, /.') into an open cut (C, D), it transforms the point associated with (A, B) into the point associated with (C, D).
Proof* (1) Any real number x determines a class Kx of numbers of the form ax -f b where a and b are any rationals. Kx is the same ** K** + t, f°r all rational values of a and b. Hence, if x and y are two irrationals, Kf and A'f are either identical or mutually exclusive. Thus the class of all real numbers falls into a set of mutually exclusive
• This :ir-Hiii.-iit makes une of portions of the theory of classes which could not »»• treated adequately without a long digression. Hence we assume knowledge of thi- mi-thodn and terminology of this branch of mathematics without further explanation.
|i:J] INDEPKNDKNCK PROOFS 27
classes [A*]. With each class K we associate a particular one of its numbers,* k, and thus obtain a set of numbers [k] such that every real number can be written uniquely in the form ak + b.
Now consider the number system whose elements are the complex numbers of the form ai + b, where a and b are rational and i = V^l. If we take as pseudo-points and pseudo-lines the points and lines of a three-space based (as in the proof of Theorem 15) on this number system, it is clear that all the assumptions except C are satisfied. If we also take as the pseudo-points HQ) H^ Jfn those having the coordinates (0, 1, 0, 0), (1, 1, 0, 0), (1, 0, 0, 0), the net of rationality R(//0//rffa)) consists of Hx and the points whose coordinates are (.!', 1, 0, 0), where x is rational. Suppose now that we associate the pseudo-point (ai + b, 1, 0, 0) with every cut in this net which in the ordinary geometry would determine an irrational point (ak -f b, 1, 0, 0). Every point is thus associated with an infinity of cuts, contrary to Assumption C(l). Moreover, the cuts with which any point is asso- ciated occur between every two pseudo-points and hence between every two cuts of R(/To/f1/roo). Therefore Assumption 0(2) remains true in this space.
(2) For the second half of the theorem the pseudo-points and pseudo-lines shall be the points and lines of a three-space based on a commutative number system whose elements are the ordinary rational numbers and all open cuts in the rational numbers. The laws of combination shall be such that addition is precisely the same as for the ordinary number system and multiplication is the same between rationals and rationals or rationals and irrationals, but different l>etween irrationals and irrationals. Thus the product of the num- bers associated with two open cuts will not, in general, be the number associated with the cut given by the usual rule. Hence the pro- jective transformation x'=ax will not preserve order relations, and A -sumption C (2) must be false. On the other hand, C(l) and the other assumptions are obviously true.
* We do not show how to set up the correspondence. The assumption that this correspondence exists is a weaker form of the assumption used by Zennelo (Muthematische Annalen, Vol. LIX, p. 514) in his proof that any class can be well ordered. Our proof of the second part of the theorem is dependent on the validity of Zermelo's result that the continuum can be well ordered. The whole theorem is therefore subject to the doubts that attach to the Zermelo process because of the lack of explicit methods of setting up the correspondences in question.
•js FOUNDATIONS [CHAP. I
The existence of the required new number system can be inferred from Hamel's theorem* that there exists a well-ordered set of real numbers
(5) «!• «,, at> -•,"«>'-
such that every real number can be given uniquely by an expression of the form
(6) «0 + «!«,, + <*t\ +•- + «nain,
containing only a finite number of terms, the a's all being rational. The ordinary rules of combination for cuts determine a multiplication table for the a's ; that is, a set of rules of the form
(7) aflj = £0+ fipki + 0^+ • • • + Pnakm,
where the /3's are rational The laws of combination for the number system in general may now be stated as follows: Express the two numbers to be added or multiplied in the form (6) ; add or multiply by the rules for addition and multiplication of polynomials, reducing the result in the case of multiplication by means of the multiplication table for the a's.
Now suppose we denote by
(8) a{, a!2, •••, «/, •••
the same set of numbers [a] arranged in a different order of the same type as (5). Such an order would be obtained, for example, by inter- changing al and aa and leaving the other a's unaltered. There is therefore a one-to-one correspondence in which every a. corresponds to the a\ having the same subscript. Moreover, since the set of all a's includes the same elements as the set of all a"s, every real number is expressible in the form
(9) vi- ai<+«iai+ ••• + *.<•
A new law of multiplication, which we shall denote by x , is now defined by setting up a multiplication table for the a"s according to the rule that
(10) a\ x a'j = a0+ a^ +•-- + «X
whenever
= ac -f ajait +•• • Mathematische Annalen, Vol. LX, p. 469
§§13,14] ASSUMPTION R 29
The product, according to the new law of combination, of two real numbers is obtained by expressing each in the form (9), multiplying according to the rule for polynomials, and reducing by the multipli- cation table for the a"s.
Since the set of all expressions of the form
ao+aA +«*«,-,+ ••• forms a number system, the set of all expressions of the form
st*X+*v?i+'*;
forms a number system isomorphic with the first. For if we let each ai correspond to the a( with the %ame subscript, the sum of any two elements of the first number system corresponds, by definition, to the sum of the corresponding two elements in the second number system. Similarly for the product of a rational by a rational or of a rational by an irrational. The product of two irrationals in the first system corresponds to the product of two irrationals in the second, because the two polynomials in the a's are multiplied by the same rules as the two in the a"s, and are also reduced by corresponding entries in the respective multiplication tables.
We may insure that the two number systems shall be distinct by selecting the a's, in the first place, so that ax = V2 and «2= V3, and then choosing the a"s so that a( = av
* 14. Foundations of the complex geometry. Let us add to Assump- tions A, E, H, C the following assumption :
ASSUMPTION R On some line, I, not all points belong to the same chain.
Let P0, Plf j£ be three points of I. The geometric number system determined by the method of Chap. VI, Vol. I, by the scale PQ, Plt P^ is commutative for all the points in the chain C(P0PJPa) but not neces- sarily for other points. However, it is clear, without assuming the commutativity of multiplication, that
x' = x~l, x' = x -f- a, x?= ax, d =xa (a = constant)
define projectivities. For x' = x~l this follows from § 54, Vol. I; for x1 = x + a it reduces to Theorem 2, Chap. VI, Vol. I ; and for the other two cases, to Theorem 4, Chap. VI, Vol. I.
Let i/be any point of I not in C (/*/*£), and let [X] be the set of all points in C(P0P1PX). Then, by Theorem 11(1), the set of points
;;,» iol NDATIONS [CHAP. I
[A'+./J is a rhain. This chain has no point except II in common with C( /:/;/;), because, if A' + </= A"=£ 7^, it would follow that A'= «/, and thus «/ would be a point of C(PJ[1^). Let us denote the chain [-V + .7] hy C'.
In order to continue this argument we need the following assump- tion of closure :
I. Tlimuyh a point P of any chain C of the line I, and any print J on I but not in C, there is not more than one chain of I ichifh It 'is no other point than P in common with C.
Now let P be any point of / not, in C(P0P1PX) or C'. Such points exist, because, for example, the chain C(P0P1J) does not coincide with C(P0PlPm) or C. The chain C(PJ%) has, by Assumption I, a point different from 7i in common with C(P0P1PX). Let Xl be this point. In case Xl =£ P0, the projectivity
(12) X' = X+J(Pl-X-l-X)
transforms P^ into J, X} into itself, and Px into itself. Hence it trans- forms C(PQP1K)=C(P0X1P*) into C(JX^). Hence every point of C^Vj/i), and in particular P, is of the form X+JX", where A" and X" are in [A']. If A't = ^, the projectivity
(13) X'=JX
transforms C(^, ^, %) into C(7£<7£), which contains P. Hence, in this case P is of the form JX. Thus we have
LEMMA 1. Every point of the line I is expressible in the form A + .in, where A and B are in C(/^j£).
LEMMA 2. Two points A + JB and A'+JB', where A, B, A', B' are in C(/^/f/i), are identical if and only if A = A' and B = B'.
For if B*#,A + JB = A'+ JB1 implies J= (A1 -A)(B- /;')- ', and thus J would be in C(P0P1PX) ; and if B = B', it implies directly that A = A'.
Karh of the projectivities X'=JA'fiw\ X' = XJ transforms the chain C(/0'/?£) into C(P0J%). Hence, if A be any point of
(14) AJ=JA',
where A' is also in
§14] COMPLEX SPACE 31
Each of the projectivities A'' = (J[ — J) X and X' = X(Pl — J) trans- forms C(P0PlPa>) into C(PQ(Pl-J)J^). Hence, if A be any point of
c (/;/;£),
where A" is also in C(7jJ/f7£). By the distributive law (Theorem 5, Chap. VI, Vol. I) it follows that
A-AJ=A"-JA". By (14), this reduces to
A-JA' = A"-JA".
By Lemma 2, it follows that A=A" = A'. Hence AJ= JA. From this we can deduce, by the elementary laws of operation,
(.4 + JB) (C + JD) = A (C + JD) + JB(C + JD) = AC + AJD + J/i C + JBJD = CA + CJB + JDA + JDJB
= C(A + JB) + JD (A + JB) = (C + JD)(A+JB).
Hence the geometric number system determined by any scale on I is commutative. Since chains are transformed into chains by any protective transformation, it follows that the geometric number sys- tem determined by any scale on any line in a space satisfying A, E, H, C, R, I satisfies the commutative law of multiplication. Hence, by Theorem 1,
THEOREM 19. Assumption P is satisfied in any space satisfying Assumptions A, E, H, C, R, I.
Since every point in the geometric number system is expressible in the form A + JB, we have
(15) J* = A0+JBQ,
where AO and BQ are in C(P0PlPa). Thus J is one of the double points of the involution
(16) XX'-IB9(X + X')-A9=0,
which transforms C(/^/i) into itself. Any two points of C(P0P^) which are conjugate in this involution may be transformed projec- tively into 7* and P, by a transformation which carries C(P0P1P,) into itself. This reduces the involution to
(17) .V.V' = ^,
;;.j |-'(»rNPATH>\> [CHAP. I
win-re A must be negative relatively to the scale PJ[P,, since the tlnuble points are not in C(PQP1PJ. The transformation A' = V- ./A" now reduces (17) to vv' — P
A .\ /j
aiul thus transforms J to a point satisfying the equation
J*=-P. Hence we have
THKOREM 20. The geometric number system in any space satisfying
, • nS A. V.. 11. < '. II. ! <••< ixi'f/K'r/1/iii- irilli tin' nuil/ilf.i- nUlllltT
system of analysis, i.e. with the system of numbers a + ib, where t*= — 1 and a and b are real.
•15. Ordered projective spaces. There is an important class of projective spaces which may be referred to as the ordered projective spaces and which are characterized by the Assumptions S given below. Tliis class of spaces includes the rational and real projective spaces and many others. The set of assumptions, A, E, S, is not categorical, but it may be made so by adding a suitable continuity assumption or by some other assumption of closure.
These assumptions introduce a new class of undefined elements, called senses* in addition to the points and lines which are the undefined elements of Assumptions A and E. The senses are denoted by symbols of the form S(ABC), where A, B, C denote points.t
S 1. For any three distinct collinear points A, B, C there is a sense S(ABC).
S 2. For any three distinct collinear points tJiere is not more than one sense S(ABC).
S3. S(ABC) = S(BCA).
S4 S(ABC)* S(ACB).
:.. // S(ABC) = S(A'B'C') and S(A'B'C') = S(A"B"C"), then S(ABC)=S(A"B"C").
S 6. If S(ABO)=S(BCO), then S(ABO) = S(ACO).
87. IfOA and OB are distinct lines, and S(OAA^)= S(OAA^ and OAA^A% ^ OBB^Bf then S(OBB^= S(OBB}.
•Seta of awuniptionH more or less related to these have been given by A. R. Schweitzer, American Journal of Mathematics, Vol. XXXI, p. 866, and A. N. White- bead, The Axioms of Projective Geometry, Cambridge Tracts, Cambridge, 1906.
t With respect to the intuitional basis of these assumptions, cf. figs. 6-12, Ch»p. II.
§§ir,,io] ORDERED SPACES 33
If S(ABC) be identified with the sense-class which is discussed below m § 19, Chap. II, it will be seen that S 1 and S2 are immedi- ately verified and S 3, • • •, S 7 reduce to Theorems 2-6, Chap. II. This shows that the assumptions S are satisfied by a rational or a real protective space.
These assumptions are capable, as is shown in Chap. II, of serving as a basis for a very complete discussion of geometric order relations. Assumption P is not a consequence of A, E, S alone.
EXERCISES
1. Prove that Assumption H is a consequence of A, E, and S.
2. Prove that with a proper definition of the symbol < (less than) the geometric number system in an ordered projective space satisfies the following conditions :
(1) If a and b are distinct numbers, a < b or b < a.
(2) If a < b, then a * b.
(3) If a < b and b < c, then a < c.
(4) If a < b, there exists a number, x, such that a < x and x < b.
(5) If 0 < a, then b < a + b for every b.
(6) If 0 < a and 0 < b, then 0 < a • b.
(Cf. E. V. Huntington, Transactions of the American Mathematical Society, Vol. VI (1905), p. 17.)
3. Introduce an assumption of continuity, and with this assumption and A, E, S prove Assumption P.
4. Prove that P is not a consequence of A, E, S alone.
* 16. Modular projective spaces. We have seen (§ 7) that, in any space satisfying Assumptions A and E, any two harmonic sequences are projective. Hence, if one harmonic sequence contains an infinity of points, every such sequence contains an infinity of points, and by § 8 these points are in one-to-one reciprocal correspondence with the ordinary rational numbers. On the other hand, if one harmonic sequence contains a finite number of points, every other harmonic sequence in the same space contains the same finite number of points. Hence the spaces satisfying Assumptions A and E fall into two classes — those satisfying Assumption H and those satisfying the following:
ASSUMPTION H. If any harmonic sequence exists, at least one con- tains only a finite number of points.
INUNDATIONS L«'HAP. I
The spaces satisfying H may be called modular, and those satisfy- ing H non modular.
It follows, just as in Theorem 5, that the principle of duality is inn- for any modular space.
l^et IT be any parabolic projectivity on a line, and let Hn be its invariant point. It' //o l>e any other point of the line, the points
form a harmonic sequence, by definition. If this is to contain only a tinit*' number of points, there must be some positive integer n such that II"(//0)= II"(W0), where m is zero or a positive integer less than n. If n — m = k, we have
ami hence II* = 1.
Hence all the points of the harmonic sequence are contained in the set
In case k is not a prime number, that is, if there exist two positive integers, klt k^ different from unity such that k = kt> &2, let us con- sider the parabolic projectivity II*'. The points
HO, IP'(#0), iF*'(7/-0), ..., n*-'>*(/g
satisfy the definition of a harmonic sequence. Since any two harmonic sequences contain the same number of points, it follows that the given sequence could not have contained more than k^ points. In case /. ; breaks up into two factors, the same argument shows that the given harmonic sequence could not contain a number of points larger than either factor. This process can be repeated only a finite number of times and can stop only when we arrive at a prime number. Hence we have
TH K< nt K M 'J 1 . The number of points in a harmonic sequence is prime. The points of a harmonic sequence may be denoted by
HO, U(HO), ..., n-
where II is a parabolic projectivity. The period, p, of any parabolic projectivity is a prime number.
§i«] MODULAR SPACES 35
With reference to a scale in which Jfo= 0, II (Jf0) = 1, and the limit point of the harmonic sequence is oo, II has the equation
x'= x + 1.
Hence the coordinates of the points in the harmonic sequence are 0, 1, 2, "-,p-\,
respectively, where 2 represents 1 + 1,3 represents 2 + 1, etc. Since IP = 1, we must have that p = 0, p + 1 = 1, tip + k = k, etc. In other words, the coordinates of the points in a harmonic sequence are ele- ments of the field obtained by reducing the integers modulo p, as explained in § 72, Vol. I.
By Theorem 14, Chap. VI, Vol. I, the net of rationality determined by the points whose coordinates are 0, 1, <x> consists of the point oo and all points whose coordinates are obtainable from 0 and 1 by the operations of addition, subtraction, multiplication, and division (except division by zero). Since all numbers of this sort are con- tained in the set
0, I, •••, p-1, we have
THEOREM 22. The number of points in a net of rationality in a modular space is p + 1, p being a prime number constant for the space in question.
Obviously, if Assumption Q (§ 10) be added to the set A, E, H, the number of points on any line must be p + 1, p being prime. A space satisfying A, E, H shall be called a rational modular space. The problem of finding the double points of a projectivity in a rational modular space of one or more dimensions leads to the consideration of modular spaces bearing a relation to the rational ones analogous to the relation which the complex geometry bears to the real geometry. The existence of such spaces follows from the considerations in Chap. IX, Vol. I (Propositions K2 and Kn). The geometric number systems for such spaces may be finite* (Galois fields) or infinite.!
»E. H. Moore, The Subgroups of the Generalized Finite Modular Group, Decennial publications of The University of Chicago, Vol. IX (1008), pp. 141-190 ; L. K. Dickson, Linear Groups, Chap. I.
t L. E. Dickson, Transactions of the American Mathematical Society, Vol. VIII (1007), p. 880. See also the article by E. Steinitz referred to in § 02, Vol. I.
H FOUNDATIONS [CHAP, i
17. Recapitulation. The various groupings of assumptions which we have considered thus far may be resumed as follows: A space satisfying Assumptions
A, E is a general projective space ;
\ K, P is a proper projective space;
A, E, H is a noninodular projective space ;
A, E, H is a modular projective space ;
A, E, S is an ordered projective space ;
A, E, H, Q is a rational modular projective space ;
A, E, H, Q is a rational noumodular projective space ;
A, E, H, C, R! . , . ..
f is a real proiective space ; or A, E, K
A, E, H, C, K, I\ .
f- is a complex projective space.
The first six sets of assumptions are not, and the remaining ones are, categorical The set of theorems deducible from any one of these sets of assumptions is called a projective geometry, and the various geometries may be distinguished by the adjectives applied above to the corresponding spaces.
CHAPTER II
ELEMENTARY THEOREMS ON ORDER
18. Direct and opposite projectivities on a line. In § 9 a point A was said to precede a point B relative to a scale P0, Plf P^ if the coordi- nate of .4 in this scale was less than the coordinate of B. Supposing the coordinate of A to be a and that of B to be b, the projectivity changing P0 to A and Pl to B and leaving P» fixed has the equation
In this transformation the coefficient of x is positive if and only if A precedes B. But the transformations of the form
(2) x = ax + fi,
where a is positive, evidently form a group. This group is a subgroup of the group of all projectivities leaving Px invariant, for the latter group contains all transformations (2) for which a =£ 0.
The group of transformations (2) for which a is positive is, by what we have just seen, such that whenever a pair of points A and B are transformed to A' and B' respectively, A precedes B if and only if A' precedes B'. The. discussion of order relative to a scale could therefore be based on the theory of this group.
The order relations defined by means of this group have all, how- ever, a special relation to the point Px , and they can all be derived by specialization from a more general relation defined by means of a more extensive group. We shall therefore enter first into the discussion of this larger group, and afterwards (§ 23) show how to derive the rela- tions of " precede " and " follow " from the general notion of " sense." The definitions for the general case, like those for the special one, will be seen to depend simply on the distinction between positive and negative numbers.
A projective transformation of a line may be written in the form
«00 «0,
a a
A 10 0 ' 11 1' 10 11
where the a0.'s are numbers of the geometric number system
37
M Kl.KMKNTAKV THKOKKMS ON <>KI>KK VP. II
Iml.-i A.-uiiiptioii- A. K. II. ('. It (or A, E, K) the a^'s are real If attention IK- restricted t<> a single net of rationality satisfying As- II. the ",/s may be taken (Theorem 6, Cor. 2, Chap. I) as Tin- discussi-m which follows is valid on either hypothesis.*
1>KHMTH>N. The projectivities of the form (3) for which A>0 are called «/im7, and those for which A < 0 are called opposite.
Since the determinant of the product of two transformations (3) is the product of the determinants, the direct projectivities form a subgroup of the protective group. The same transformation (3) cannot be both direct and opposite, for two transformations (3) are identical only if the coefficients of one are obtainable from those of the other la multiplying them all by the same constant p; but this merely changes A into /rA.
In form, the definition is dependent on the choice of the coordinate -\-tem which is used in equations (3). Actually, however, the defi- nition is independent of the coordinate system, for if a given projec- tivity has a positive A with respect to one scale, it has a positive A with respect to every scale. This may be proved as follows :
Let the fundamental points of the scale to which the coordinates in (3) refer be P0, Pv P^, and let QQ, Qlt Qx be the fundamental points of ;my other scale. By § 56, Vol. I, the coordinates yQ, y^ of any point R with respect to any scale Q0, Qv Q^ are such that yjy^ = 1$ (QXQQ, Qfl)- Suppose that, relative to the scale TjJ, f^, R>, the projectivity which transforms <?0, Qlt Qx to TjJ, 7J, JF£ respectively has the equations
(4)
0.
Thus any point R whose coordinates relative to the scale ^, Pv Px are (x0, xj is transformed by this projectivity to a point R' whose coordinates relative to the scale P0, P19 jg are (y0, y^).
Since cross ratios are unaltered by projective transformations,
Henc-e it follows that if xo and xl are the coordinates of any point R relative to the scale P0, /,', 7i, the corresponding values of yn and y^given
• It IK, in fart, valid in any spare ssitisfyin.tr AsMinn>timis A. K. S. I'. The purely ordinal tlipnrcnis an- iiulccd valirl in any ordered projective sjisice (§ 15), but those regarding involutioim, conic- si-ctioiis, etc. necessarily involve Assumption P also. Cf. tin- tine print at the end of § 19.
§i«] DIKECT PROJECTIVITIES 39
In/ (4) are the coordinates of R relative to the scale Qo, QI} #„. Let us indicate (4) by (yQ, y,) = T(x9, xj, and (3) by « aft = S(*0, xj.
Now a direct transformation (3) carries a point whose coordinates relative to the scale PQ, 1[, 1* are (xo, xj into one whose coordinates rela- tive to the same scale are (x'0, x[), where (x'0, x[) = S(xQ, xj. The coordi- nates of these two points relative to the scale Q0, Qlt Qx are (yo, yj = T(x0, xv) and (y'0, y{) = T(x'0> x[} respectively. Hence, by substitution,
<tf, yi) = T(S(xQ, xj) = T(S(T-*(y0, y,))),
or ,'
where T~ l indicates, as usual, the inverse of T. The determinant of the transformation 2'ST~l is
where K is real (or rational), and A' therefore has the same sign as A. Thus the definition of a direct projectivity is independent of the choice of the coordinate system.
This result can be put in another form which is important in the sequel:
DEFINITION. Two figures are said to be conjugate under or equiva- lent with respect to a group of transformations if and only if there exists a transformation of the group carrying one of the figures into the other.
THEOREM 1. If two sets of points are conjugate under the group of direct projectivities on a line, so are also the two sets of points into which they are transformed by any projectivity of the line.
Proof. Let S be a direct projectivity changing a set of points [A] into a set of points \_E], and let T be any other projectivity on the line, and let T(A)=A' and T(B) = B'. Since T~l(A')=A, S(A) = B, and T(B)=&, it follows that TST~l(A') = B'. But the discussion above shows that TST~l is a direct projectivity. Hence [A'] and [/?'] are conjugate under the group of direct projectivities, as was to be proved.
According to the definition in § 75, Vol. I (see also § 39, below), the group nf direct projectivities is a self-conjugate subgroup of the group of all projec- tivities on a line. Since this is the only relation between the two groups which we have employed in the proof of the theorem above, this theorem can be generalized to any case in which we have one group of transformations appearing as a self-conjugate subgroup of another.
40 ELKMKNTVKY Til K<>K K.MS ON ORDER [CHAP. II
EXERCISES
1. Within the field of all real numbers the positive numbers may be defined a* those numbers different from /ero which possess square roots. Generalize this definition to oth.T fields, and thus generalize the definitions of direct proj.vtmties. In each case, determine how far the theorems ou sense and ,.r.l.-r in thi- following sections can be generalized (cf. § 72, Vol. I).
2. The group of projectivities which transform a net of rationality into itnt» If has a self-conjugate subgroup consisting of those transformations which are products of pairs of involutions having their double points in the net of rationality. This group contains all projectivities for which the determinant is the square of a rational number.
•3. Work out a definition and theory of the group of direct projectivities inde|M-ndent of the use of coordinates. This may be done by the aid of theorems in Chap. VIII, Vol. I (cf. §§ 69 and 70, below).
19. The two sense-classes on a line. DEFINITION. Let A0, B0, CQ be any three distinct points of a line. The class of all ordered* triads of points ABC on the line, such that the projectivities
are direct, is called a sense-class and is denoted by S(AyB^C^. Two ordered triads in the same sense-class are said to have the same sense or to be in the same sense. Two collinear ordered triads not in the same sense-class are said to have opposite senses or to be in opposite senses.
One sense-class chosen arbitrarily may be referred to by a particular name, as right-handed, clockwise, positive, etc.f
The term " sense," standing by itself, might have been defined as follows : 44 The senses are any set of objects in one-to-one and reciprocal correspondence with the sense classes." This is analogous to the definition of a vector given in § 42. When there is question only of one line, any two objects whatever may serve as the two senses — for example, the signs + and — . This agrees with the definition of sense as " the sign of a certain determinant." When dealing with more than one line, it is no longer correct to say that there are IPO lenses; tlu-re an-, in fact, two senses for each line.
• "Order," here, is a logical rather than a geometrical term, just as in the defi- nition of " throw " (§ 28, Vol. I). It is a device for distinguishing the elements of a «et. For example, when we say that ABC cannot be transformed into AC It by any transformation of a given group, it is a way of saying that the group contains no transformation changing A into A, B into C, and C inco B.
t A partial list of references on the notion of sense in one and more dimensions would include : M obi us, Barycentrische Calcul, note, in § 140 ; Gauss, Werke, Vol. V 1 1 1 . p. 248 ; von Staudt, Beitrage zur Geometric der Lage, §§ 8, 14 ; Study, Archiv IT Mathematik und Physik, Vol. XXI (1913), p. 193: Encyclopadie der Math. Wl». Ill AB 7, p. 618.
5 1«] SENSE-CLASSES 41
When one adopts, as we do, the symbol S(ABC} to stand for a sense-class, there is no occasion for attaching a separate meaning to the word " sense." It
may be regarded as an incomplete symbol,* like the — in the — of the calculus.
dx dx
THEOREM 2. If the ordered triad ABC is in the sense-class S'(A0BQCQ), then S(ABC)=S(ASB9CI). If S(ABC} = S(A'B'C') and S(A'B'C') = S(A"B"C"), then S(ABC) = S(A"B"C").
€' ABC A" B" C" A' B'
FIG. 6
Proof. Both statements are consequences of the fact that the direct projectivities form a group.
THEOREM 3. If S (ABC} * S(A'B'C'} and S(A'B'C'}^ S(A"B"C"}} then S(ABC) = S(A"B"C").
C" C' B1 ABC A" B" A'
Proof. If S(ABC)=f- S(A'B'C'}, the projectivity ABC-^A'B'C' is opposite. Hence the theorem follows from the fact that the product of two opposite projectivities is direct.
COROLLARY. There are two and only two sense-classes on a line.
THEOREM 4. If A, B, C are distinct collinear points, S(ABC} = S(BCA} and S(ABC) * S(ACB).\
Proof. Let A, B, C be taken as (1, 1), (1, 0), (0, 1) respectively. Then , _ x
r' — r
iCj — ^Q
is an opposite projectivity interchanging B and C and leaving A invariant. Hence S(ABC) =£ S(ACB). In like manner, we can prove that S(ACB)*S(BCA). It follows, by Theorem 3, that S (ABC}
= S(BCA).
* The term " incomplete symbol " appears in Whitehead and Russell's Principia Mathematica, Vol. I, Chap. Ill, of the Introduction, together with a discussion of its logical significance.
t This may be expressed by the phrase " Sense is preserved by even and altered by odd permutations." A transposition is a permutation in which two and only two elements are interchanged, and an even (odd) permutation is the resultant of au even (odd) number of transpositions. Cf. Burnside, Theory of Groups of Finite Order, Chap. I.
l-J KI.KMKNTAKY Til K< >K KMS < >N ORDER (CHAP.
TMK..KKM & // 9(ABI>)~S(BC0)t th.;i S(AIiD) = S(A('l>).
Proof. Chnose the coordinates so that />=(0, 1), A = (l, 0), # = (I, 1). The transformation of ABD to #£/> may be written in
the form o o o o
.1 BCD
1»' — 1» ' // *' "Pt/^ H
** I ^— •*.« |^ Mf»t/jy X 1O» O
because (0, 1) is invariant and (1,0) goes to (1, 1). This transforma- timi will be direct if and only if a > 0. The point C, being the trans- form of (1, 1), is (1, 1 4- a). The transformation carrying ABD to A CD is x' = x
x[= (1 +a)xv
which is direct because (14- «) > 0.
As an immediate consequence of Theorem 1 we have
THEOREM 6. IfS(ABC)=S(A1BlCl)andABCA1B1C1-^A'B'CrA'lB'lC[,
then
FIG. 9
Theorems 2-0 contain the propositions given in § 15, Chap. T, as Assunij*- tion.s S. ThiMircui <i is slightly more general than S 7 but is directly deduriMe from it. The developments of the following sections will be based entirely on thf-i- projMisitioiis, and hence belong to the theory of any ordered projective »|>ace, «'xc«'|it when* reference is made to figures whose existence deix-mls on AKsuinptimi I'. 'I'licuri-ins of the latter sort hold in any space satisfying A, K, I', S.
'I'li.-s.- ]ir(i|Nisitions have the advantage, as assumptions, of correspond u\% to some of our simplest intuitions with regard to tin- lim-ar order relations. The reader may verify this by constructing the figures to which they correspond (cf. fig*. 0-0). Each proposition will be found to correspond to a number of riflually distinct figures.
§§ it), 20] SENSE-CLASSES 43
20. Sense in any one-dimensional form. DEFINITION. If 1, 2, 3, 1', 2', 3' are elements of the same onr-dimensional form, and A, B, C, A', B', C' are colliuear points such that
then the ordered triad 123 is said to have the same sense as 1'2'3' if and only if S(ABC) = S(A'B'C'). The set of all ordered triads having the same sense as 123 is called a sense-class and denoted by £(123).
In. view of Theorem 6 this definition is independent of the choice of the points A, B, C, A', B', C'. It is an immediate corollary of the definition that the plane and space duals of Theorems 2~6 all hold good (cf. figs. 10 and 13).
By the definition of a point conic there is a one-to-one correspond- ence between the points [P] of the conic and the lines joining them to a fixed point P0 of the conic. We now define any statement in
terms of order relations among the points of the conic [P] to mean that the same statement holds for the corresponding lines [-^P]. By Theorem 6, above, together with Theorem 2, Chap. V, Vol. I, it follows that this definition is independent of the choice of the point P0. The definitions of the order relations in the line conic, the cone of lines, and the cone of planes are made dually.*
The propositions with regard to sense are perhaps even more evident intu- it ioually when stated with regard to a conic or a flat pencil than with regard to the points of a line (cf. figs. 10 and 11).
* These definitions arc in reality special cases of the definition given above for any one-dimensional form, since the cones and runic sections are onc-iiiinensional forms of the second degree (§ 41, Vol. I) and since the notion of projectivity between one-dimensional forms of the first and second degrees has been defined in § 76, Vol. I. However, at present we do not need to avail ourselves of the theorems in Chap. VIII, Vol. I, on which the latter definition is based.
44 i;i. i:\l KNTAi; Y THKOREMS ON ORDER [CHAP. II
21. Separation of point pairs. DEFINITION. Two points A and B of
a liue are said to separate two points C and D of the same line if and
'. (') * S(ABD). This is indicated by the symbol A H II CD.
THEOREM 7. (1) The relation ABICD implies the relations CD II AB .I/; in '. ,tnd excludes the relation AC\\BD. (2) Given any /ottr distinct points of a line, we have either AB II CD or AC\\ BD or ADIBC. (3) From the relations AB\\CD and AD II BE follows thf relation AD\\CE. (4) If AB\\CD and ABCD^A'B'C'D', then A'&IC'D1*
Proof. (1) If AB II CD, we have
(5) S(ABC} * S(ABD),
which, by the definition of separation, implies AB II DC. By Theorems '2-6 we obtain successively, from (5),
S(ABC) = S(ADB), S(ABC) = S(ADC), S(ACB) = S(DAB), S(A CB) = S(DCB),
S(CDA)*S(CDB),
the last of which implies CD II AB. The relation AC\\ BD is excluded because it means S(ACB) 3= S(ACD), which contradicts the second of the equations above.
(2) By the corollary of Theorem 3 we have either S(ABC)=£ S(ABD) (in which case AB II CD) or S(ABC) = S(ABD). In the latter case either S(ABC)*= S(ADC) or S(ABC) = S(ADC). The first of these alternatives is equivalent to S(ACB) =£ S(ACD) and yi.-lds 'AC\\BD; the second implies S(ADC) = S(ABC) = S(ABD) *S(ADB), and thus yields AD\\BC.
(3) The hypotheses give S(ABC) * S(ABD) and S(ADB) ¥= S(ADE). The first of these gives S(BCA) = S(DBA), which, by Theorem 5, implies S(DBA) = S(DCA), and thus S(ADB) = S(ADC). Hence. 1 >y the second hypothesis, S(ADC) =£ S(ADE), and therefore ^47) II r/;.
(4) This is a direct consequence of Theorem 6.
• The properties expressed in this theorem are sufficient to define abstractly the relation of separation. Of. Vailati, Revue de Math^matiques, Vol. V, pp. 70, 183; alao Padoa, Revue de Mathe'inatiques, Vol. VI, p. 35.
§§2i,±»] LINEAR ORDER 45
THKOREM 8. If A and B are harmonically conjugate with regard to C and D, they separate C and D.
Proof. By Theorem 7 (2) we have either AB II CD or A C \\BD or AD \\BC. We also have ABCD-^BACD. Hence AC\\BD would imply HC II AD, contrary to Theorem 7 (1) ; and AD II BC would imply /.'/> \\AC, contrary to Theorem 7 (1). Hence we must have AB II CD.
THEOREM 9. An involution in which two pairs separate one another has no double points.
Proof. Suppose that the given involution had the double points M, N, and that the two pairs which separate one another are A, A' and B, B1 respectively. Since the involution would be determined by the projectivity MNA _ MNA^
in which, by Theorem 8,
S(MNA) ¥= S(MNA'),
it would follow, by Theorem 6, that every ordered triad was carried into an ordered triad in the opposite sense. Since the involution carries AA'B to A'AB', we should have
S(AA'B) =£ S(A'AB') ;
and hence S(AA'B) = S(AA'B'),
contrary to hypothesis.
This theorem can also be stated in the following form :
COROLLARY 1. An involution with double points is such that no two pairs separate one another.
COROLLARY 2. If an involution is direct, each pair separates every other pair. If an involution is opposite, no pair separates any other pair.
22. Segments and intervals. DEFINITION. Let A, B, C be any three distinct points of a line. The set of all points X such that
S(AXC} = S(ABC]
is called a segment and is denoted by ABC. The points A and C are called the ends of the segment. The segment ABC, together with its ends, is called the interval ABC. The points of ABC are said to be interior to the interval ABC, and A and C are called its ends.
COROLLARY 1. A segment does not contain its ends.
COROLLARY 2. If D is in ABC, then
ABC = A DC.
4i! KI.K.MKNTAKY THEOREMS OK OBDEB [<'HAP.II
COROLLARY 3. If D i* in ABC, then B and D are not separa A and C.
THKORKM 10. If A and B are any two distinct points of a line, there art two and only two segments, and also two and only two intervals, of which A and B are end*.
Proof. Let C and D be two points which separate A and B har- monically. If A' is any point of the line distinct from A and B, either
or S(AXB) = S(ADB).
In one case X is in ACS, and in the other case in ADB.
I>KKIXITION. Either of the two segments (or of the two intervals) whose ends are two points A, B may be referred to as a segment AB (or an interval AB). The two segments or intervals AB are said to be complementary to one another.
COROLLARY. If A, B, C are any three distinct points of a line, the line consists of the three segments complementary to ABC, BCA, CAB, together with the points A, B, and C.
Proof. Any point X distinct from A, B, C satisfies one of the rela- tions AC II BX or AB II CX or AX II BC.
THEOREM 11. If Alt Az, • • •, AK is any set ofn(n> 1) distinct points of a line, the remaining points of the line constitute n segments, each of which has two of the points Alt A_2, • • •, An as end points and no two of which have a point in common.
Proof. The theorem is true for n = 2, by Theorem 1 0. Suppose it true for n = k. If A- -|- 1 points are given, the point At + l is, by the theorem for the case n = k, on one of the k segments determined by the other k points, say on the segment whose ends are A, and Af By the corollary to Theorem 10, this segment consists of At + ,, together with two segments whose ends
are respectively At + v A{ and At + l, Ar Hence the theorem is valid for n = k + 1 if valid for n = k. Hence the theorem is established by mathematical induction.
DEFINITION. A finite set of collinear points, At(i — l, • • •, n), is in the geometrir.nl «,-,//•/• {A^i^A^ • • • An} if no two of its points are
§§2-2,23] LINEAR ORDER 47
separated by any of the pairs A^A^ AaAt, • • •, AHAr As an obvious consequence of Theorem 11 we now have
THEOREM 12. To any set [A] of n points of a line the notation A^, Aa, • • •, An may be assigned so that they are in the order {A A - • - An}. A set of points in the order {A^A^ • • • Au} is also in the orders A4 • - • AA and A, • • . A.
EXERCISES
1. If AB II CD and A C II BE, then CD II BE.
2. The relations AB II CD, AB\\ CE, A B \\DE are not possible simultaneously.
3. Any two points A, B are in the orders {AB} and {BA}. Any three collinear points are in the orders {ABC}, {ACB}, {CAB}.
23. Linear regions. The set of all points on a line, the set of all points on a line with the exception of a single one, and the segment are examples (cf. Ex. 1 below) of what we shall define as linear regions on account of their analogy with the planar and spatial regions con- sidered later.
DEFINITION. A region on a line is a set of collinear points such that (1) any two points of the set are joined by an interval consisting entirely of points of the set and (2) every point is interior to at least one segment consisting entirely of points of the set. A region is said to be convex if it satisfies also the condition that (3) there is at least one point of the line which is not in the set.
DEFINITION. An ordered pair of distinct points AB of a convex region R is said to be in the same sense as an ordered pair A'B' of R if and only if S(ABAX) = S(A'B'AaD), where Ax is a poin,t of the line not in R. The set of all ordered pairs of R in the same sense as AB is denoted by S(AB) and is called a sense-class. The segment comple- mentary to AA^B is called the segment AB. The corresponding inter- val is called the interval AB. A set of points of R is said to be in the order {A^ • • • A^ if they are in the order {A1A3 - • • AnAx}. If C is separated from Ax by A and B, C is "between A and B with respect to R. If S( AB) = S(CD}, then C is said to precede D, and D to follow C, in the sense AB.
If there is a point B*,, other than Ax, which is not in the convex region R, the sense S(ABAX) is the same as the sense S(ABB*,), and the segment AA^B is the same as the segment ABXB. Hence
4s Kl. i:\lKNTAKY THKOKEMS ON ORDER [CHAP. II
IKKM 13. /'<"• a given convex region R the above definition has t\e tame meaning if any other point collinear with R but not in R be tubttituied for Am.
COROLLARY 1. // S(AB) = S(A'BI) and S(A'B') = S(A"B"), then
COROLLARY 2. If S(AB) * S(A'B') and S(A'B') * S(A"B"), then S(AB)=S(A"B"). COROLLARY 3. S(AB) * S(BA). COROLLARY 4. // S(AB) = S(BC), then S(AB) = S(AC).
These corollaries are direct translations of Theorems 2-5 into our present terminology. Theorem 7 translates into the following state- ments in terms of betweenness :
THEOREM 14. (1) If C is between A and B, then B is not between A and C. (2) If three points A, B, C are distinct, C is between A and B or B is between A and C or A is between C and B. (3) If C is between A and B and A is between B ami E, then C is between B and E.
Theorem 7 translates into the following statements in terms of " precede " and " follows."
THEOREM 15. (1) If C precedes B in the sense AC, then B does not precede C in this sense. (2) In the sense AC, either B precedes C or C precedes B. (3) If, in the sense AB, A precedes C and E precedes A, then E precedes C.
DEFINITION. If A and B are any two points of a convex region R, the set consisting of all points which follow A in the sense AB is called the ray AB. The point A is called the origin of the ray. The ray consisting of all points which precede A in the sense AB is said to be opposite to the ray AB. The set of all points which precede A in the sense AB is sometimes called the prolongation of the segment AB beyond A.
EXERCISES
1. A convex region on a line is either a segment or the set of all points on tin- line with the exception of one point.*
2. If three points of a convex region are in the order {ABC}, they are in th»- order {CBA} but not in the order {ACB} or {CAB}.
3. In a convex region, if A is between B and C, it is between C and B.
4. Between any two points there is an infinity of points.
•This exercise requires the use of an assumption of continuity (C and R, or K).
. '.'4]
LINEAR ORDER
49
5. If B is on A (' and C is on BD, then C is on AD and B is on AD.
6. The relations B is on AC, B is on AD, B is on CD are not possible simultaneously.
7. If 5 and C are on J Z>, then Bis on A C or on CD.
8. Choosing a system of nonhomogeneous coordinates in which A* is oo, show that the sense AB is the same as the sense A'B' if and only if B — A is of the same sign as B' — A'; also that two point pairs have the same sense if and only if they are conjugate under the group
x' — ax + 6, where a > 0.
24. Algebraic criteria of sense. If A = (a0, a^, JB=(b0, 6,), and C= (CQ, Cj) are any three distinct points of the line, the transformation
(6)
changes (1, 0), (0, 1), and (1, 1) into A, B, and C respectively if and only if pQ and pl satisfy the equations
that is, if
&= Pi
With this choice of P0/p1 the determinant of the transformation (6) is of the same sign as
S =
By definition the projectivity is direct if and only if S is positive. Now if A' = (a'Q, a{), B' = (b'0, b'J, C' = (c'0, c'J are any three points of the line, and
S' =
' a'
0 "o
' a'
1 "i
two cases are possible. If S' is of the same sign as S, the projectivities in which
(7) (1,0) (0, 1)(1, V-fiABC,
(8) (1,0)(0,1)(1,1)X^5'C"
are both direct or both opposite, and hence the projectivity in which (9)
KI.KMKNTAKY TH K< >KKMS ( >N ORDKII [CHAP. II
is direct. If N' is opposite in sign to S, one of the projectivities (7) and - direct and the otlier opposite, and hence (9) is opposite. Hence
TIIKMKKM I-'-, M J = («0, «,), B=(t>o> &,)> £ = (<v c,), A' = (a'0, a{), # = (fl'ot i,'i)t < " = (f jf c|) to collinear points. Then S(ABC) = S(A'li'< ") ul only if the expressions
b(c{
c' a
c. a
',/!> .s /<//<.
COROLLARY 1. Three points given by the finite nonhomogeneous coor- dinates a, b, c are conjugate under the group of all direct projectivities to three points given by the finite nonhomogeneous coordinates a', b', c', respectively, if and only if (a — b)(b — c) (c — a) and (a1 — b') (b' — c') (c' — a') have the same sign.
Proof. Set a = «j/«0, b = b1/b0, c = CI/CQ, and apply the theorem.
COROLLARY 2. Two points given by the finite nonhomogeneous coor- ili notes a and b are conjugate under the group of all direct projectiritiis l> 'i ring the point oo of the nonhomogeneous coordinate system invariant t<> the two points given by the finite nonhomogeneous coordinates a' and V respectively if and only if a — b and a' — b' have the same sign.
Proof. Set a = al /aQ, b = bl /bQ, CQ = 0, c1 = 1, and apply the theorem.
THKOKKM 17. A, B separate C, D if and only if the cross ratio Ii i.l//, CD) is negative.
Proof. By the last theorem, A, B separate C, D if and only if
and
d
are opposite in sign. But the quotient of these two expressions has the same sign as B (AB, CD} (cf. p. 165, Chap. VI, Vol. I).
With the aid of this theorem the proof of Theorem 7 can be made much more simply than in § 21.
25. Pairs of lines and of planes. THEOREM 18. The points of space not on either of two planes a and ft fall into two classes such that two points Oj, Oa of the same class are not separated by the points in ?'•// irk the line Op^ meets the planes a and ft, while two points O, P of differ- • a f i lasses are separated by the points in which the line OP 'meets a and ft.
Proof. By the space dual of Theorem 10 the planes of the pencil aft are separated by a and ft into two segments. Let [0] be the set
§ -'.r>J
SKl'A RATION liY PLANKS
FIG. 13
of points on the planes of one of these segments but not on the line aft, and let [P] be the set of the points on the planes of the other segment but not on the line aft.
The two planes CD and TT of the pencil aft which are on any two points 0 and P are separated by a and ft. Hence, by Theorem 7 and § 20, the points in which the line OP meets a and ft are separated by 0 and P. In like manner, any two points Olt #2 de- termine with the line aft a pair of planes (or a single plane) not sepa- rated by a and ft, and hence the line 0^0Z meets a and ft in points (or a single point) not separated by Ol and 02. By the same reasoning, any line P^ meets a and ft in points (or a point) not separated by Pl and Pr
COROLLARY I. If I and m are two coplanar lines, the points of the plane which are not on I or m fall into two classes such that two points Oj, 02 of the same class are not sepa- rated by the points in which the line 0^02 meets I and m, while two points 0, P of different classes are separated by the points in which OP meets I and m.
COROLLARY 2. There is only one pair of classes [0] and [P] satisfying the conditions of the above theorem (or its first corollary) determined by a given pair of planes (or lines).
FJ<;. 14
DEFINITION. Two points in different classes (according to Corol- lary 1) relative to two coplanar lines are said to be separated by the two lines; otherwise they are said not to be separated by the lines. Two points in different classes (according to Theorem 18) relative to two planes are said to be separated by the two planes; otherwise are said not to be separated by the planes.
KI.K.MKNTAUY TH K» HI K.MS < >N <>R1>KK L«'HAP. 11
EXERCISES
1. If /, and /2 are two coplanar lines and O any point of their common plane, all triads of points in a fixed sense-class Sl on /j are projected from o into triads in a fixed sense-class St on /8 (Theorem (J). If P is any other point of the plane, it is separated from 0 by ll and /2 if and only if triads in the sense S. are not projected from P into triads in the sense 52.
This problem can l>e stated also in terms of the sense of pairs of points in the region obtained on /, or /2 respectively by leaving out the common point. The theorem in this form is generalized in § 30. In the form stated in Ex. 1 it has the following generalization.
2. If /t and /, are two noncoplanar lines, and o is any line not intersect ing them, all triads in a fixed sense S± on ll are axially projected from o into triads in a fixed sense 52 on /2 (Theorem 6). The lines not intersecting /j and L fall into two classes : those by which triads in the sense Sl are projected into triads in the sense 52, and those by which triads in the sense Sl are projected into triads in the sense opposite to S¥
3. Obtain the definition of separation of two coplanar lines by two points as the plane dual of the definition of separation of two points by two coplanar lines. Prove that if two coplanar lines separate two points, then the points separate the lines. State and prove the corresponding result for pairs of points and of planes.
26. The triangle and the tetrahedron.
THEOREM 19. If a line I not passing through any vertex of a triangle ABC meets the sides BC, CA, AB in A^ Br Cl respectively, then any other line m which meets the segments BAf, CB^A also meets the segment AC ^B.
Proof. Suppose first that m At<-
passes through Al ; then t x ^
C*
and hence, if Bl and B^ do not separate A and C, Cl and C2 do not separate A and B. Similarly, the theorem is true if m passes through Bf
If m does not pass through Al or BV let m' be a line joining Al to the point in which m meets CA. By the argument above we have first that m' meets all three segments BA^C, CB^A, and AC\B, and then that m meets them.
Let us denote the segment AC^B by 7, BA^C by a, and CB^A by 0, and the segments complementary to a, ft, 7 by a, ft, 7 respectively. The
§26]
THE TRIANGLE
53
above theorem then gives the information that every line which meets two of the segments a, ft, 7 meets the third. Any line which meets a and ft meets 7, for, as it does not pass through A or B, it meets either 7 or 7 ; but if it met 7, and by hypothesis meets a, it would meet ft. Hence the theorem gives that a, ft, 7 are such that any line meeting two of these segments meets the third. By a repetition of this argu- ment it follows that every line of the plane which does not pass through a vertex of the triangle meets all three segments of one of the trios afty, afiy, afty, afty, and no line whatever meets all three segments in any of the trios afty, afty, afty, afty.
The lines of the plane, exclusive of those through the vertices, therefore fall into four classes:
(1) those which meet a, ft, 7,
(2) those which meet a, ft, 7,
(3) those which meet a, ft, 7,
(4) those which meet a, ft, 7.
No two lines l^ /2 of the same class are separated by any pair of the lines joining the point ljn to the vertices of the triangle, while any two lines Jj, m1 of different classes are separated by two of the lines joining the point I1ml to the vertices. This result is perhaps more intuitively striking when put into the dual form, as follows :
THEOREM 20. The points of a plane not on the sides of a triangle fall into four classes such that no two points L , L of the same class are separated by any pair of the points in which the line L^L^ meets the sides of the triangle, while any two points L^, Ml of different classes are separated by two of the points in which the line LJI^ meets the sides of the triangle.
DEFINITION. Any one of the four classes of points in Theorem 20 is called a triangular region. The vertices of the triangle are also called vertices of the triangular region.
FIG. 16
KI.KMKNTAIIY 'I'll K< »K KM S < >N OKDKK [CHAP. II
42'
Tin- property «»f tin- triangle stated ii\ Theorem 19 can also serve as ft basis for a discussion of the ordinal theorems on the tetrahedron and for those of tin- (/<• + l)-i">int in n-space. SUPJMIM- \u- have a tetra- hedron whose vertices are Alt At, At, A4. Let us denote its faces by alt a , a , a , the face al being opposite to the vertex Alt etc. ; let us denote the edges by au, aw, au, an, aM, «42, the edge ay being the line A{Aj. h edge ay is separated by the vertices A(, Aj into two segments, which we shall denote by o^ and o^. Let TT be a plane not passing through any vertex ; the six segments which it meets may be denoted by <rjs, <ru, • • •, <r4J, and the complementary segments by <r12 Then as a corollary of Theorem 19 we have that any plane which meets three nonco- planar segments of the
864 **> '* •"> ff* meets all the rest of
them, and, moreover,
no plane meets all the
segments aJ2, <FJg, • • •,
<r4J. If we observe that
any plane not passing
through a vertex must
meet the edges a12,
au» au *u tnree distinct points, it becomes clear that the planes not
passing through any vertex fall into eight classes such that two planes
of the same class are not separated by a pair of vertices, whereas
two planes of different classes are separated by a pair of vertices.
Under duality we have
TMK.IIKKM 21. The points not upon the faces of a tetrahedron fall into eight classes such that two points of the same class are not sepa- rated by the points in which the line joining them meets the faces, whereas two points of different classes are separated by two of the points in which their line meets the faces of the tetrahedron.
DEFINITION. Any one of the eight classes of points in Theorem 21 is called a tetrahedral region. The vertices of the tetrahedron are also called vertices of any one of the tetrahedral regions.
. 17
$§26,27] THE TETRAHEDRON 55
It would be easy to complete the discussion of the triangle and the tetrahedron at this point — for example, t<> define the term "boundary" and to prove that the boundary of any one of the classes of points in Theorem 20 is composed of A, B, C and three segments having the property that no line meets them all. We shall defer this discussion, however, to a later chapter, where the results will appear as special cases of more general theorems.
27. Algebraic criteria of separation. Cross ratios of points in space. The classes of points determined (Theorems 18~21) by a pair of inter- secting lines, a triangle, a pair of planes or by a tetrahedron can be discussed by means of some very elementary algebraic considerations. As these are similar in the plane and in space, let us carry out the work only for the three-dimensional cases.
Suppose that the homogeneous coordinates of four noncoplanar points Av A3, A8, At are given by the columns of the matrix,
(10)
and let (XQ, x^ x2, #8) be the homogeneous coordinates of any other point X. Let us indicate by \x, «2, ag, at the determinant of the matrix obtained by substituting #0, xjt x2, xa respectively for the ele- ments of the first column in the matrix above; by a^ x, «g, aj the determinant obtained by performing the same operation on the second column, etc. The expressions \y, a2, ag, aj etc. have similar mean- ings in terms of the coordinates of a point (y0, y^ y2, ys) = Y. The following expressions are formed analogously to the cross ratios of four points on a line (cf. § 58, Vol. I):
a , a , a \ \y, fi^, ci , a
— 2" 8 — 4
" = \a a a ^
U., U/^, I*., «t-
„
> a* «8» x > aa> a8>
k =|alt a,, x, at\ Jalt a,, y, aj
. M |«lt «2, <v «1 | «,. «,. «8» FT
Clearly there are twelve numbers ky which could l>e denned analo- gously to these ; and if the notation Alt A3, At, A4, A', Y be permuted among the six points, 720 such expressions are defined. Eaeh number k^
Kl.KMKN TAKY THEOREMS OX ORDER [CHAP, n
is an absolute invariant of the six points, for it is unaltered if the coordinates of any point be multiplied by a constant or if all six points be subjected to the same linear transformation.
If}' be,not upon any of the planes determined by the points At, A^ A , A , there exists a projectivity which carries Y into (1, 1, 1, 1) and the IH .mis .1 At,As,A4 into the points represented by the columns of
(12)
Let (X0> Xlt Xt, Xt) be the point into which (x0, xt, #2, xj is carried by this projectivity. By substituting in (11) we see that
From this it follows that \x, «2, ag, aj, \alt x, a8, aj,etc. could be taken as the homogeneous coordinates with respect to the tetrahedron of reference whose vertices are A^ A^, A^ A4.
The line (Xo, Xlt X2, -X.)- X(l, 1, 1, 1)
meets the planes determined by the four points represented by (12) in four points given by the values X = -3T0, X = Xlf X = X^, X = A'g. The cross ratios of pairs of these points with (X0, Xr Xs, A'g) and (1, 1, 1, 1) are XQ/X^, XjX^ and XjX^. Hence ku, Jc^, kM are cross ratios of X and Y with pairs of points in which the line joining them meets the faces of the tetrahedron A,AA,At.
v v 1284
By Theorem 17, the points X and Y are separated by the planes AtAtA4 and AfAaAl if and only if &H is negative. They will be sepa- rated by A^A^AI and A^A^A^ if and only if kot is negative, and by AlAfA4 and AlAtAi if and only if k^ is negative. Hence, by Theorem 21, we have
THEOREM 22. The points X and Y w/ill be in the same class urith rexpect to the tetrahedron AlA,tAgAt if and only if ku, ku, k^ are all positive.
COROLLARY. The eight regions determined by the tetrahedron AlA,2AtA4 are those for which tlie algebraic signs of kl4, ku, kM appear in tlie following combinations: (+, +, +), (+, +, -), (+, -, +), (-, +, +),
§ 27 J CRITERIA OF SEPARATION 67
Recalling that \x, «a, ag, «4| = 0 is the equation of the plane A^A^A^ (cf. § 70, VoL I), we see that if
a(x) = aft + aft + aft + or,*, = 0
^sft^+^+^+fta^O
are the equations of two planes, the formula given above for the cross ratio of two points X and Y with the points of intersection of the line A'y with these planes becomes
«(y)
Thus two points are in the same one of the two classes determined by the planes a (x) and $ (x) if and only if this expression is positive. This result assumes an even simpler form when specialized somewhat with respect to a system of nonhomogeneous coordinates. Suppose that XQ = 0 be chosen as the singular plane in a system of nonhomo- geneous coordinates ; then the same point is represented nonhomo- geneously by (x, y, z) or homogeneously by (1, x, y, z), and the plane represented above by a (x) = 0 has the equation
If /8 (x) = 0 be the plane x0 = 0, the expression for the cross ratio written above becomes , .
which reduces in nonhomogeneous coordinates, when (x0, xlt #2, xg) and (yc» y\» y*> y8) are rePlaced by (1, a/, y', z') and (1, *", y", z"), to
a^ + a^ + aaz' + «0 «/' + «/' + «/' + ao
Hence two points (xr, y', z') and (x", y", z"} are separated by the sin- gular plane, and a^x + azy + agz + «o = 0 if and only if the numerator and denominator of (14) are of opposite sign. For reference we shall state this as a theorem in the following form :
THEOREM 23. The two classes of points determined, according to Theorem 18, by the singular plane of a nonhomogeneous coordinate system and a plane ax + by + cz + d = 0 are respectively the points (x> y> z) for which ax + by + cz + d is positive and the points for i'-/i ich it is negative.
58 Kl.KMKNTAKV THEOREMS ON ORDEB [CH.U-. n
EXERCISES
1 ' .irry out the disrussion analogous to tlie above in the two-dimensional case. (ifiH-ntli/*1 to a dimensions.
2. How many of the 720 numbers analogous to fcu are distinct?
28. Euclidean spaces. DKKINITION. The set of all points of a pro- jtvtive sjHiri'* of n dimensions, with the exception of those on a single (w _ 1). space S" contained in the ?i-space, is called a Euclidean space of n iliiiH-itxioit*. Thus, in particular, the set of all but one of the JH tints of a projective line is called a Euclidean line, and the set of all the points of a projective plane, except those on a single line, is called a Euclidean plane.
DEFINITION. The projective (n— 1) -space S50 is called the singular (n — 1) -space or the (n — I) -space at infinity or the ideal (n — 1) -space associated with the Euclidean space. Any figure in S* is said to be ideal or to be at infinity, whereas any figure in the Euclidean Ti-space is said to be ordinary.
The ordinary points of any line in a Euclidean plane or space form a Euclidean line and thus satisfy the definition (§23) of a linear convex region. The definitions and theorems of that section may therefore be applied at once in discussing Euclidean spaces. Thus, if A and B are any two ordinary points, we shall speak of " the seg- ment AB" « the ray AB" etc.
The first corollary of Theorem 18 yields a very simple and impor- tant theorem if the line in be taken as the line at infinity, namely :
THEOREM 24. The points of a Euclidean plane which are not on a line I fall into two classes such that the segment joining two points of the same class does not meet I and the segment joining two points of different classes does meet I.
COROLLARY. If a is any ray whose origin is a point of /, all points of a are either on I or on the same side of I.
In like manner Theorem 18 yields
THEOREM 25. The points of a Euclidean three-space which are not on a plane IT fall into two classes such that the segment joining two point* of the same class does not meet IT and the segment joining two points <>f Different classes does meet TT.
• We shall refer to a line, plane, or n-space in the sense of Chap. I, Vol. I, as a l>n>j»Ttivf line, plane, or n-«pace whenever there is possibility of confusion with other types of spaces.
§§->K,29] EUCLID KAN STACKS 59
p
DEFINITION. The two classes of points determined by a line / in a Euclidean plane, according to Theorem 24, are called the two sides of /. The two classes of points determined by a plane TT in a Euclidean three-sj>ace, according to Theorem 25, are called the two sides of TT.
The two sides of TT are characterized algebraically in Theorem 23.
DEFINITION. An ordered pair of rays h, k having a common origin is called an angle and is denoted by 4 hk. If the rays are AB and A C, the angle may also be denoted by 4 BA C. If the rays are opposite, the angle is called a straight angle ; if the rays coincide, it is called a zero angle. The rays h, k are called the sides of 4 hk, and their common origin the
vertex of 4M.
EXERCISES
1. The points of a Euclidean plane not on the sides or vertex of a nonzero angle 4 hk fall into two classes such that the segment joining two points of different classes contains one point of h or k. In case 4 hk is not a straight angle, one of these two classes consists of every point which is between a point of h and a point of k.
2. Generalize Theorem 25 to n dimensions.
29. Assumptions for a Euclidean space. A Euclidean space can be characterized completely by means of a set of assumptions stated in terms of order relations. Such a set of assumptions is given below. It is a simple exercise, which we shall leave to the reader, to verify that these assumptions are all satisfied by a Euclidean space as defined in the last section.
The reverse process is also of considerable interest. This consists (1) in deriving the elementary theorems of alignment and order from Assumptions I— VIII below, and (2) in defining ideal elements and showing that these, together with the elements of the Euclidean space, form a projective space. For the details of (1) and an outline of (2) the reader may consult the article by the writer, in the Transactions of the American Mathematical Society, Vol. V (1904), pp. 343-384, and also a note by R L. Moore, in the same journal, Vol. XIII (191 2), p. 74. On (2) one may consult the article by R Bonola,diornale di Matematiche, Vol. XXXVIII (1900), p. 105, and also that by F. W. Owens, Trans- actions of the American Mathematical Society, Vol. X I (1910), p. 141. Compare also the Introduction to VoL I.
This set of assumptions refers to an undefined class of elements called points and an undefined relation among points indicated by saying "the points A, B, C are in the order {ABC}"
60 KU:MI:NT\I;Y Til K< MiK.MS ON ORDER [CHAP. U
The assumptions are as follows:
I. // inputs A. /•', <' in-f in the order {ABC}, they are distinct.
II. If points A, B, C are in the order {ABC}, they are not in the vrdtr [BOA}.
1 >KHMTI« >N. If A and B are distinct points, the segment AB consists of all points X in the order {AXB}\ all points of the segment AB are said to be between A and B; the segment together with A and B is called the interval AB- the line AB consists of A and B and all points X in one of the orders {ABX}, {AXB}, {XAB}\ and the ray AB consists i >f H and all points X in one of the orders {AXB} and {ABX}.
•III. If points C and D (C 3* D) are on the line AB, then A is on the line CD.
IV. If three distinct points A, B, and C do not lie on the same line, and D and E are two points in the orders {BCD} and {CEA}, then a point F exists in the order {AFB} and such that D, E, and F lie on the same line.
V. If A and B are two distinct points, there exists a point C such that A, B, and C are in the order {ABC}.
VI. There exist three distinct points A, B, C not in any of the orders {ABC}, {BCA}, {CAB}.
DEFINITION. If A, B, C are three noncollinear points, the set of all points collinear with pairs of points on the intervals AB, BC, CA is called the plane ABC.
VII. If A, B, C are three noncollinear points, there exists a point D not in the same plane with A, B, and C.
VIII. Two planes which have one point in common have two distinct points in common.
IX. If A is any point and a any line not containing A, there is not more than one line through A coplanar with a and not meeting a.
XVII. If there exists an infinitude of points, there exists a certain pair of points A, C such that if [<r~\ is any infinite set of segments of the line AC, having the property that each point of the interval AC if a point of a segment <r, then there is a finite subset, <rlt <rt, • • •, <rn, with the same property*
• The proposition here stated about the interval AC is commonly known as the H»-iiie-Borel theorem. The continuity a«sumption is more usually stated in the form of the " Dedekind Cut Axiom." Cf . R. Dedekind, Stetigkeit und irrationalen Zahlen, Braunschweig, 1872.
§§-.», 30] KrCLIDKAN 1'LANK lil
Assumptions I to VIII are sufficient to detine a three-space which is capable of being extended by means of ideal elements into a pro- jective space satisfying A, E, S. This space will not, in general, sat- isfy Assumption P. If the continuity assumption, XVII, be added, the corresponding protective space is real and hence properly projective. Assumption IX is the assumption with regard to parallel lines. Assumption VIII limits the number of dimensions to three.
30. Sense in a Euclidean plane. Suppose that L is the line at infinity of a Euclidean plane. Every collineation transforming the Euclidean plane into itself effects a projectivity on L which is either direct or opposite (§ 18). Since the direct projectivities on /„ form, a group, the planar collineations which effect these transformations on lx also form a group.
DEFINITION. A collineation of a Euclidean plane which effects a direct projectivity on the line at infinity of this plane is said to be a direct collineation of the Euclidean plane. Any other collineation of the Euclidean plane is said to be opposite. Let A, B, C be three non- collinear points; the class of all ordered triads A'B'C' such that the collineation carrying A, B, and C to A', B', and C' respectively is direct, is called a sense-class and is denoted by S(ABC). Two ordered triads of noncollinear points in the same sense-class are said to have the same sense or to be in the same sense. Otherwise they are said to have opposite senses or to be in opposite senses.
Since the direct projectivities form a group, it follows that if a triad A'B'C' is in S(ABC), then S(ABC) =S(A'B'C'}.
THEOREM 26. There are two and only two sense-classes in a Eicclidean plane. If A, B, and C are noncollinear points, S(ABC) = S(BCA) =£ S(ACB).
Proof. Let A, B, C be three noncollinear points. If A', B', C' are any three noncollinear points such that the projectivity carrying A, B, C to A', B', C' respectively is direct, S(ABC) contains the triad A'B'C'. Because the direct projectivities form a group, S(ABC) = S (A'B'C'). The triads to which ABC is carried by collineations which are not direct all form a sense-class, because the product of two opposite collineations is direct. Thus there are two and only two sense-classes.
Suppose we denote the lines BC, CA, AB by a, b, c respectively and let A', B', C' be the points of intersection of a, b, c respectively
KLKMKNTAKY T! I K<>]; KMS ON (HiDKK [r,,Ap. II
with /.. Tin- i'lMjcctivity carrvin^ M'><' t<> /•'('. I evidently carries ;tinl r tn />. r, and <i ivspcctivi-ly, and thus carries A' B'C' to /:'<''. I', and thus is dinvt ($ ID). Ik-nce
S(.UK')=S(BCA).
Th- j.mjr.-tivity carrying .l/.'C'to ^4(7# carries A'B'C' to A'C'B*, and hence is not iliivrt : and hence
S(ABC)* 8(ACB).
THKUKKM L'7. 7Vv> jHiint* C and D are on opposite sides of a line Mi ,/ and „„/./ if s(ABC) * S(ABD).
This theorem can be derived as a consequence of Ex. 1, § 2.">. It can also be derived from the following algebraic considerations.
Let us choose a system of nonhomogeneous coordinates in such a way that the singular line of the coordinate system is the same as the singular line of the Euclidean plane. The group of all projec- tive collineations transforming the Euclidean plane into itself then reduces (§ 67, Vol. I) to
x' = a.x + bv + c,, (15) , A =
1 1
«2 \
0.
If we change to the homogeneous coordinates for which x = xjx^ and y = XZ/XQ, the line at infinity has the equation a;Q=0, and the equations (15) reduce to
/•I /»v / /i'<y>_l_/»-»>_I_Ji/yi
yio^ Xj — Cj»c0-(- ttjXj -f- OyKn,
On the line at infinity this effects the transformation
3u — rt >jT -{-- J) X
which is direct if and only if A > 0.
Let the nonhoraogeueous coordinates of three points A, B, C be («,. «..). ('',. ''2)» (^,, c2) respectively. The determinant
H7) S =
is uiultijilit'd by A whenever the points A, B, C are subjected to the IriMisfurmation (15). This is verified by a direct substitution. Hence
§§ao,»ij EUCLIDEAN PLANK 63
the algebraic sign of S is left invariant by all direct collineations and changed by all others. Hence we have
THEORKM 28. An ordered triad of points (a^ <»2), (blt £>2), (ct, c2) has the same sense as an ordered triad (a(, a2'), (b[, b!2), (c^, c£) if and only if
the determinants „ „ \
and
b[ V, 1
c( c'2 1
have the same sign.
Theorem 27 now follows as a corollary of Theorem 23, § 27.
EXERCISES
1. If 4 ABC = 4A'BC', S(ABC) = S(A'BC').
2. Let 4 hk be said to have the same sense as 4h'kf if S(ABC) = S(A'B'C'), where B is the vertex of 4 hk, A a point of h, C a point of k, and A', H', < " points analogously defined for 4 h'kf. Define positive and negative angles and develop a theory of the order relations of rays through a point.
3. Let p and a be two planes of a projective space which meet in a line /» ; let us denote the two Euclidean planes obtained by leaving /» out of p and <r by pr and <rl respectively; and let Sp be an arbitrary sense-class in pv All ordered point triads of Sp are projected from a point 0 not on p or or into triads of a fixed sense-class S0 in <rr Any other point P not on p or a is separated from O by p and a- if and only if triads in the sense-class Sp are not projected from P into triads of $„.
*31. Sense in Euclidean spaces. The definition given above of direct transformations in a Euclidean plane, based on the concept of direct transformations on the singular line, cannot be generalized to three dimensions. This is because the plane at infinity is projective and, as will be proved in the next section, does not admit of a dis- tinction between direct and opposite projectivities. Nevertheless, the algebraic criterion A > 0 does generalize and is made the basis of the definition which follows.
With reference to a nonhomogeneous coordinate system, of which the singular (n — l)-space is the (n — 1) -space at infinity, the equations of any projective collineation of a Euclidean 7i-space take the form *
where the determinant |ay| is different from zero. The resultant of
* The reader may, if he wishes, limit attention to the case n = 3. We have not actually developed the theory of coordinate systems in n dimensions, but as there is no essential difference in this theory between the three-dimensional case and the n-dimensional, we do not intend to write out the details.
64 Kl.KMKNTAKV Tl I KOKKMS ( >N ORDER [CHAP. II
two transformations of this luim has a determinant which is the {.nxlurt of the determinants of the two transformations. Since the coefficients appear non homogeneously in (18), it is clear that a self- conjugate subgroup of the group of all transformations (18) is defined I iy the condition jay| > 0. It follows by the same reasoning as used in § 18 that this subgroup is independent of the choice of the frame of reference, so long as the singular (n — l)-space coincides with the singular (n — l)-space of the corresponding Euclidean Ti-space.
DEFINITION. The group of all transformations (18) for which the determinant |a<,-|>0 is called the group of direct collineations. In a Kuclidean w-space let Av A^ • • •, An+1 be n + 1 linearly independent points ; the class of all ordered (w--f-l)-ads* A[A'Z • • • A'n+1 such that the collineation transforming Alt Az,---,An+1 into A[, A!2,---,A^+l respec- tively is direct is called a sense-class and is denoted by S(A1A^ • • • An+l).
THEOREM 29. There are two and only two sense-classes in a Euclidean n-space. The sense-class of an ordered n-ad is unaltered by even per- mutations and altered by odd permutations.
Proof. The argument for the three-dimensional case is typical of the general case. Let the coordinates of four points A, B, C, D be («!» «,. «,)» (*V &2> &8). (<v <v cs)» (di> d* da) respectively. The determinant
<19)
2
is multiplied by | a~ | whenever the points are simultaneously subjected to a transformation (18). Hence the algebraic sign of (19) is left in- variant by all direct collineations.
Since an odd permutation of the rows of (19) would change the sign of (19), no such permutation can be effected by a direct collineation. The remaining statements in the theorem now follow directly from the theorem that any ordered tetrad of points can be transformed by a transformation of the form (18) into any other ordered tetrad.
* 32. Sense in a protective space. Let us consider the group of all linear transformations • n
(20) *;=i*,> (t=0,...,n)
for which the determinant \a^ is different from zero. * An n-ad is a get of n objects (cf . § 19).
§:«J PROJECTIVE SPACE 65
If (£Q, •••,#„) is a set of homogeneous coordinates, the equations (20) continue to represent the same transformation when all the a</s are multiplied by the same constant p ; and two sets of equations like (20) represent the same transformation only if the coefficients of one are proportional to those of the other.
If each ay. be multiplied by p, |«0.| is multiplied by p*+1. Hence, if |a0 is negative and n is even, we may multiply each a{J by — 1 and thus obtain an equivalent expression of the form (20) for which \atj\ is positive. If, however, n is odd, pn+1 = k < 0 has no real root. Hence, if 71 is odd, a transformation (20) for which |at;/| is negative is not equiv- alent to one for which \afj\ is positive. Hence the condition |a,y|> 0 determines a subset of the transformations (20) if and only if n is odd. This subset of transformations forms a group for the reason given in § 18 for the case n = l.
DEFINITION. If n is odd, the group of transformations (20) for which |a0.| > 0 is called the group of direct collineations iii 7i-space.
This definition of the group of direct collineations is independent of the choice of the frame of reference, as follows by an argument precisely like that used to prove the corresponding proposition in § 18.
In a space of three dimensions, let us inquire into what sets of five points the set (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 1, 1, 1) can be transformed by direct collineations. If the initial points are to be transformed respectively into the points whose coordinates are the columns of the matrix
(21) the collineation must take the form
4 = PtVo + />!««•*! + /W, + f where the p's satisfy the equations
(23)
I
»;.; l l.l.MI-M \i:V THEOREMS ON ORDER [CHAP, n
Substituting the values of pf determined from these equations in the determinant of the transformation (22), we see that the value of this determinant is
where the expressions in parentheses are abbreviations for deter- minants formed from the matrix (21) having these expressions as their main diagonals. The number (24) has the same sign as
( -'*) («M«n«*a3.) <aoo W«> (aooana*°») (°ooaiia«a J (a»auanaj> which is entirely analogous to the expression found in Theorem 16. The initial set of points is transformable into the points whose coor- dinates are the columns of (21) by a direct transformation if and only if (25) is positive.
This result may be stated in the form of a theorem as follows :
THEOREM 30. If a set of jive points whose homogeneous co&rdinates are the columns of the matrix (21) be such that the product of the fun /'-rowed determinants obtained by omitting columns of this matrix •in positive, it can be transformed by a direct collineation into any other set of points having the same property, but not into a set for finch the analogous product is zero or negative.
COROLLARY. Any even permutation but no odd permutation of the vertices of a complete Jive-point can be effected by a direct collineation.
DEFINITION. Let A, B, C, D, E be five points no four of which are coplanar. The class of all ordered pentads obtainable from the pentad A, B, C, D, E by direct collineations is called a sense-class and is denoted by S(ABCDE).
Theorem 30 and its corollary now give at once the following :
THEOREM 31. There are two and only two sense-classes in a real pro- jectice three-space. The sense-class of a set of Jive points is unaltered by even permutations and altered by odd permutations.
If an analogous definition of sense-class had been made in the plane, we should have had that all planar colliueations are direct, and hence that there is only one sense-class in the plane. This remark, together with Theorem 31, expresses in part what is. meant by the proposition :
The real protective plane is one-sided and the real protective three- space is two-sided.
PROJECTIVK I 'LANK
67
Although we have grounded this discussion upon propositions regarding certain groups of colliueations, the notion of sense is connected with a much more extensive group. We shall return to this study, which will give a deeper insight into the notions of sense and of one- and two-sidedness, in a later chapter.
33. Intuitional description of the projective plane. We may assist our intuitive conception* of the one-sidedness of the real projective plane by a further consideration of the regions into which a plane is separated by a triangle. These are represented in fig. 10. Since any tri- angular region is projectively transform- able into any other, it follows that any triangular region may be represented like Region I in fig. 16. In fig. 18 the four regions are thus represented, together with a portion of the relations among them.
The representation is more complete if the two segments labeled ft are superposed in such a way that the end labeled A of one coincides with the end labeled A of the other. This is represented in fig. 19 and may be realized in a model by cutting out a rectangular strip of paper, giving it a half twist, and pasting together the two ends.
FIG. 18
FIG. 19
To complete the model it would be necessary to bring the two edges labeled ft in fig. 18 into coincidence. This, however, is not possible in a finite three- dimensional figure without letting the surface cut itself.f
The twisted strip as an example of a one-sided surface is due to Mobius. J It has only one boundary AftCftA. An imaginary man OP on the surface (fig. 19) could walk, without crossing the boundary, along a path which is the
« It would not be difficult to give a rigorous treatment of the propositions in this section, but it is thought better to postpone this to a later chapter.
t Plaster models showing this surface are manufactured by Martin Schilling of Leipzig. J Gesammelte Werke, Vol. II, p. 619.
r,s
Kl.r.MKNTAKV TMKHKKMS <>N <>KI»KK [CHAP, n
of a straight line in tin- project ive plane, till he arrived at the antipodal position '"A It ii --mall triangle J!ST were to be moved with the man with- out In-iiii,' lifted fr.'iu the surface or being allowed to touch the man, it would be foun.!. when the man arrived at the position OQ, that the triangle could be superposed upon itself, R coinciding with itself, but S and T interchanged. in other words, the boundary of the triangular region containing 0 would .-oin.-i.le with itself with sense reversed.
It ix not essential that the triangular region RST be email, but merely that the figure URST move continuously so that the triangle JIST remains a triangle ami the jioint 0 is never on one of its sides. The possibility of making this transformation of the figure ORST into "/.''/ '.s1 is not affected by joining the two /hedges together, because none of the paths need meet the boundary of the strip. Therefore a corresponding continuous deformation can be made in the pro- jective plane.
If we think of the figure ORST in the projective plane, the four points enter symmetrically. Thus, since 5 and T can be interchanged by continuously moving the complete quadrangle, any two vertices can be interchanged by such a motion, and hence any }>erimitation of the four vertices can be effected by such a motion. This is intimately associated with the fact that all projectivities in the plane are direct (§ 32), as will be proved in a later chapter, where the notion of continuous deformation of a complete quadrangle in a projective plane is given a precise formulation.
The triangle RST may be replaced by any small circuit containing 0, and
it still remains true that O and the circuit may be continuously deformed till O coincides with itself and the circuit coincides with itself reversed. For • •\;iniple, the circuit may be taken as a conic section, and the projective plain- imaged as the plane of elementary geometry plus " a line at infinity " (see the introduction to Vol. I, §§ 3, 4, 5, and also § 28 above). The ellipse I (fig. 20) may be deformed into the parabola II, this into the hyperbola III, this into the parabola IV, and this into the ellipse V. The reader can easily verify that the sense indicated by the arrow on I goes continuously to that indicated mi V. The figures may be regarded as the projections from a variable center of an ellipse in a plane at right angles to the plane of the paj>er.
FIG. 20
PROJECTIVE SPACE
69
This deformation of an ellipse and also the corresponding one of the quadrangle OUST depend on internal properties of the surface; i.e. they are i in It-pendent of the situation of the surface in a three-dimensional space. They are sharply to be distinguished from the property expressed by saying that the man (>/' comes back to the position OQ, for the latter is a property of the space in which the surface lies.* In fact, the closely related proposition, that if the man OP walk along a straight line in a projective plane till he cdines back to the position OQ, the triangle JIST comes back to RTS, implies that if a tetrahedron (e.g. PQRS) be deformed into coincidence with itself so that two vertices are interchanged, the other two vertices will also be inter- changed. And the last statement is a manifestation of the theorem (§ 32) that although the projective plane is one-sided, the projective three-space is two-sided. .
A sort of model of the pro- jective three-space may be obtained by generalizing the discussion of the plane given above. Any one of the eight regions determined by a tetra- hedron is protectively equiva- lent to any other. Hence we pass from fig. 17 to fig. 21, which represents in full only the relations among the seg- ments, triangular regions, and tetrahedral regions having A^ as an end, or vertex. Each of the triangles having A2, A3, At as vertices is represented by two triangles in fig. 21. Thus, in
order to represent the projective space completely we should have to bring each of the triangular regions A^A^A^ into coincidence with the one which is symmetrical with it with respect to Ar In other words, fig. 21 would represent a projective three-space completely if each point on the octahedral surface formed by the triangular regions AzAsAt were brought into coinci- dence with the opposite point.
EXERCISE
Show that the octahedron in fig. 21 may be distorted into a cube so that the projective three-space is represented by a cube in which each j>oint coin- cides with its symmetric point with respect to the center of the cube.
» E. Steinitz, Sit/ungsberichte der Berliner Mathematischen Gesellschaft, Vol. VII (1908), p. 35.
CHAPTER III
THE AFFINE GROUP IN THE PLANE
34. The geometry corresponding to a given group of transformations. The theorems which we have hitherto considered, whether in general projective geometry or in the particular geometry of reals, state prop- erties of figures which are unchanged when the figures are subjected to collineations. For example, we have had no theorems about indi- vidual triangles, because any two triangles are equivalent under the general projective group, and thus are not to be distinguished from one another. On the other hand, there does not, in general, exist a collineation carrying a given pair of coplanar triangles into another given pair of coplanar triangles ; and thus we have the theorem of Desargues, and other theorems, stating projective properties of pairs of triangles. We have thus considered only very general properties of figures, and so have dealt hardly at all with the familiar relations, such as perpendicularity, parallelism, congruence of angles and seg- ments, which make up the bulk of elementary Euclidean geometry. These properties are not invariant under the general projective group, but only under certain subgroups. We shall therefore approach their study by a consideration of the properties of these subgroups.
There are, in general, at least two groups of transformations to con- sider in connection with a given geometrical relation : (1) a group 1>\ means of which the relation may be defined, and (2) a group under which the relation is left invariant. These two groups may or may not be the same.*
We have already had one example of a definition of a geometrical relation by means of a group of transformations. In § 10 two collmcar triads of points are defined as being in the same sense-class if they art- conjugate under the group of direct projectivities on the line. The relation between pairs of triads which is thus defined is invariant under the group of all projectivities (§ 18).
• The proup (1) will always be a self -con jugate subgroup of (2), as follows directly fnnii the dftinitinn of a self-conjugate subgroup. See § 39, below, where the r61e of wlf-r,,rijugate subgroups is explained and illustrated.
70
§§;u,35] THE AFF1XE GROUP 71
The system of definitions and theorems which express properties invariant under a given group of transformations may be called, in agreement with the point of view expounded in Klein's Erlangen Programm,* a geometry. Obviously, all the theorems of the geometry corresponding to a given group continue to be theorems in the geometry corresponding to any subgroup of the given group ; and the more restricted the group, the more figures will be distinct rela- tively to it, and the more theorems will appear in the geometry. The extreme case is the group corresponding to the identity, the geometry of which is too large to be of consequence.
For our purposes we restrict attention to groups of protective collineations,t and in order to get a more exact classification of theorems we narrow the Kleinian definition by assigning to the geometry corresponding to a given group only the theory of those properties which, while invariant under this group, are not invariant under any other group of protective collineations containing it. This will render the question definite as to whether a given theorem belongs to a given geometry.
Perhaps the simplest example of a subgroup of the projective group in a plane is the set of all projective collineations which leave a line of the plane invariant. The present chapter is concerned chiefly with the geometry belonging to this group.
The chapter is based entirely on Assumptions A, E, P, HQ. In fact, the theorems of §§ 36, 38, 39, 40, 42, 45, 46, 48 depend only on A, E, HQ. The class of theorems which depend on assumptions with regard to order relations has already been touched on in §§ 28~30.
35. Euclidean plane and the affine group. Let I* be an arbitrary but fixed line of a projective plane IT. In accordance with the definition in § 28 we shall refer to lv as the line at infinity. The points of lx shall be called ideal\ points or points at infinity, whereas the remaining points and lines of TT shall be called ordinary points and lines. The set of all ordinary points is a Euclidean plane. In the rest of this chapter the term " point," when unmodified, will refer to an ordinary point.
*Cf. F. Klein, Vergleichende Betrachtungen tiber neuere geometrische For- srlmngen, Erlangen 1872 ; also in Mathematische Annalen, Vol. XLIII (1893), p. 68.
t From some points of view it would have been desirable to include also all projective groups containing correlations.
J There is some divergence in the literature with respect to the use of this word and the word "improper." On the latter term sec § 85, Vol. I.
72 THK A K KINK UIMUT IN THE PLANE [CHAP, in
I IKKIM ii«»N. Any projective collineation transforming a Euclidean plain- into itst-lf is said to be affine ; the group of all such collineations is called tin- njfine group, and the corresponding geometry the affine
THEOREM 1. There is one and only one affine collineation transform- ing three vertices A, B, C of a triangle to three vertices A', B', C' respec- tively of a triangle.
Proof. Since /. is transformed into itself, this is a corollary of Theorem 18, § 35, Vol. I.
With respect to any system of nonhomogeneous coordinates of which /. is the singular line, any affine collineatiou may be written in the form (§ 67, Vol. I)
(1) y'lICt&y + o1!
where A = ^ j1 ^ 0.
36. Parallel lines. DEFINITION. Two ordinary lines not meeting in an ordinary point are said to be parallel to each other, and the pair of lines is said to be parallel. A line is also said to be parallel to itself.
Hence, in a Euclidean plane we have the following theorem as a consequence of the theorems in Chap. I, Vol. I :
THEOREM 2. In a Euclidean plane, two points determine one and only one line ; two lines meet in a point or are parallel ; two lines parallel to a third line are parallel to each other ; through a given point there is one and only one line parallel to a given line I.
I >KKINITION. A simple quadrangle ABCD such that the side AB is parallel to CD and BC to DA is called a parallelogram.
DEFINITION. The lines AC and BD are called the diagonals of the simple quadrangle ABCD.
In terms of parallelism, most projective theorems lead to a con- siderable number of special cases. Moreover, since the affine geom- etry is not self-dual, theorems which are dual in projective geometry may have essentially different affine special -cases. A few affine theo- rems which are obtainable by direct specialization are given in the following list of exercises, and a larger number in the next section.
5*36,37] PARALLEL LINES 73
EXERCISES
1. If the sides of two triangles are parallel by pairs, the lines joining corre- sponding vertices meet in a point or are parallel.
2. If in two projective flat pencils three pairs of corresponding lines are parallel, then each line is parallel to its homologous line.
3. With respect to any system of nonhomogeneous coordinates in which /. is the singular line, the equation of a line parallel to ax + l>y + c = 0 is ax + by + c' = 0.
4. A homology (or an elation) whose center and axis are ordinary trans- forms la, into a line parallel to the axis.
5. If the number of points on a projective line is/> + 1, the number of points in a Euclidean plane is p2, the number of triangles in a Euclidean plane is P3(p — 1)2(/»+ l)/6, and the latter is also the number of projective collinea- tions transforming a Euclidean plane into itself.
37. Ellipse, hyperbola, parabola. DEFINITION. A conic meeting /„ in two distinct points is called a hyperbola, one meeting it in only one point a parabola, and one meeting it in no point an ellipse. The
t.
Ellipse Parabola Hyperbola
FIG. 22
pole of lx is called the center of the conic. Any line through the center is called a diameter. The tangents to a hyperbola at its points of intersection with /« are called its asymptotes. A conic having an ordinary point as center is called a central conic.
EXERCISES
1. An ellipse or a hyperbola is a central conic, "but a parabola is not.
2. The center of a parabola is its point of contact with /«.
3. No two tangents to a parabola are parallel.
4. The asymptotes of a hyperbola meet at its center.
5. Two conjugate diameters (cf. §44, Vol. I) of a hyperbola are harmon- ically conjugate with respect to the asymptotes.
6. If a simple hexagon be inscribed in a conic in such a way that two of its pairs of opposite sides are parallel, the third pair of opposite sides is parallel.
71 THK All INK GROUP IN THE PLANE [CHAP, in
7. If a jiarallflogram be inscribed in a conic, the tangents at a pair of ,.|.|M.Mte v. -Fliers an- parallel.
8. If tin- vertices of a triangle are on a conic and two of the tangents at the vertices are parallel to tin- respectively opposite sides, the third tangent is parallel to the third side.
9. If a j>arallelc)<;raiii lie circumscribed to a conic, its diagonals meet in the center and are conjugate diameters.
10. If a parallelogram be inscribed in a conic, any pair of adjacent sides are parallel to conjugate diameters. Its diagonals meet at the center of the conic.
11. Let /' and /*' be two points which are conjugate with respect to a conic, let p be the diameter parallel to PIV, and let Q and Q' be points of intersection with the conic of the diameter conjugate to p. The lines PQ, and P'Q,' meet on the conic.
12. If a parallelogram OAPB is such that the sides OA and OB are conju- gate diameters of a hyperbola and the diagonal OP is an asymptote, then the other diagonal AB is parallel to the other asymptote.
13. If two lines OA and OB are conjugate diameters of a conic which they meet in A and B, then any two parallel lines through A and B respectively meet the conic in two points A' and B' such that OA' and OB' are conjugate diameters.
14. Any two parabolas are conjugate under a collineation transforming /« into itself.*
15. Any two hyperbolas are conjugate under a collineation transforming /,. into itself.*
16. Derive the equation of a parabola referred to a nonhomogeneous coor- dinate system with a tangent and a diameter as axes.
17. Derive the equation of a hyperbola referred to a nonhomogeneous coor- dinate system with the asymptotes as axes.
18. Derive the equation of an ellipse or a hyperbola referred to a nonhomo- geneous coordinate system with a pair of conjugate diameters as axes.
38. The group of translations. DEFINITION. Any elation having /. as an axis is called a translation. If I is any ordinary line through the center of a translation, the translation is said to be parallel to I.
COROLLARY. A translation carries every proper line into a parallel line and leaves invariant every line of a certain system of parallel lines.
TM KOREM 3. There is one and only one translation carrying a point A to a point B.
Proof. Any translation carrying A to B must be an elation with /, as axis and the point of intersection of the line AB with lx as center. Hence the theorem follows from Theorem 9* Chap. Ill, Vol. I.
* On the corresponding theorem for ellipses, see § 76, Ex. 7.
§38]
TRANSLATIONS
75
Til Km; KM 4. An ordered point pair AB can be carried by a trans- lation to an ordered point pair A'B' such that A' is not on the line AB, if a nd only if ABB' A' is a parallelogram.
Proof. Let L* and Mx be the points at infinity on the lines AA' and AB respectively. The translation carrying A to A' must carry the line AMX to A'MX and leave the line BL^ invariant. Hence the point B, which is the intersec- tion of AM* with BL*,, is carried to B', which is the intersection of A'MxviithBLx. Hence the points A' and B' to which A and B respec- A i lively are carried by a translation are such that ABB' A' is a parallelogram. Since there is one and only one translation carrying A to A', the same reasoning shows that when- ever ABB' A' is a parallelogram there exists a translation carrying A and B to A1 and B' respectively.
THEOREM 5. An ordered point pair AB is carried by a translation to an ordered point pair A'B', where A' is on the line AB, if and only if Q(L*AA', L*,B'B), Lx being the point at infinity o/AB.
Proof. Let P be any point not on the line AB, and let M* and section of PA and PA' with I,
respectively be the points of inter- Let Q be the point of intersection of
EM* with PL*. Then, by the last theorem, the translation carrying
TIU: AITIM: <;uorr IN TIN-: IM.ANK irnAP. m
AtoB carries P to Q, and hence carries A' to the point of intersection vV. with .I/.'. Hence Nx, Q, and It' are collinear, and hence we havi- Q(LmAA',LmI?J?).
TIIKOKKM 6. If A, B, C are any three points, the resultant of the translations carrying A to B and B to C is the translation carrying .( /.
/'roof. Let A», B*,, C. be the points of intersection of the lines BC, CA, AB respectively with L. Suppose first that the three points A*, Bm, C» are all distinct. The translation carrying A to B changes the line AB* into the line BBX, and the translation carrying B to C changes the line BBX into CBn. Hence the line AB* is invariant under the resultant of these two translations.
Consider now any other line through !?„, and let it meet A A,, in A' and B C in C' ; also let B1 be the point of intersection of A'Cn with /?<7(fig. 25). We then have that the translation carrying A to B carries A' to B' (Theorem 4), and on ac- count of Q(AXBB', AvC'C) (Theorem 5) the translation carrying B to C carries B'
to C'. Hence the resultant of the two translations carries A' to C' and thus leaves the line A'B* invariant; that is, it leaves all the lines through /?„ invariant. Since it obviously leaves all points on /, invariant, it is a translation (Cor. 3, Theorem 9, Chap. Ill, VoL I).
If two of the three points A*, Bx, C,, coincide, they all coincide, and in this case the theorem is obvious.
By definition, the identity is a translation. Hence we have
COROLLARY. TJie set of all translations form. a group.
THEOREM 7. The group of translations is commutative.
Proof. Given two translations Tt and T2 and let A be any point, rTl(A)=A' and Tt(A') = B'. If B' = A, T2 is the inverse of T,, and hence T, and Ta are obviously commutative. If B' =£ A and B' is not
§:«] TRANSLATIONS 77
(»n the line AA', let B (fig. 23) be the point of intersection of the line through A parallel to A' B' with the line through B' parallel to A A', then ABB' A' is a parallelogram, and it is obvious that T1(B) = B' and 1^(A)=B. Hence T/T^-l) =#'. But, by the definition of A' and B', T9Tl(A) = If. Hence, in this case also, Tx and T2 are commutative.
In case B' is on the line AA', let P and Q (fig. 24) be two points such that A'B'QP is a parallelogram, let B be the point of inter- section of AA' with the line through Q parallel to AP, and let Z-., J/,,jVcobe the points at infinity of P(), /M, and fM' respectively. Then, since Ti(A') = B', it is obvious that Ta(P) = Q, and hence that TZ(A) = B. Moreover, on account of ^(LXAB,L^B'A'), T,(^) = ^' implies that Tfl(B)=S1. Hence TlT9(A) = Jil, and thus, in this case also, Tj and T2 are commutative.
THEOREM 8. If OX and OY are two nonparallel lines and T is any translation, there is a unique pair of translations TI? T2 such that TX is parallel to OX, T2 parallel to OY, and T1T2 = T.
Proof. In case T is parallel to OX or 0 Y the theorem is trivial. If T is parallel to neither of them, letP=T(0) and let A\ and Yl be the points in which the lines through P parallel to 0 Y and OX respectively meet OX and 0 Y respectively. Then OXf Y^ is a par- allelogram, and if Tt be the translation carrying 0 to X^ and T2 the translation carrying 0 to Ft, it follows, by Theorems 4 and 6, that TtTt=*T.
On the other hand, if T{ is any translation parallel to OX, and Tj any translation parallel to OY, and T{(0) =A'/ and TJ(0) =Y{, the product T(T2 carries 0 to a point P' such that OX[P'Y[ is a par- allelogram. But P' = P if and only if X[ = Xl and Y[= Yf Hence T determines TX and T2 uniquely.
THEOREM. 9. With respect to a nonhomogeneous coordinate system in which lx is the singular line a translation parallel to the x-axis
has the equations
x'=x + a,
(2)
y-y-
Proof. The point into which (0, 0) is transformed by a given trans- lation parallel to the z-axis may be denoted by (a, 0). By Theorem 5 and § 48, Vol. I, it then follows that any point (x, 0) of the a>axis
78 THK All INK GROUP IN THE PLANK [CHAP, in
is transformed into (<- + <i, 0). Since lines parallel to the y-axis are transformed into lines parallel to the y-axis, and since lines parallel to the j-axis are invariant, it follows that the given translation takes the given form (-).
' inversely, any transformation of the type (2) leaves all lines par- allel to the .r-axis invariant and transforms any other line into a line parallel to itself. Hence it is a translation parallel to the a>axis.
THEOREM 10. With respect to a nonhomogeneous coordinate system in irhich /. is the singular line, any translation can be expressed in
the form
x = x + a,
'
Proof. By Theorem 8 any translation is the product of a translation parallel to the avaxis by one parallel to the y-axis. Hence it is the product of a transformation of the form
x'= x + a
y'=y>
by a transformation of the form
x' = x,
EXERCISE Investigate the subgroups of the group of translations.
39. Self -con jugate subgroups. Congruence. DEFINITION. Any sub- group G' of a group G is said to be self-conjugate or invariant* under G if and only if 2T2~ * is an operation of G' whenever 2 is an operation of G and T of G'.
The geometric significance of this notion is as follows: Suppose that two figures F^ and Fy are conjugate under G', and T is a trans- formation of G' such that F2 = rY(Fl)
If /', and F% are changed into F[ and i F' ) ~ ^ *"'
l-\ by any transformation 2 of G, then 2~ ! (F[) = Ff Hence t T2~ ' (F[) = Ff
• These terms have already been defined in
t These relations may be illustrated by the VLx/ s
i.vins diatrrain (probably due to S. Lie). FIG. 26
§:wj CONOR IKN(K 79
and STS-^Y) = .Ff'. Therefore, if G' is self-conjugate under G, the figures /'/ and F^ are conjugate under G'. Hence the property of being conjugate under the self -conjugate subgroup G1 is a property left invariant by the group G. Thus the theory of figures con- jugate under G' belongs to the geometry corresponding to G, pro- vided that G is not a self-conjugate subgroup of any other group of projective collineations.
THEOREM 11. The group of translations is self -conjugate under the affine group.
Proof. Let T be an arbitrary translation and 2 an arbitrary affine transformation. We have to show that 2T2"1 is a translation. If P be any point of /«, 2(P) is also on L. Therefore, since T leaves all points of /« invariant, so does ST2"1. The system of lines through the center of T is a system of parallel lines ; 2 trans- forms this system of parallel lines into a system of parallel lines ; and hence the latter system of parallel lines is invariant under 2T2-1. Hence (cf. Cor. 3, Theorem 9, Chap. Ill, Vol. I) 2T2'1 is a translation.
COROLLARY 1. The group of translations is self-conjugate under any subgroup of the affine group which contains it.
COROLLARY 2. For any affine collineation S, and any translation T, there exists a translation T' such that 2T=T'2 and a translation T" such that TS = 2T".
Proof. Let 2T2'1 = T' and 2-'T2 = T". By the theorem, T' and T" are translations. But
2T2-1 = T' and 2-IT2 = T" imply 2T = T'2 and T2 = 2T" respectively.
DEFINITION. Two figures are said to be congruent if they are con- jugate under the group of translations.
This definition will presently be extended by giving other condi- tions under which two figures are said to be congruent.* In view of Theorem 11, the theory of congruence as thus far defined belongs to the affine geometry.
• A complete definition would be of the form, "Two figures are said to be con- gruent if and only ?/•••"
80 TIIK AT FINE GROUP IN THE PLANE [CHAP, in
40. Congruence of parallel point pairs. The figure consisting of t\\.. dist met points A, /•' may be looked at in two ways with respect to congruence. We consider either the two ordered* point pairs Mi ami /:.! or the point pair AB without regard to order. In the second case AB and BA mean the same thing and AB is congruent to BA because the identity belongs to the group of translations. On the other hand, the ordered pair AB is not conjugate to the ordered pair BA under the group of translations, because the trans- lation carrying A to B does not carry B to A (this is under Assump- tions A, E, H0).
THEOREM 12. If ABDC is a parallelogram, the ordered point pair AB is congruent to the ordered point pair CD. If the condition Q(/^AC, I*DB) is satisfied where P^ is an ideal point, the ordered point pair AB is congruent to the ordered point pair CD.
Proof. This is a corollary of Theorems 4 and 5.
COROLLARY 1. Let A and B be any two distinct points and 0 the harmonic conjugate of the point at infinity of the line AB with respect to A and B. Then the pair AO is congruent to the pair OB.
DEFINITION. The point O in the last corollary is called the mid- point of the pair AB. In case B=A, A is called the mid-point of the pair AB.
COROLLARY 2. The line joining the mid-points of the pairs of vertices AB and AC of a triangle ABC is parallel to the line BC.
Proof. Let #. and C. be the points at infinity of the lines AB and AC respectively, and let Bl and Cl be the mid-points of the pairs AB and AC respectively. Then, by the definition of "mid-point,"
Hence the lines B^ BC, and BnCn concur, which means that BlCl and BC are parallel.
I JKFIMTION. The line joining a vertex, say A, of a triangle ABC to the mid-point of BC is called a median of the triangle.
THEOREM 13. The three medians of a triangle meet in a point.
* Cf . footnote on page 40.
§§40,4i] CONGRUENCE 81
Proof. Let the triangle be ABC\ let A», £„, C. be the points at infinity of the sides BC, CA> AB respectively ; and let Alt Blt Ct be the points of intersection of the pairs of lines BBX and CCnt CC* and AA*t
AA<* and BB* respectively (fig. 27). Then, by well-known theorems on harmonic sets (§ 31, Vol. I), the medians of the triangle ABC are AAlt BBlt and CClt and these three lines concur.
EXERCISES
1. The diagonals of a parallelogram bisect one another; that is, if ABCD is a parallelogram, the mid-points of the pairs A C and BD coincide.
2. Let a and b be two parallel lines. The mid-points of all the pairs AH where A is on a and B on b are on a line parallel to a and b.
3. If the sides AB, BC, CA of a triangle ABC are respectively parallel to the sides A'B', B'C', CM' of a triangle A'B'C', and the ordered point pair A B is congruent to the ordered point pair A'B', then the two triangles are congruent.
4. The mid-points of the pairs of opposite vertices of a complete quadri- lateral are collinear. Let us call this line the diameter of the quadrilateral.
5. A line through a diagonal point O of a complete quadrangle, parallel to the opposite side of the diagonal triangle, is met by either pair of opposite sides of the quadrangle which do not pass through 0 in a pair of points having 0 as mid-point.
41. Metric properties of conies. The following list of exercises contains a number of theorems on conies which involve the congru- ence of parallel point pairs and can be derived by aid of the theorems in the last sections.
>L» rin. AiTiNK cKorr IN THK PLANE [CHAP. m
EXERCISES
1. The iui<l-]M'ints of :i system of j.:iirs of {mints of a conic A A', BB', <'< ", etc. an- eollinear if the lines .11. /•'/.', CC' are parallrl. The line containing the mid-points is a diameter conjugate t<) the diametrr parallel to A A'.
2. Let .1 :in«l l\ be two |K»int,s of a parabola. If the line joining the mid- jmint (' of the p;iir AB to the pole P of the line AB meets the conic in O, th.-n O is the mid-|>oint of the pair CP.
3. If a line inert s a hyi>erbola in a pair of points //,//.,, and its asymptotes in a pair AvAt, the two pairs have the same mid-i>oint. The pair H1Alia con- gruent to the pair HtAr
4. The point of contact of a tangent to a hyperbola is the mid-point of the pair in which the tangent meets the asymptotes*.
5. Let Jj and .-la be each a fixed and X a variable point of a hyperbola, and let A", and A'2 be the points in which the lines XAl and AM2 meet one of the asymptotes. The point pairs A'jA'2 determined by different values of A' are all congruent.
6. The centers of all conies inscribed in* a simple quadrilateral A BCD are on the line joining the mid-points of the point pairs CA and BD.
7. The centers of all conies which pass through the vertices of a complete quadrangle A BCD are on a conic C'2, which contains the six mid-points of the pairs of vertices of the quadrangle, the three vertices of its diagonal triangle, and the double points (if existent) of the involution in which /«, is met by the pencil of conies through A, B, C, D. From the projective point of view, according to which /« is any line whatever, C2 is called the nine-point (or the iln-ni-/>iiiiit) conic of the complete quadrangle A BCD and the line /«. Derive the analogous theorems for the pencils of conies of Types II-V (cf. § 47, Vol. I).
8. The five diameters f of the complete quadrilaterals formed by leaving out one line at a time from a five-line meet in a point A, which is the center of the conic tangent to the five lines.
9. The six points A determined, according to the last exercise, by the six complete five-lines formed by leaving out one line at a time from a six-line an- on a conic C2.
10. The seven conies C2 determined, according to the last exercise, by the - \.-n complete six-lines formed by leaving out one line at a time from a -••\.-n-line, all pass through three points.
42. Vectors. Any ordered pair of points determines a set of pairs all of which are equivalent to it under the group of translations. In order to study the relations between such sets of pairs we introduce the notion of a vector. The term " vector " appears in the literature
• A conic is said to be inscribed in a given figure if the figure is circumscribed to the conic (cf. § 43, Vol. I).
t Cf . Ex. 4, § 40. This and the following exercises are taken from an article by W. W. Taylor, Messenger of Mathematics, Vol. XXXVI (1907), p. 118.
§42] VECTORS 83
under a multitude of guises, none of which, however, is in serious contradiction with the following abstract definition. In this definition the term " ordered pair of points " is to be understood to include the case of a single point counted twice.
DEFINITION. A planar field of vectors (or vector field) is any set of .objects, the individuals of which are called vectors, such that (1) there is one vector for each ordered pair of points in a Euclidean plane, and (L') there is only one vector for any two ordered pairs AB and A'B' which are equivalent under the group of translations. A vector cor- responding to a coincident pair of points is called a null vector or a zero vector, and denoted by the symbol 0.
For example, a properly chosen set of matrices would be a vector field according to this definition. So would also the set of all translations including the identity ; also a set of classes of ordered point pairs such that two point j-uir.s are in the same class if and only if equivalent under the group of trans- lations. However a vector field be defined, it will be found that, in most applications, only those properties which follow from the definition as stated above are actually used.
A precisely similar state of affairs exists in the definition of a number system. The objects in the particular number system determined for a given space by the methods of Chap. VI, Vol. I, are points, but a number system in general is any set of objects in a proper one-to-one correspondence with this set of points.
In the following discussion we shall suppose that one field of vectors has been selected, and all statements will refer to this one field. Thus, the vector corresponding to the point pair AB is a definite object, and we shall denote it as " the vector AB" or, in symbols, Vect ( AB}.
Since any point of a Eu- clidean plane can be carried by a translation to any other point, the set of all vectors is the same
I Hi.
as the set of vectors OA, where
0 is a fixed and A a variable point. Consequently, the following defi- nition gives a meaning to the operation of " adding " any two vectors. DEFINITION. If O, A, C are points of a Euclidean plane, the vector OC is called the sum of the vectors OA and AC. In symbols this is
84
TIIK AFF1NE GROUP IN THE PLANE [CHAP, in
in.liratotl l.y Yrrt (nC) = Vect (OA) + Vect (AC). The operation of obtaining the sum of two vectors is called addition of vectors. An obvious corollary of this definition is that
Vect (AB) + Vect (BA) = 0.
Hri ice we define:
DEFINITION. The vector Vect (BA) is called the negative of the vector Vect (AB), and denoted by — Vect (AB).
THEOREM 14. The operation of addition of vectors is associative; that is, if a, b, c are vectors, (a + b) + c = a + (b + c).
Proof. Let the three vectors be OA, AB, BC respectively ; then, by definition, both (Vect (OA) -f-Vect (AB)) + Vect (BC) and Vect (OA) + (Vect (AB) + Vect (BC)) are the same as Vect (0(7).
DEFINITION. Two vectors are said to be collinear if and only if they can be expressed as Vect (OA) and Vect (OB) respectively, where 0, A, B are collinear points.
THEOREM 15. The sum of two noncollinear vectors OA and OB is the vector OC, where C is such that OACB is a parallelogram.
Proof. By Theorem 4, the vector OB is the same as the vector AC. Hence, by definition, the sum of OA and OB is OC.
THEOREM 16. The sum of two collinear vectors OA and OB is a vector OC such that Q(%,AO, R,BC)t where Px is the point at infinity of the line AB.
Proof. Let L and M be two points such that OBML is a parallelogram. Hence Vect (OB)= Vect(LJIf). Then, by definition, C must be such that Vect (LM) = Vect (A C), that is, such that AC ML is a parallelogram. Let /-. be the ideal
point of intersection of the lines OL and BM, and let M» be the ideal point of intersection of the lines AL and MC. The complete quadrangle ^ determines Q(%AO, P»BC).
M
§§4-.', 43] YKCTORS 85
COROLLARY. If O, A, B are three collinear points, and C a point siifh that Vect(OA) + Vect (OB) = Vect(OC), then, with respect to any scale (cf. § 48, Vol. I) in which PQ is 0 and J^, the point at infinity of the line OA, A+B = C.
Proof. Cf. Cor. 1, Theorem 1, Chap. VI, Vol. I.
THKOREM 17. The operation of adding vectors is commutative ; that is, if a and b are vectors, a + b = b + a.
Proof. Let the vectors a and b be Vect (OA) and Vect (OB) respectively. If 0, A, B are noncollinear, the result follows from Theorem 15, and if they are collinear, from Theorem 16.
43. Ratios of collinear vectors. By analogy with the case of addition we should be led to base a definition of multiplication of collinear vectors upon the multiplication of points in § 49, Vol. I. There are, however, a great many ways of defining the product of two vectors, which would not reduce to this sort of multiplication in the case of collinear vectors. Hence, in order to avoid possible confusion we shall not introduce a definition of the multiplication of vectors at present, but only of what we shall call the ratio of two collinear vectors.
DEFINITION. The ratio of two collinear vectors OA and OB is the number which corresponds to A in the scale in which P^ is O, Pl is B, and Px is the point at infinity of the line OA. It is denoted by
Vect (OA) OA
- s - - or bv --- 7 OB
It is to be emphasized that the ratio of two collinear vectors as here defined is a number. By comparison with the definition in § 56, Vol. I, we have at once
THEOREM 18. If A, B, C, Z)« are collinear points, Dn being ideal,
AC *(D.A,BC)--.
Theorem 13, Chap. VI, Vol. I now gives
THEOREM 19. If Av A^, At, At are any four collinear ordinary points,
THEOREM 20. If two triangles ABC and A'B'C' are such that the sides AB, BC, CA are parallel to A'B1, B'C', C'A' respectively,
AB BC = CA A'ff ~ B'C' ~ C'A' '
M! THE AFFINE GROUP IX THE PLANE [CHAP, in
Proof. Suppose that the translation which carries A' to A carries /;' t<> /•' ami C* to < L Then Bl is on the line AB and Cl on the line ami th«- line BlCl is parallel to BC. Thus, if Bx be the point at infinity of tin- lint- .!/>', and <« the point at infinity of the line AC,
All AC CA Hence, by Theorem 18, — = = — = —
which is, by definition, the same as
AB ' CA A'B'~ C'A1'
In like manner, it follows that
AB BC
Since we have not defined the product of two vectors,* it is necessary to resort to a device in order to compute conveniently with them. This we do as follows:
1 JKKIXITION. With respect to an arbitrary vector OA, which is called a unit vector, the ratio o/;
OA'
where OB is any vector collinear with OA, is called the magnitude of OB.
Observe that the magnitude of OB is the negative of the magnitude of BO. Since the magnitude of a vector is a number, there is no diffi- culty about algebraic computations with magnitudes. In the rest of this section we sliall use the symbol AB to denote the magnitude of the vector AB. No confusion is introduced by this double use of the symbol, because the ratio of two vectors is precisely the same as the quotient of their magnitudes.
DEFINITION. If F is any collineation not leaving lx invariant, the lines r(/<B) and F"1^.) are called the vanishing lines of F. If II is any projectivity transforming a line I to a line I' (which may coincide with /), the ordinary points of I and V which are homologous with points at infinity are (if existent) called the vanishing points of II. If II is an involution transforming I into itself but not leaving the IK lint at infinity invariant, the vanishing point is called the center of the involution.
TiiK'iKKM 21. DEFINITION. If 0 and 0' are the vanishing points, on I and V respectively, of a projectivity transforming a line I to a
1 43] VANISHING LINES 87
parallel* line I', and X is a variable point of I, andX' the point of V to which X is transformed, the product OX • O'X' is a constant, called tin- power of tfte transformation.
Proof. Let P*> be the point at infinity of / and V • and let Xl and A'2 be two values of X, and A'/ and A"2' the points to which they are transformed by the given projectivity. Then, by the fundamental property of a cross ratio,
B (PJD, A>Y2) = B (0'Pn, ATX)
0 Y O'X1 and hence, by Theorem 18, — = — j— i.
Hence, by the definition of magnitude of vectors, OX2 - 0'AT2' = OXl • O'X[.
COROLLARY 1. The power of an involution having a center 0 and a conjugate pair AAl is OA • OA^
COROLLARY 2. Let II be a homology whose center is an ordinary point F and whose axis is an ordinary line, and let D be any point of the vanishing line II"1^). If P is a variable point, P' = II(P), and I)' is the point in which the line through P' parallel to FD meets the vanishing line II (lx), then pp
P'D1
Proof. Let Q and Q' be the points in which the line FP meets the vanishing lines II"1^*,) and H(L) respectively. By the theorem,
from which we derive successively
PF + FQ _ FP' +P'Q'
FQ P'Q'
PF FP1
FP _ QF FP'~ P'Q''
Since II is a homology, the two vanishing lines are parallel. Hence
QF DF
Hence
P'Q1 P'J>' FP DF FP' ~ P'D'
* With the extension of the definition of congruence in the next chapter the restriction to parallel lines may be removed.
88 THE AFFIX K GROUP IN THE PLANE [CHAP, in
EXERCISES
1. If a |>roj«vtivity .1 RCD~^ A 'H <"!>' is such that the point at infinity of the line A B corresponds to the point at infinity of the line A'B',
AB ^A'B* CD CD-'
2. If three parallel lines a, b, c are met by one line in the points A', B', C" n-si»Ttiv»'ly and by another line in A"B"C" respectively, then
A'B' ^ A"B" A'C' A"C"'
3. If ABCD&re any four collinear points,
A B - CD + A C - DB + A D • B C = 0 .
4. Six points form a quadrangular set Q (/12ZJ2C2, vljfijC^) if and only if
B (.4^,, #,(?!) . B (B&, CV4t) • B (C,C2, A&) = - 1.
5. The condition for a quadrangular set may also be written
6. If three tangents to a parabola meet two other tangents in Pv 1\, Ps and <2ii Qj. Q» respectively, then p p Q Q
PS* = ~W*'
Conversely, if five lines are such that the points in which two of them meet the other three satisfy this condition, the conic to which the five lines are tangent is a parabola.
7. Let O be the center of a hyperbola, and A^ and Az the points in which the asymptotes are met by an arbitrary tangent ; if another tangent meets the asymptotes OAV OA2 in B^ and jB2 respectively,
OAl _
8. If a fixed tangent p to a conic at a point P meets two variable conju- gate diameters in Q and Q', then PQ • PQ' is a constant. Let 0 be the center of the conic. If the diameter parallel to p meets the conic in S, then
PQ-PQ' =-
9. Let Ol and O2 be the points of contact of two fixed parallel tangents to a conic. If a variable tangent meets the two fixed tangents in A", and X3 respectively, O,A', • OtXt is constant. If O is the center of the conic and B is a point of intersection of the diameter through O parallel to the fixed tangenta,
§§4:«.44|
THEORY OF TRANS\ KUSALS
89
44. Theorems of Menelaus, Ceva, and Carnot. THEOREM 22 (MENELAUS). Three points A', B', C' of the sides BC, CA, AB, respectively, of a triangle are collinear if and only if
A'B B'C C'A
A'C B'A C'B
1
Proof. Let the points at infinity of BC, CA, AB, A'B' be Ax, Bx, C*,, Px respectively, and let A" be the intersection of APX with BC. Then,' supposing A', B', C' collinear,
P P
-*<» ... . •*<*>
and
|
V Hence and Hf>nr.fi |
T>> |
' A v A', A"C) A', BA") A'C A' |
A'C |
|||
|
B'A B-"' |J|=R(C.C", A'B B'C |
ff A\ - |
Rl A |
A'A" A'A" |
|||
|
C'A |
V--TOO A'B |
A'B A" 1 |
A'C B'A C'B A'C A'A" A'B
The converse argument is now obvious.
THEOREM 23 (CEVA). The necessary and sufficient condition for the concurrence of the lines joining the vertices A, B, C of a triangle to the points A', B', C' of the opposite sides is
A'B B'C C'A
(4)
A'C B'A C'B
= -1.
«.m THE AKFIM: <;KOUP IN THE PLANE [CHAP, m
Proof. Let C" be the point of intersection of the lines A'B' and AB. Suppose first that C" is an ordinary point. Then, by the theorem of M,n,laus, JB M C^A =
A'C ' B*A ' C"B
The point C" is harmonically conjugate to C' with respect to A and B if and only if the lines AA', BB', CC' meet in a point. Thus,
C'A C"A _ ~T*n-
is a necessary and sufficient condition that AA', BB', CC' concur. But on multiplying (5) by (6) we obtain (4).
)C
In case C" is an ideal point, the line A'B' is parallel to AB and. by Theorem 20,
(7} — - — = 1
A'C B'A
The condition that C" be harmonically conjugate to C' with regard to A and B now takes the form
C'A
C'B On multiplying this into (7) we again obtain (4).
THEOREM 24 (CARNOT). Three pairs of points, A^AZ, B^B^ CtCtt respectively, on the sides BC, CA, AB, respectively, of a triangle are on the same conic if and only if
g ^ j , ,
AVC ' BVA ' C^B ' AtC
§44]
THEORY OF TRANSVERSALS
91
Proof. Suppose first that the conic reduces to two lines containing
y/j, />t, Cl and A9, Bf, C3 follows directly from Theorem 22 when we multiply together the conditions that Alt Blt Cl and A2, BZ, €„ be re- spectively collinear.
Now consider any proper conic through Alt At, Bv B^ meeting the line A B in C[ C2'. By the theorem of Desargues (Theorem 19, Chap. V, Vol. I) the pairs AB, C'jCj, and C{C[ are in involution. Hence
and hence
respectively. The formula (8) in this case
FIG. 32
or
Hence the formula (8) is equivalent to the formula obtained from it by substituting C[, C£ for Clt Ca respectively. Hence the formula holds for any conic. The converse argument is now obvious.
The last three theorems are the most important special cases of the "theory of transversals." A few further theorems of this class and some other propositions which can readily be derived from them are stated in the exercises below. Further theorems and references will be found in the Encyclopadie der Math. Wiss. Ill AB 5, § 2, and III C 1, § 23.
EXERCISES
1. The six lines joining the vertices A, B, C of a triangle to pairs of points ! . I.,, /•',/'-• f\C"t on tne respectively opposite sides are tangents to a conic if
and only if the relation (8) is satisfied.
2. If th«- sides ]',<', ('A, AB of a triangle-are tangent to a conic in AA BVC1 respectively, CA, AB, BC, =
BAi' CB, AC,
mi: AI-TIM-: <;K<.>UP IN THE PLANE [CHAP.IH
3. If a lin«- />'< ' mr.-t- :i ••<.nir in .1, and .!„, and two parallel lines through B and (', rej»j*ftiv.-lv, ni«-«-t it in tin- pairs ( ',, < '2 and Bv #2 res|>ectively,
AjB AjB BiC B^C _
' ' '
4. L«-t two linrs >» ami A through a point 0 meet a conic in the pairs Av At and /*,, Bt ifsjKvtiv.-ly. If 0, n, \> are variable in such a way that a and b remain n-sjHH-tively parallel to two fixed lines,
is a constant.
5. If the sides of a triangle meet a conic in three pairs of points, the three pairs of lines joining the pairs of points to the opposite vertices of the triangle are tangents to a second conic. State the dual and converse of this theorem.
6. If two points are joined to the vertices of a triangle by six lines, these lines meet the sides in six points (other than the vertices) which are on a conic. Dualize.
7. If a line meets the sides A9AV A^AV • • •, AnA0, respectively, of a simple polygon A9AiAt • • • An in points B0, Bv •••, BH respectively,
'IpBo AIBI dn^» — i . ... _ _!_.
8. If a conic meets the lines A0AV -4r<l2, •••, AnA0, respectively, in the pairs of points B0C0, B^C-p •••, BHCn respectively,
_ -
~
9. If a conic is tangent to the lines A0AV A^v • • •, AnA0, respectively, in the points B0, Blt • • •, Bn respectively,
A0B0
AB
= (-!)«
45. Point reflections. DEFINITION. A homology of period two whose axis is /. is called a point reflection.
From this definition there follows at once :
THEOREM 25. A point reflection is fully determined by its center. The center is the mid-point of every pair of homologous points. Ect'nj two homologous lines are parallel.
I'liKou KM 2fi. The product of two point reflections whose centers 'liftinct is a translation parallel to the line joining their centers.
Proof. The product obviously leaves fixed all points of I*, and also the line joining the two centers. Let Cl and C3 be the two centers,
§45] POINT KEFLECTIONS 93
and let /' be any point not on the line C^Cf Also let P' be the trans- form of P by the point reflection with Cl as center, and let Q be the transform of P' by the point reflection with C'2 as center. Since Cv is the mid-point of the pair PP', and ('„ of the pair P'<Q, the line PQ is parallel to C^C, (Theorem 12, Cor. 2). Thus the product of the two point reflections leaves invariant all lines parallel to C\Ca, and hence is a translation.
COROLLARY. The product of any even number of point reflections is a translation.
THEOREM 27. Any translation is the product of two point reflections one of which is arbitrary.
Proof. Let T be any translation, Cl the center of any point reflec- tion, Ct = T((71), and C^ the mid-point of the pair C,*?,. The product of the reflections in Cl and <?a is a translation, by Theorem 26, and since it carries Cl to C3, it is the translation T, by Theorem 3.
COROLLARY 1. The product of any odd number of point reflections is a point reflection.
Proof. Let the given point reflections be Pr P2, • • •, P2n + 1. By Theorem 26 the product PJ*.2 - • • P2n reduces to a translation, which, by Theorem 27, is the product of two point reflections one of which is P2(| + 1. Hence there exists a point reflection P such that
p p . . p — pp p — p
x lx 2 A2n + l~ i2n+li2n + l~
COROLLARY 2. The product of a translation and a point reflection is a point reflection.
COROLLARY 3. TJie set of all point reflections and translations form a group.
THEOREM 28. TJie group of point reflections and translations is a self-conjugate subgroup of the affine group.
Proof. It has been proved, in Theorem 11, that if T is a trans- lation and 2 an aflfine collineation, 2T2"1 is a translation. Precisely similar reasoning shows that if T is a point reflection, 2T2"1 is a point reflection.
COROLLARY. The group G of point reflections and translations is self -conjugate under any subgroup of the ajfine group which contains G.
94 THE AFFINE GROUP IN THE PLANE [CHAP, in
THKOKKM •_".>. ll'itlt respect to any system of nonhomogeneous «><>r(li- natts in which /. is the singular line. Hie equations of a point reflection
have the form
ar = — x -f a,
'
Proof. The point reflection whose center is the origin is of the form
a/ = — x,
y'=-y>
because this transformation evidently leaves (0, 0) and /. pointwise invariant and is of period two. Since any other point reflection is the resultant of this one and a translation, it must be of the form (9).
EXERCISES
1. An ellipse or a hyperbola is transformed into itself by a point reflection whose center is the center of the conic.
2. Let [C2] be a system of conies conjugate under the group of translations to a single conic. Under what circumstances is [C*2] invariant under the group of translations and point reflections?
3. Investigate the subgroups of the group of translations and point reflections.
4. Any odd number of point reflections Pa, P2, • • -, Pn satisfy the condition,
(PJV ••?.)'=!*
5. Let T be the point reflection whose center is the pole of /« with respect to the n-point whose vertices are the centers of n point reflections PI, P2, ••-, Pw. Then*
P,TP,TP,T • • • PnT = 1.
i
46. Extension of the definition of congruence. DEFINITION. Two figures are said to be congruent if they are conjugate under the group of translations and point reflections.
This definition is obviously in agreement with that given in § 39. It will be completed in § 57, Chap. IV. The main significance of the present extension of the definition is that it removes any necessity of distinguishing between ordered and nonordered point pairs in state- ments about congruence.
• Cf . pp. 40, 84, Vol. I. The center of T is the " center of gravity " of the cen- ters of Pj, . . ., p.. ('f. H.Wiener, Berichte der Gesellschaft der Wissenschaften zu Leipzii;. Vol. XI.V (1W«). p. f>08.
§§4«,47] HOMOTHETIC GROUP 95
THEOREM 30. Any ordered point pair AB is congruent to the ordered point pair BA.
Proof. Let O be the mid-point of the ordered point pair AB. The point reflection with 0 as center interchanges A and B.
COROLLARY. If a point reflection transforms an ordered point pair ABto A'B',
Proof. By Theorem 26 the given point reflection is the product of the point reflection in the mid-point of AB and a translation. The point reflection in the mid-point of AB interchanges A and B, and the translation leaves all vectors unchanged.
47. The homothetic group. DEFINITION. A homology whose axis is lm is called a dilation. Dilations and translations are both called homothetic transformations. Two figures conjugate under a homo- thetic transformation are said to be homothetic.
Homothetic figures are also called, in conformity with definitions introduced later, " similar and similarly placed."
The point reflections are evidently special cases of dilations. Since the product of two perspective collineations (§28, Vol. I) having a common axis is a perspective collineation, the set of all homothetic transformations form a group; and by an argument like that used for Theorem 11 this group is self-conjugate under the afline group. Hence we have
THEOREM 31. The set of all homothetic transformations form a group which is a self -conjugate subgroup of the affi,iie group.
Further theorems on the homothetic group are stated in the exercises below.
EXERCISES
1. The ratios of parallel vectors are left invariant by the homothetic group.
2. If two point pairs AB and CD are transformed by a dilation into A'B' and C'jy respectively, . „ „„
~ArB'~ err'
3. If two triangles are homothetic, the lines joining corresponding vertices iii*'<-t in a point or are parallel.
4. The equations of the homothetic group with respect to any nonhomo- geneous coordinate system of which /» is the singular line are
x' = ax + b,
M
THK Al-TINK (iKol I' IN THE PLANE [CHAP, ill
48. Equivalence of ordered point triads. Although the theory of congruence as based on the ^roup of translations and point reflections does not yield metric relations between pairs of points unless they are on parallel lines, yet when applied to point triads it yields a complete theory of the equivalence (in area) of triangles.*
In this section we shall give the definitions and the more important sufficient conditions for equivalence, using methods somewhat analo- gous to those in the first book of Euclid's Elements. Instead of tri- angles, however, we shall work with ordered triads of points. This permits the introduction of algebraic signs of areas, though, as we do not need to refer to the interior and exterior of a triangle, we shall not actually employ the word "area." The triads of points which are referred to are all triads of noncollinear points.
Our definitions have their origin in the intuitional notions : that any triangle ABC is equivalent in area to the triangle BCA, that two triangles are equivalent in area if one can be transformed into the other by a transla- tion or point reflection, and that two triangles which can be obtained by adding equivalent triangles are equivalent.
DEFINITION. If ABC and ACD are two ordered point triads, and B, C, and D are colliuear, and B =£ D (fig. 33), the point triad ABD is called the mm of ABC and ACD and is denoted by ABC + ACD or by ACD + ABC. A A
DEFINITION. An ordered point triad t is said to be equivalent to an ordered point triad t' (in symbols, t^t'} (1) if t can be carried to t' by a point reflection, or (2) if t and t' can be denoted by ABC and
• The idea of building up the theory of areas without the aid of a full theory of congruence is due to E. B.Wilson, Annals of Mathematics, Vol. V (2d series) (11)03), p. 29. His method is quite different from ours, being based on the observation (cf. § 62, below) that an equiaffine collineation is expressible as a product of simple shears. Still another treatment of areas based on the group of translations and employing continuity considerations is outlined by Wilson and Lewis, "The Space- time Manifold of Relativity," Proceedings of the American Academy of Arts and Sciences, Vol. XLVIII (1912). We shall return to the subject in later sections.
§48] EQUIVALENCE OF POINT TRIADS 97
BCA respectively, or (3) if there exists an ordered point triad t such that t *^ t and i ^ t', or (4) if there exist ordered point triads tlf tt, t{, t't such that t^t{, t^t!2 and < = <,+ <„ and t'=t[ + t't. An ordered point triad t is not said to be equivalent to an ordered point triad t' unless it follows, by a finite number of applications of the criteria (1), (2), (3), (4), that t ^ t'.
Since any translation is a product of two point reflections, Criteria (1) and (3) give
THEOREM 32. Two ordered point triads are equivalent if they are conjugate under the group of translations and point rejections.
THEOREM 33. If A, B, and C are noncollinear points, ABC ^ ABC, ABC ^ BCA, ABC^ CAB.
Proof. From (2) of the definition it follows that ABC *= BCA and BCA ^ CAB. Hence, by (3), ABC ^ CAB. But, by (2), CAB ^ ABC. Hence, by (3), ABC *= ABC.
From the last two theorems and from the form of the definition we now have at once
THEOREM 34. If t^ t9, then tz *= tv
THEOREM 35. If A, B, C are any three noncollinear points and 0 the 'mid-point of the pair AB, then AOC^OBC.
Proof. Let C' be the point to which C is changed by the translation shifting A to 0, and let M be the point of intersection of the non- parallel lines BC and OC'. Since COBC1 is a parallelogram, M is the mid-point of the pairs CB and C'O. Thus we have
AOC »e OB C' ^ BC'0 = BC'M+ BMO and OB C = OEM + 0 M C.
But the point reflection with M as center carries OMC into C'MB.
OMC ^ C'MB ^ BC'M, and OBM^BMO,
and hence, by comparison with the equivalences and equations above,
AOC^OBC.
THE AFFINE GROUP IN THE PLANE [CHAP. HI
CD A
o
C..AO.
*
FIG. 35
THEOREM 36. Two ordered point triads ABCl and ABC\, where C =£ C , are equivalent if the line C^ is parallel to the line AB.
Proof. Let Ct be such that B is the mid-point of t\Ca, and let the line C9Ct meet the line AB in 0, which is an ordinary j>oint because Ct is not on the line C^. It follows (§ 40) that 0 is the mid- point of the pair C^C3.
By Theorems 34 and 35, ABC^BAC^C^BA. By definition, CtBA = CSBO + C.OA. By Theorem 3 5,C.BO
V O
•^ Cf)B and Hence C9BA* = CSAB ^ ABC\. Hence
COROLLARY. If a point B' is on a line OB and a point C' on a different line OC, and the lines BC' and B' C are parallel, BOC^B'OC'.
Proof. By hypothesis,
BOC=BOC' + BC'C and C'B'0 = C'B'B+C'BO.
But C'B'B^C'CB^BC'C, by Theorems 36 and 34, and C'BO *e BOC', by Theorem 34. Hence BOC *e C'B'O ^ B'OC' FlG> 36
THEOREM 37. If A, B, and C are any three noncollinear points, and P and Q are any two distinct points, there exists a line r parallel to PQ such that if R is any point of r, ABC ^ PQR.
Proof. Let T be the translation such that T (A) — P, and let T /:) = & and T(C) = C". If B' is not on the line PQ, let R' be the intersection (fig. 37) of the line through C' parallel to PB' with the line through P parallel to QB'. If B' is on the line PQ, let R' l)e the point of intersection with PC' of the line through B1 parallel to QC'.
c'
§§ 48, 49]
EQUIVALENCE OF POINT TRIADS
99
In both cases the lines which intersect in R' are by hypothesis non- parallel, so that R' is always an ordinary point. By Theorem 32, ABC^PB'C'. In case B' is not on PQ, it follows, by Theorem 36, thatP^'C"^ PB'R'^PQR'. In case B' is on PQ, it follows, by the corollary of Theorem 36, that PB'C'^PQR'. By Theorem 36 the line r through R' parallel to PQ is such that for every point R onr,ABC^PQR. FIG. 37
EXERCISES
1. Two ordered point triads ABC and AB'C' are equivalent if the points B, C, B', C' are collinear and Vect (BC) = Vect (B'C').
2. Let 0 be the point of intersection of the asymptotes / and m of a hyperbola, and let L and M be the intersections with / and m respectively of a variable tangent to the hyperbola. Then the ordered point triads OLM are all equivalent.
49. Measure of ordered point triads. The theorems of the last sec- tion state sufficient conditions for the equivalence of ordered point triads. In order to i
obtain necessary con- c, o (
ditions, we shall in- troduce the notion of measure, analogous to the magnitude of a vector.
DEFINITION. Let 0, P, Q be three non- collinear points. The measure of an ordered point triad ABC rela- tive to the ordered triad OPQ as a unit is a number m(ABC) deter- mined as follows : If the line BC is not parallel to OP, let B1 and Cl be the points in which the lines through B and C respectively, parallel
FIG. 38
100 THE AFFINE GROUP IN THE PLANE [CHAP. HI
to OP, meet the line OQ, and let Al be the point in which the line through A, parallel to OP, meets the line-BC. Let AAl denote the magni- tude of the vector .I./, relative to the unit OP (§ 43), and B^ the magnitude of the vector B1L\ relative to the unit OQ. The measure of the ordered triad ABC is* ^ . g ^
and is denoted by m (ABC). If the line BC is parallel to OP, CA is uot parallel to OP, and the measure of ABC is denned to be m (BCA).
If this definition be allowed to apply to any ordered point triad whatever (instead of only to noncollinear triads, cf. § 48), we have m(ABC) — 0 whenever the points A, B, C are collinear.
THEOREM 38. If ABC^A'B'C', then m(ABC) = m(A'B'C').
Proof. Let us examine the four criteria in the definition of equiva- lence in § 48.
(1) In case ABC is carried to A'B'C' by a point reflection, each of the vectors AAl and BlCl is transformed into its negative (Theorem 30, corollary), and hence the product of their magnitudes is unchanged.
(2) According to the second criterion, ABC^BCA. Suppose, first, that neither BC nor CA is parallel to OP, and let A^, Blt C{ have the meaning given them in the definition above. Then
m(ABC)=AA1-BlCl.
Let /?, (fig. 38) be the point in which the line through B, parallel to OP, meets the line CA, and let A2 be the point in which OQ is met by the parallel to OP through A. Then if BB^ and CVAZ represent the magnitudes of the corresponding vectors relative to OP and OQ as units, m (BCA) = BB^ ^
By Theorem 20, g.M.
But since the lines CClt A^A^, BBi are parallel, it follows from § 43 that
A^C _ A& BC~ B^i
Hence ^l» = d£,
or m(ABC) =AAl - B^^
• The factor ). in lacking in this expression, because we are taking a triangle rather than a parallelogram as the unit.
MEASURE OF POINT TRIADS
101
In case BC is parallel to OP, the last clause of the definition states m(ABC) = m(BCA).
In case CA is parallel to OP, AB and BCare not parallel to OP, and hence the argument above shows that
But, by definition, Hence
m(CAB)=m(ABC). m(BCA) = m(CAB). m(ABC)=m(BCA).
(3) Corresponding to the fact that if tl ^ t2 and t2 ^ t3, then ^ ^ ts, we have that, since m (t) is a uniquely defined number, if m (£j) = m (t2), and m (<2) = m (£3), then m (^) = m (2g).
(4) Let ^, C, D be three col- linear points and A any point not on the line BC (fig. 39). In case the line BC is not parallel to OP, let Al be the point in which the line through A, parallel to OP, meets BC, and let Blt Clt Dl be the points in which the lines through B, C, D respectively, parallel to OP, meet OQ. Then
m(ABD)
= AA. -B.D.
= m(ABC) + m(ACD).
In case the line BC is parallel to OP, let S be the point in which BC meets OQ, and A^ be the point in which the line through ^parallel to OP, meets OQ. Then
m (ABD} = m (BDA) = BD • SA^=BC • SA^+ CD • SA, ( CD A) = m(ABC} + m (A CD).
Thus, in every case, if
= t
wi(<2)= m(ta).
Comparing the results proved in these four cases with the definition of equivalence, we have at once that whenever ^— ^2, m(t^— m(ta).
I<IL' THK A1TINK <iK»>l 1' IN THE PLANK L< HAI-. in
Tn KI >KKM 39. If B, C, and Dare collinear points, and the point A is not on the line BCt m(ABC) _ S£
m(ABD)~ BD
Proof. In case the line BC is not parallel to OP, let Alt B^ Cl have the meaning given them in the definition of measure, and let Dl be the point in which the line through D, parallel to OP, meets OQ (fig. 39).
m(ABC) = AAl • B& = B& m (ABD) ~ AAV • B^ ~ BVDV '
But, by § 43,
In case BC is parallel to OP, let A^ be the point in which the line through A, parallel to OP, meets OQ, and S the point in which BC meets OQ. Then
m(ABC) m(BCA) _ BC • SAt BC m (ABD) , , m (BDA) ~ BD • SAt ~ BD '
COROLLARY 1. If B, C,D,E are points no two of which are collinear with a point A,
B (AB. AC, AD, AE) =
m(ABE) m(ACE)
(COROLLARY 2. If B, C, D are points no two of which are collinear with a point A, and if P* is the point at infinity of the line CD (the latter not being parallel to AB),
THEOREM 40. If m(ABC) = m(A'B'C')* 0, then ABC^A'B'C'.
Proof. By Theorem 37 there exists a point C" on the line A'C' such that ABC^A'B'C". Hence A'B'C' ^A'B'C", and by the last theorem, C' = C".
In consequence of the last two theorems the unit point triad may be replaced by any equivalent triad without changing the measure of any triad.
THEOREM 41. If ABC '« ABC', andC* C', the line CC1 is parallel to the line AB.
§49] MKASUJK OK POINT TK1ADS 103
Proof. The unit triad OPQ may be chosen so that OP is parallel to AB. Then if Cl is the point in which the line through C, parallel to OP, meets OQ, and Bl the point in which AB meets OQ,
If < ',' is the point in which the line through C', parallel to OP, meets OQ,
By Theorem 38, m(ABC)=m(ABC'), and hence C\= C[. Hence the line CC' is parallel to AB.
THEOREM 42. If ABC^AB'C', and B' is on the line AB, and C' on the line AC, then the line BC' is parallel to the line B'C.
Proof. By the corollary of Theorem 36, if C" is a point of AC' such that BC" is parallel to B'C, then
ABC^AB'C".
By Theorem 41 the only points C such that ABC^AB'C are on the line through C", parallel to AB'. Hence C'=C".
It is notable that although the sufficient conditions for equivalence given in § 48 are all proved on the basis of Assumptions A, K, H0, the discussion of the ratios of vectors, and hence all the necessary conditions for equivalence, involve Assumption P in their proofs. This is essential,* as we can show by proving that Assumption P is a logical consequence of these theorems, together with the previous theorems on equivalence. As was pointed out in § 3, Assump- tion P is a logical consequence of the theorem of Pappus, Theorem 21, § 36, Vol. I. When one of the lines of the configuration is taken as /«, this theorem assumes the form :
If a simple hexagon AB'CA'BC is such that A, B, C are on one line and A', B', C' on another line, and ifAB' is parallel to A'B and BC parallel to B'C, then C A' is parallel to C A.
lu case the lines containing ABC and
A' B'C', respectively, are parallel, this can be proved from the Desargues tlit-oreni on jK-rspective triangles; so that we are interested only in the
» The rOle of Assumption P(or rather of the equivalent theorem of Pappus) in the theory of areas was first determined in a definite way by D. Hilbert, Grundlagen «1»T (Jeometrie, Chap. IV.
104 1111 All INK (IROUP IX THE PLANE LCHAP. in
when AB and A I: intcrs.M-t in a point 0. By Theorem 36, since Ml' is panilM tu .1 /.'. ".I 1 '^olil: ; an<l sinr.- !',(" is parallel to B'C, UJili' s= lly tin- lirtiiiition (:•) of .-.iiiivaleuce it follows that OAA'^O< < . But by Theorrm »-' this implies that. 1C" is parallel toyl'C.
This is perhaps the simplest way of proving the fundamental theorem of projective geometry if it be desired to base projective geometry upon eleiiini- tary Kurlulraii geometry (cf. Ex. 3, § 54).
The notion of measure can be extended to any ordered set of n points, i.e. (cf. § 14, VoL I) to any simple n-point. The details of this discussion are left to the reader. An outline is furnished by the problems below. The principal references are to A. F. Mobius, Der barycentrische Calcul, §§ 1, 17, 18, 165; Werke, Vol. I, pp. 23, 39, 200; VoUI,p.485. See also the Encyclopadieder Math. Wiss., Ill AB9,§ 12. It is to be borne in mind in using these references that our hypotheses are narrower than those used by the previous writers.
EXERCISES
1. For any three points A, B, C,
m(ABC) + m(ACB) = 0.
2. For any four points 0, A, B, C,
m(ABC) = m(OAB) + m(OBC) + in (OCA).
3. For any n points Av A2,- • •, An the number
I8) + • •• + m(OAn_lAH) +
is the same for all choices of the point 0. We define it to be the measure of the simple n-point A j/!2 • • • An and denote it by m^A^A^ • • • A^).
4. »«(.-! lAt • • • An_lAH) = m(A2As • • • AnAJ.
5. m(AlAz-"Aa)+m(AlAnAH + l"-An + k) = m(A1A2-"AH + k-).
6. Derive a formula for m(A^A2' • • An) analogous to the definition of m(ABC) in terms of vectors collinear with two arbitrary vectors OP and OQ.
7. Prove the converse propositions to those stated in the exercises in § 48.
8. If ABCD&ud A'B'C'D' are two parallelograms whose sides are resj>ec- lively parallel, m(ABCD) _ AB_ BC_
m (A'B'C'l?) A'B' ' B'C" '
9. The variable parallelogram two of whose sides are the asymptotes of a hyperbola and one vertex of which is on the hyperbola has a constant measure.
10. If a variable pair of conjugate diameters nreets a conic in point pairs A A', BB", the parallelogram whose sides are the tangents at A, A', It, B' has a constant measure. The parallelogram ABA'B' also has a constant measure.
5§49,so] THE EQUIAFFINE GROUP 105
50. The equiaffine group. THKOKKM 43. If two equivalent ordered point triail* t and t are transformed by an affine collineation into t( and t'v then t[ ^ t'r
Proof. It is necessary merely to verify that the relation used in each of the criteria (1), • • •, (4) in the definition of equivalence (§ 48) is unaffected by an affine collineation. For Criterion (1) this reduces to Theorem 28. For Criteria (2), (3), (4) it is a consequence of the fact that an affine collineation transforms ordered triads into ordered triads and collinear points into collinear points.
THEOREM 44. If an affine collineation transforms one ordered point triad into an equivalent point triad, it transforms every ordered point triad into an equivalent point triad.
Proof. It follows from Theorem 43 that if ABC is transformed by a given collineation into an equivalent ordered point triad A'B'C', then every point triad equivalent to A B C is transformed into a point triad equivalent to A'B'Cf and thus into one equivalent to ABC. By Theorem 37 any ordered point triad whatever is equivalent to some point triad ADC, where D is on the line AB. Hence the present theorem will be proved if we can show that ADC is transformed into an equivalent point triad.
Denote the point to which D is transformed by the given collinea- tion by D'. By Theorem 39,
m(ADC) _AD , m(A'D'C') = A'D' m(ABC) ~ AB* m(A'B'C1) ~ A'B' '
By §43,
where J^ is the point at infinity of the line AB. But since the given collineation is affine, R, is transformed to the point at infinity P^ of the line A'B', and
Since m(ABC) = m(A'B'C'), it follows that m(ADC) = in(A'D'C'). Hence
DEFINITION. Any affine collineation which transforms an ordered point triad into an equivalent point triad is said to be equiajffine.
106
THE AK1 INK (JROUP IN THE PLANE [CHAP, in
THKOREM 45. The equiajfine collineations form a self -conjugate subgroup of //»• ojfinf </r<>i>j>.
Proof. By the last theorem an equiaffine colliiieation transforms every ordered point triad into an equivalent point triad. Hence, by Condition (3) in the definition of equivalence, the product of two equiaffine colliueations is equiaffine. By Theorem 43, 2T2"1 is equi- affine whenever T is equiaffine and 2 affine.
THEOREM 46. Let A, B, A', B' be points such that A^B and A' =£ B' ; let a be a line on A but not on B, and let a' be a line on A1 but not on B1. There is one and only one equiaffine collineation transforming . I to A', B to B', and a to a'.
Proof. Let C be any point distinct from A on a. By Theorem 37, there is a point C' on the line a' such that
ABC^A'B'C'.
By Theorem 1 there is one and only one affine transformation carrying A, B, C to A', B', C' respectively, and by definition this transformation is equiaffine. By Theorem 41, C' is the only point on a' such that ABC^A'B'C'. Hence (Theorem 44) there is only one equiaffine transformation carrying A, B, a into A', B', a' respectively.
EXERCISE
Any affine collineation leaves invariant the ratio of the measures of any two point triads.
*51. Algebraic formula for measure. Barycentric coordinates. Consider a nonhomogeneous coordinate system in which L is the singular line. Let the unit of measure for ordered triads be OPQ, where 0 = (0, 0), P = (1, 0), Q = (0, 1). Let A = (alf «2), B = (^ &2), C = (c,, c2) ; the line through A, parallel to OP, consists of the points (ctj+X, a2), where X is arbitrary, and the line BC has the equation (§ 64, VoL I),
y
= o.
In case the line BC is not parallel to OP, and therefore bz =£ c2, the point Al which appears in the definition of measure (§ 49) is (at+X, a,),
where X satisfies
X 0 0
0.
§51]
Hence
FORMULA FOR ME AST UK AA.-^-
107
The points BI and Cl of the definition of measure are (0, &2) and (0, ct), respectively, so that rt r - ;,
Hence
(10)
m(ABC) =
\ \ -
C, C. 1
That the same result holds good in case BC is parallel to OP is readily verified.
Now if A, B, C are transformed to A', B', C' respectively by a transformation f
1 l' A = l '
of the affine group,
(11)
«2 ft
(12)
ci C2 1 Hence we have
THEOREM 47. A transformation (11) of the a/ine group is equiajfint if and only if* a
1 l =1.
Let A = (alt a2), B=(blf b2), C = (ct, cg) be the vertices of any triangle, and P = (x, y) any point. In the homogeneous coordinates for which xl/x0=x, xjx^=y, these points may be written A = (1, alt a^, etc. Hence by the result established in § 27 for the three-dimensional case, the numbers proportional to
I x y
1 x y
' £» =
i 2 I x y
may be regarded as homogeneous coordinates of P in a system for which ABC is the triangle of reference.
* By comparison with § 30 this condition yields the result that, in an ordered space, the equiaffine collineations are all direct.
ln.s THK AITINK CKoll' IN TIIK I'LANK [CHAP. Ill
This is a particular one of the homogeneous coordinate systems for \\liii-h A lie is the triangle of reference, and of course corresponds to a particular choice of the point (1, 1, 1). Other particular systems may be obtained by replacing (1, alt aj by (k, kalt kaj and like changes. The coordinates written down, however, have (in view of (10)) the remarkable property that
Also, in view of Ex. 2, § 49, they satisfy the condition
for all ordinary points P. If ABC be taken as the unit of measure, this condition assumes the form
Since all ordinary points satisfy this condition, the equation
M-li+fc-o,
which can always be satisfied by properly chosen homogeneous coordi- nates, must represent L. Therefore the point (^, ^, £), which is polar to /. relatively to the triangle ABC, must be the point of intersection of the medians of this triangle.
DEFINITION. Given a homogeneous coordinate system with respect to which the line at infinity has the equation
*0+*l+*2=°>
the three numbers x^ xlt #2, which are homogeneous coordinates of an ordinary point P and satisfy the condition
30+^+^=1, are called the/ barycentric coordinates of P, relative to the triangle
*o=°' *i=°> *a=°-
EXERCISES
1. Defining the barycentric coordinates of a point P, relative to a triangle ABC,&» ^m(ABP) . =m(BCP) . ^m(CAP)
*° m(ABC)' ** m(ABC)' ** ~ m(ABC)' prove that a line is represented by a linear equation.
2. If A, B, C, D are four fixed points of a conic, and P a variable point, the ™tio m(ABP)-m(CDP) .
m(ADP)-m(CBP) i« constant (cf. Cor. 1, Theorem 30).
§§51,52]
LINE REFLECTIONS
109
3. Show that the equation of a conic through five |Kiintfc A, B, C, D, E may be written in the form
( . I 1>E) (BCE) (ABX) (CDX) - (A BE) (CDE) (ADX) (BCX) = 0, where (ADE) stands for
and the other parenthetical triads have analogous meanings.
*52. Line reflections. DEFINITION. A homology of period two whose center is on /„ is called a line re/lection ; if its center is L and its axis I, we shall denote the line reflection by {Ll}.
This definition could also be expressed by saying that a line reflec- tion is a transformation having an axis such that (1) if P' be the transform of a point P and P^P', the mid-point of the pairPP' is on the axis of the reflection ; and (2) if Pl and P[ are any other pair of homologous points, the line PjP{ is parallel to PP'.
THEOREM 48. A product of tico line reflections is an equiajfflne collineation.
Proof. Let the given line reflections be {£/,} and (A/J. Let I be any line meeting both ^ and /„, and let L be any point at infinity not on I. Then {L i} . {L i} = {L t} . {Li} • {Ll} -{LI}.
Let A be the point of intersection of I and llt-B any other point of llt C any other point of I, Cl the point to which C is transformed by {LJ^s, and 0 the point in which the line CCl meets lf Since 0 is the
mid-point of CC,, Theorem 35
o»
gives in case A =£ 0 3=- B,
Since CA 0 + COB = CAB, and C^BO + C^OA = CJiA, it follows that
Fl.:. 41
In case A=O or 0=B the same result follows directly from Theorem 35. In like manner, if Bl be the point to which B is transformed by
\ / > /" 1 4 I> ^- /~* T> 4
LAif — ltfA.
Hence
CVBA
•CBVA.
110 mi: All INK GROUP IN THE PLANE [CHAP, in
The product {///} • {L^} transforms C^A to CB^A and is therefore an equiattme collineation. In like manner, {Lj,^ • {LI} is also equiathnu Hence the product {LJJ • {L{$ ™ equiaffine.
THEOREM 49. An equiajfine collineation is a product of two line reflections.
Proof. Let F be any equiaffine collineatiou. If there be any point which is not on an invariant line of F, let A t be such a point. Let Af Af AI be defined by the conditions
By the hypothesis on Al the points A0, A^ A.2 are noncoUinear, and by the hypothesis that F is equiaffiue
Hence, by Theorem 41, the line AQAa is parallel to A^A^ or else A0= Af
Let Ml be the mid-point of the pair AQAZ, and M2 of the pair A^A^.
Let .Z/j be the point at infinity of the line AQA2, Lz of the line A^A^ and
K of the line A^A%. Since A0AS is parallel to AlAa, it follows that
Ls
A^AyL^A^A^K, and hence, by the definition of mid-point, that Mlf M2,
and Lt are collinear. Since A0, A2, and the point at infinity of the line AQAt are transformed by F to Alf A3, and the point at infinity of the lmeAlA,tT(Ml)=Ma.
Let /j be the line A^M^ and 13 the line joining the mid-point of A^At to the mid-point of M^Mf By the above,
Hence {LJJ • {L
But since T(A0AlMl) = A1AaM2, it foUows, by Theorem 1, that
§52] LINE REFLECTIONS 111
In case there is no point not on an invariant line of F, the invariant lines all meet in a point O. For the point of intersection of any two of them is invariant, and any three nouconcurrent ordinary lines have at least two ordinary points in common. Thus we should be led to a contradiction with Theorem 46 if the invariant lines were not concurrent.
Let A^ be a point which is not invariant, and let AZ = T (AJ. Also let Bl be another point which is not invariant and not on the line Al AZ, and let F (BJ = Bn. The lines A^A^ and BlBz neces- sarily meet in 0.
If 0 is ordinary, then since any line through it is invariant, all points of /_, are invariant, and hence A1Bl is parallel to AzBf Since F is equiaffine,
Hence, by Theorem 42, A^BZ and A2Bl are parallel, and A1B1AZBZ is a parallelogram. Hence 0 is the mid-point of A^A^ and B^B^ and F is a point reflection.
Let a be the line A^A^ and A the point at infinity of a, and let b be the line BJ$n and B the point at infinity of b. The product {Ab} • {Ba} transforms Alt
BI} 0 into Az, Bz, 0 respec- Flo 43
tively, and hence is F.
If 0 is an ideal point, let I be the line A^B^, and let m be the line joining the mid-points of A^AZ and B^BZ. Then {Om} - {01} transforms 0, Alf Bl into 0, A2, Bz respectively, and hence, by Theorem 46, is F.
COROLLARY 1. An equiaffine collineation F such that A, T(A) and F2(^4) are collinear for all choices of A is either a point reflection or a translation or an elation whose center is at infinity and whose aj>ns is an ordinary line.
Proof. In the argument al>ove it was proved that if the point 0 is ordinary, F is a point reflection; and that if O is ideal, F = {Om} • {01}. If m and I are parallel, F is evidently a translation ; and if m and I are not parallel, it is an elation with 0 as center and the line joining 0 to the point Im as axis.
112 TIN: AITIM-: GROUP IN THE PLANK [< HAP. m
DKKIM rii'N. An rlutiou whose center is at infinity and whose axis is an ordinary line is called a simple shear.
COROLLARY 2. 7/T = {LJJ - {£/,}, then for every line I concurrent with I and I which is not a double line of F there exist points L and M and a line m such tint I
F = {Mm} • {LI}. Thtre also exist a point M' and a line m' such that
F = {LI} - {M'm'}. If I be taken as variable,
Proof. The first conclusion follows from the arbitrariness in the choice of A l in the proof of the theorem above. The second conclusion follows from the first, combined with the fact that
The projectivities follow from the constructions given in the proof of theorem for AQ, A2, Mlf etc.
COROLLARY 3. If F = {L2l^ - {LJJ, then for every point L of L which is not a double point of F, there exists a point M of /„ and two lines I and m concurrent with l^ and lz such that
T = {Mm} • {LI}.
There also exist a point M' and a line m' such that
{M'm'}.
THEOREM 50. The set of all ajjjine collineations which are products of line re/lections form a group. Every transformation of this group is either an equiajfine transformation or the product of an equiajfine transformation by a line reflection.
Proof. By Theorems 48 and 49 the product of an even number of line reflections is equiaffine and reduces to a product of two line reflections. Hence the product of an odd number of line reflections reduces to a product of three line reflections. The statements above follow in an obvious way from this.
§.v_>] LINK REFLECTIONS 113
EXERCISES
1. Let the points at infinity of /,, /.,, / rcs]><>etively in Theorem 49, Cor. 2, be denoted l>y L{, /-..!, //. If tin- jK)ints LVL{,LVL^ are distinct, the pairs /-,/.]• A..A.J. LL' are in involution.
2. In case L^ is on /2 and Z.2 is not on lv {Z2/,} • {L^} = T is a collineation of Type // (cf. § 40, Vol. I), parabolic on /. and of period two on the line joining Ll to the point of intersection of /x and /2. If / be any line, except lv through the point /,/2, P the point in which / meets /«, and L the harmonic conjugate of Ll with respect to P and T(P),
If ^f be the harmonic conjugate of L^ with respect to P and T~ '(/'),
T=;/v:-;j//2}.
3. The product {L2/2} • {LJJ is a point reflection if and only if Zt is on /2 and Z2 on lr A point reflection with O as center is the product of any two line reflections {A1/1} and ;A2/2} for which /: is on 0, /2 on O, L^ on lz, and Z2 on lr
4. The product {Z2/2} • {L^l^} is a translation if and only if Zt = Z2 and /t is parallel to /2. The ideal point L1 is the center of the translation. If T is any translation, T<* its center, Pj any ordinary point, P = T(/>1), P2 the mid- point of the pair PPV and pl and p2 two parallel lines through Pl and Ps respectively, T = { T., p2} . { T*,Pl}.
5. The product {Z,2/2} • {Lj/,} is a simple shear if Ll ^ L2 and /t = /a, or if Ll = Z2 and /j intersects 12 in an ordinary point, but not in any other case.
6. Let 2 be a simple shear whose axis is / and whose center is L. Let Pj be any point of /«, P = 2(/\), and P2 the harmonic conjugate of L with respect to P and P2. Then 2 = {P2l} • {Pt/}. If ;>, be any line meeting / in an ordinary point, p = ^(p^, and j92 the harmonic conjugate of / with respect to p and pv
2 = {LpJ • {LPl}.
7. Let PP1P2P8P4 be a simple pentagon. Let C\, C%, C8, C4, Cs be the mid-points of the pairs PPj, PjP-j, P2P8, ^8^*4' PJ* respectively. If the line PPt is parallel to P8P4, and PP4 is parallel to P^-j, the three lines CtC4, C3C6, PC8 are concurrent or parallel. Discuss the degenerate cases.
8. Every eqiiiaflfine transformation is either the identity or a point reflec- tion or an elation whose center is at infinity (i.e. a translation or a simple shear) or expressible as a product of two elations whose centers are at infinity.
9. Prove Cors. 2 and 3 of Theorem 49 directly, without using the theory of equivalence.
10. A necessary and sufficient condition that a planar collineation be the product of two harmonic homologies is that it transform ordered point triads into equivalent point triads relative to a fixed line of the collineation regarded as /. (E. B. Wilson, Annals of Mathematics, Vol. V, 2d series (1903), p. 45).
114 TIIK .UTl\E GROUP IX THE PLANE [CHAP, in
11. Let us denote an involution whose double points are L and M l»y /. M . If /I = -;/,,J/1( and 72 = {Z.2J/2} are two distinct involutions on tin-
nine line, then fur t-vcry point La of this line, L3 not being a double jxiint of /, • 72, there exists a unique point Ms and involution {LtM4} such that if we denote {LtMt} by 7, and {LtMt} by 74,
/3/j/i = 74, and 727j = 73/4.
The pairs 7M.Uj, 7.2.172, 7>,J78, 7,4.174 are all pairs of the same involution, unless the pairs 7,,.1/j and LtMt have a point in common, in which case all four pairs have this point in common.
12. The projectivities on a line which are expressible in the form /.,. I/,; •;/,.,. U,; form a group.
The last two exercises connect with the following algebraic considerations. An involution in a net of rationality is always of the form (§ 54, Vol. I)
where a, b, c, d are rational. The double points are the roots of
ex2 - 2 ax - I = 0,
and both will be rational if k is rational in
«2 + be = P.
Now any projectivity is the product of two involutions, a double point of one of which may be chosen arbitrarily. The projectivity may therefore be written
, ax + b a -- \- b „ _ ex— a _ (aa? + b'c) x 4- (ofb — ab')
ax 4- b (ac' — a'c) x 4- (be' 4- ««') '
e1 -- a ex — a
and so has the determinant
aa'bc' 4- a2a'2 4- bb'cc' 4- b'caa' — (aa'bc' — a?b'c' — a'*bc 4- aa'fi'c) = a2 (a'2 4- b'c) 4- be (b'c' 4- a'2) = H*P*,
where fc'2= a'* + b'c'. Hence (1) the product of two involutions whose double points have rational coordinates is a projectivity whose determinant is a per- fect square; and (2) if the determinant of a projectivity is a perfect square, and one of two involutions of which it is a product has rational double points, then the other has rational double points. Hence there is a subgroup of the group of colli neat ions of a linear net of rationality generated by the involu- tions with rational double points. This is the group of transformations whose determinants are perfect squares.
LINE REFLECTIONS
115
*53. Algebraic formulas for line reflections. Let us employ the nonhomogeneous coordinates for which /„ is the singular line and the corresponding homogeneous coordinates for which
X. X9
J — /p » — «»
«£n Xn
The line /. now has the equation XQ = 0, and the equations (1) of the atfine group become
x =
A =
=£0.
On the line L this effects the transformation
According to § 54, Vol. I, this is an involution if and only if al = — bf Thus al = — b2 is a necessary condition that (13) represent a line reflection.
The ordinary double points of (13) are given by the following equations, in which we have put a = av = — bz.
tfi _ I i nf* *-- « 11 ' /» — 1 1 \ / /
If (13) is to be a line reflection, it must have a line of fixed points. Hence the two equations (14) must represent a single ordinary line, which requires
(15) 0 =
a-1 a, —
a — 1 c.
The first of these conditions is equivalent to A = — 1.
Since the coefficients of x and y in (14) cannot all vanish, the conditions (15) are also sufficient that (14) represent a single ordinary line. Hence
THEOREM 51. A transformation of the form
x' = ax + y' = azx —ay
is a line reflection if and only if
(16)
1 c.
= 0.
im; AITIM: <;K<>I i- IN TIIK PLANE [CHAP. m
From this it follows that a product of two line reflections is such that A = 1, and a product of three line reflections is such that A = — 1. By Theorems 47 and 49 any transformation for which A = 1 is a product of two line reflections. Any transformation T for which A= — 1, when multiplied by a line reflection A yields a transforma- tion 2 for which A = 1, i.e. an equiaffine transformation. From T A = 2 follows T = 2A. Hence T is a product of three line reflections. Tims we have (cf. Theorem 47)
THEOREM 52. The group of affine transformations which are prod- ucts of line reflections has the equations
a. b
i i
= 1.
EXERCISES
1. The set of all affine transformations which are products of equiaffine transformations by dilations form a group which is a self-conjugate subgroup of the affine group. Its equations are
x = QjZ + b^y T Cj, y' = a2x + bzy + c2,
where k is any number in the geometric number system.
2. The set of all affine transformations which are products of line reflections and dilations form a group which is self-conjugate under the affine group. Its equations are . , ,
where k is any number in the geometric number system.
54. Subgroups of the affine group. We give below a list of the principal subgroups of the affine group which we have considered in this chapter and in § 30 of Chap. II. These are all self-conjugate subgroups. We also include the groups which will be considered in the next chapter in connection with the Euclidean geometry.
The groups are all described by means of the conditions which must be imposed on the coefficients of the equations of the affine group to reduce it to each of the other groups. In some spaces, i.e. when the variables and coefficients are in certain number systems, these groups are not all distinct. However, they are all distinct in case the variables and coefficients are ordinary rational numbers.
554] LIST OF SUBGROUPS 117
With respect to a system of nonhomogeneous coordinates of which L is the singular line, the equations of the attine group are
v' = a x where A= ai
The principal subgroups connected with the affine geometry are :
(2) A>0;
the transformations satisfying this condition are direct (§ 30).
(3) A = *',
where k is in the geometric number system (§ 53, Ex. 1).
(4) A = ±*2,
where k is in the geometric number system (§ 53, Ex. 2).
(5) A2 = l;
these are products of two or of three line reflections (Theorem 52).
(6) A = l, the equiaffine group (§ 51).
/^7\ f- I C\ ff __ t
\ / 2 1 ' 1 2'
the homothetic group (§ 47).
\ / 2 1 ' 1 t* 1 ""
the group of translations and point reflections (§ 45).
(9) «», = &!= 0, ^=^=1,
the group of translations (§ 38).
The principal groups connected with the Euclidean geometry are :
i •« f\\ 2 I 2 Z. 2 I T~ 2 i f\ ft 1 i ~ T f\
\HJ) (*i ~T W2 i"lt/2'^V> ll' 22 '
the Euclidean group (§§ 55 and 62). Its transformations are called similarity transformations.
(11) o » + al = ll + 622 =,fc 0, a161 + aj>, = 0, A > 0, the direct similarity transformations.
where k is in the geometric number system, where k is in the geometric number system.
118
THE All INK <JROUP IN THE PLANE [CHAP, ill
(14) a« + o*«l, «1 = ±&2, «, = * the group of displacements and symmetries (§ 62).
(15) a« + a«s the group of displacements.
The relations among these groups may be indicated by the follow- ing diagram, in which we have included only those groups which are distinct in case of the real geometry. A dotted line indicates that the lower of the two groups joined is a subgroup of the upper, and a solid line that it is a self- conjugate subgroup.
The fundamental importance of the group of translations is indicated by the fact that it is a self-conjugate subgroup of each of the other groups.
10) Euclidean
Homothetic
Translation* FIG. 44
EXERCISES
1. Supposing the number of points on a line to be p + 1, what is the number of transformations in each of the groups listed above?
2. Supposing the geometric number system to be («) the ordinary real, or (6) the ordinary complex number system, how many parameters are there in the equations for each of the groups listed above?
3. Prove that the plane affine geometry as a separate science could be based on the following assumptions with regard to undefined elements, called ]>i!'mt>, and undefined classes of points, called lines:
I. Two points are contained in one and only one line.
I 1. For any line / and any point /*, not on /, tin-re is one and only one line containing P and not containing any point of /.
I II. Every line contains at least two points.
IV. There exist at least three noncollint-ar jx.ijits.
V. The special case of the Pappus theorem given in the fine print in § 49; or Theorem 41.
CHAPTER IV
EUCLIDEAN PLANE GEOMETRY
55. Geometries of the Euclidean type. We come now to the extension of the definition of congruence which was promised in §§ 39 and 46. This requires the consideration of groups which are not self-conjugate under the affine group. Not being self-conjugate, these groups are not determined uniquely by the affine group, and hence our definitions will contain a further arbitrary element.
DEFINITION. Let I be an arbitrary but fixed involution on /.. This involution shall be called the absolute or orthogonal involution. The group of all projective collineations leaving I invariant shall be called a parabolic * metric group. The transformations of the group shall be called similarity transformations. Two figures conjugate under the group shall be said to be similar. The geometry corresponding to the group shall be called the parabolic metric geometry.
The absolute involution is supposed to be fixed throughout the rest of the discussion, but of course there are as many parabolic metric groups as there are choices of I. We nevertheless speak of the para- bolic metric group in order to emphasize the fact that we are fixing attention on one group.
In case the plane in which we are working is' a real plane and the absolute involution is without double points, the parabolic metric geometry is the Euclidean geometry. It is for this reason that we refer to the parabolic metric geometries as geometries of the Euclidean type.
The investigations in the following sections are arranged in order of increasing specialization. First we consider a perfectly general involution, I, in a projective plane satisfying A, E, P, Ho. Then we consider a particular type of involution in an ordered plane, and finally limit the plane to be the real plane.
» The reason for the term " parabolic " in this connection is explained in a later chapter, where the elliptic and hyperbolic metric groups are defined.
119
120 EUCLIDEAN PLANK <;i:<>MKTRY [CHAI-. iv
Wlu-n the plane and the involution are fully specialized, it is a theorem (§ 70) that the real plane is contained in a complex plane in which the absolute involution has double points. Thus the theorems on the general type of involution (where the possible existence of double points is taken into account) come to have a new application.
56. Orthogonal lines. DEFINITION. Two lines are said to be orthog- onal or perpendicular to each other if and only if they meet /„ in conjugate points of the absolute involution.
The following consequences of this definition are obvious :
THEOREM 1. The pairs of perpendicular lines through any point, 0, are the pairs of an involution. Through any point there is one and but one line perpendicular to a given line. A line perpendicular to one of two parallel lines is perpendicular to the other. Two lines perpen- dicular to the same line are parallel.
DEFINITION. In case the absolute involution I has two double points, /t and /2, they are called the circular points. Any line through /j or /2 is called an isotropic line or a minimal line.
Any isotropic line has the property of being perpendicular to itself. The circular points are so called because all ordinary points of any circle (cf. § 60) are on a conic through II and /2. The ordinary points of the conic section referred to in the following lemma will later be proved to be on a circle.
DEFINITION. A homology of period two whose center L is on lx, and whose axis / meets /„ in the point conjugate to the center with regard to the absolute involution, is called an orthogonal line reflection, and is denoted by {LI}.
Since the center of a homology is not a point of the axis, the center cannot be a double point of the orthogonal involution, nor can the axis pass through such a point. An orthogonal line reflection is of course a special case of a line reflection as defined in § 52.
I.F.MMA. Let O and Pl be two points not collinear with either double point of the absolute involution. There is one and only one conic, C*, having O as center, passing through P}, and having the pairs of the absolute involution as pairs of conjugate points.
Proof. Let Pt be the harmonic conjugate of Pt with respect to 0 and the point at infinity, P*,, of the line OPr Any conic containing Pl
§66]
121
and having 0 as center must contain I\t by the definition of center. Let A' be a variable point of /., and Y the conjugate of X in the absolute involution. Any of the triangles OXY must be self-polar to any conic satisfying the required conditions. But if P is the point of intersection of the lines PtX and P.t Y, and Q the point of intersection of P! AT and 0r,
Y \
FIG. 45
and hence the points Pv and P are harmonically conjugate with re- spect to X and Q. Hence P must be on any conic through P^ with regard to which X is the pole of QY. Hence P must be on any conic satis- fying the hypothe- ses of the lemma.
Since P^[X} -^P.2[Y], the points P, together with P^ and P2, consti- tute a unique conic (§ 41, Vol. I); and this conic, by its construction, satisfies the condition required by the lemma.
COROLLARY. In case the absolute involution has double points the conic C* passes through them.
THEOREM 2. An orthogonal line reflection leaves the absolute in- volution invariant.
Proof. If I is the axis of an orthogonal line reflection and L its center, let 0 be any point on I and Pl any point not on I. The conic Ca(cf. Lemma), which contains^, has 0 as center, and has the absolute involution as an involution of conjugate points, must have L and / as pole and polar. Hence, by the definition of pole and polar (§ 44, Vol. I) (7s is transformed into itself by the harmonic homology having L and I as center and axis. Hence the absolute involution is transformed into itself by the orthogonal line reflection {LI}.
1-2-2
KIVLIDKAN IM.ANK <;K<>MKTKV
[CHAP. IV
THF.ORKM 3. The product of two ortliogonal line reflection* >/7/o.sr axes are parallel is a translation parallel to any line perpendicular to the axes.
Proof. Let the given line reflections be {Lfy and {A/J. Their axes meet in a point L' of /., and L^ and Lz must be conjugate to L' with respect to the absolute involution. Hence Ll = LZ. The product there- fore leaves all points on /, invariant and also all lines through L^ Hence it is a translation parallel to any line through L^.
THEOREM 4. A translation, T, whose center is not a double point of the absolute involution, is a product of two orthogonal line reflection*, {Ll }, {fy }, where L is the center of the translation. If 0 is an arbi- trary ordinary point and P the mid-point of the pair 0 and T(0), / may be chosen as OL' and lz as PL', where L' is the conjugate of L with respect to the absolute involution. Or l^ may be chosen as /'// and I as the line joining T (0) to L'. A translation whose center is a double point of the absolute involu- tion is a product of four orthogonal line reflections.
Proof. If ^ = OL' and 12 = PL', the reflection {Ll^ leaves O inva- riant and {Ll^f carries 0 to Hence the translation {Llz} carries O to T (O), and, by Theorem 3, Chap. Ill, is identical with T.
If /t = PL' and l^ = QL', where
Q = T (0), the reflection {Ll^ carries 0 to Q and {Ll£ leaves Q invariant. Hence, as before, {L12} • {Ll^ = T.
A translation whose center is a double point of the absolute involu- tion can be expressed as a product of two translations with arbitrary points of /, as centers (Theorem 8, Chap. Ill), and hence is expressible as a product of four orthogonal line reflections.
DEFINITION. If the axes of two orthogonal line reflections intersect in an ordinary point, 0, the product is called a rotation about 0, and the point O is called its center.
THEOREM 5. A rotation which is the product of two orthogonal line reflections whose axes are orthogonal is a point reflection.
Fir.. 40
7] 1>IS1'1.A< K.MKNTS !:>:}
I'roof. Let the two line reflections be {/-,/,} and {Lflt} and letO be the point of intersection of /t and /2. Since l^ and /s are orthogonal, Ll is on /a and L3 on /t. The product {LJlz} • {£/,} therefore leaves 0 and every point of /„ invariant. Moreover, it is of period two on the axis of either of the line reflections. Hence it is a homology of period two with 0 as center and lx as axis, i.e. a point reflection.
DEFINITION. If a line I is perpendicular to a line m, the point of intersection of the two lines is called the foot of the perpendicular /. A line I is said to be the perpendicular bisector of a pair of points A and B if it is perpendicular to the line AB and its foot is the mid-point of the pair AB.
DEFINITION. A simple quadrangle AB CD is said to be a rectangle if and only if the lines A B and CD are perpendicular to AD and BC.
EXERCISES
1. A parallelogram A BCD is a rectangle if and only if the lines AB and A I) are perpendicular.
2. The perpendicular bisectors of the point pairs AB, BC, CA of a- tri- angle ABC meet in a point.
3. The perpendiculars from the vertices of a triangle to the opposite sides meet in a point.
4. The lines through the vertices of a triangle parallel to the transforms of the opposite sides by a fixed orthogonal line reflection are concurrent.
57. Displacements and symmetries. Congruence. DEFINITION. The product of an even number of orthogonal line reflections is called a displacement. The product of an odd number of orthogonal line reflections is called a symmetry.
THEOREM 6. The set of all displacements form a self-conjugate subgroup of the parabolic metric group.
Proof. That the displacements form a group is evident because (cf. § 26, Vol. I): (1) the identity is a displacement, being the prod- uct of any orthogonal line reflection by itself; (2) the inverse of a product of orthogonal line reflections is the product of the same set of line reflections taken in the reverse order; (,S) the product of an even number of orthogonal line reflections by an even number of orthogonal line reflections is, by definition, a displacement.
The group of displacements is contained in the parabolic metric group by Theorem 2.
ti>4 l.l ' l.lhKAN 1M.ANK (JEOMETRY [CHAP. IV
If {U} is an orthogonal line reflection, 2 a similarity transforin;iLion, and L'= 2(L), /'= 2(/), then 2 • {LI} • 2"1 is a harmonic homology with V as center and /' as axis. But since L and the point at infinity of / are paired in the absolute involution, so are L1 and the point at infinity of /'. Hence 2 • {LI} • 2"1 = {L'l1} is an orthogonal line reflection.
If A, and A, are any two line reflections 2 A^.,2"1 = 2 A^"^ A,2~l. A similar argument shows that 2AjA2 • • • AB • 2"1 is a product of n orthogonal line reflections whenever At, • • •, AB are orthogonal line reflections and 2 is in the parabolic metric group. Hence the group of displacements is a self-conjugate subgroup of the parabolic metric group.
COROLLARY 1. The set of all displacements and symmetries form a self-conjugate subgroup of the parabolic metric group.
DEFINITION. Two figures such that one can be transformed into the other by a displacement are said to be congruent. Two figures such that one can be transformed into the other by a symmetry are said to be symmetric.
COROLLARY 2. If a figure Fv is congruent to a figure F^ and F^ to a figure Ft, then F^ is congruent to F3.
COROLLARY 3. If a figure Fl is symmetric with a figure F2, and Fy is symmetric with a figure F3, then Fl is congruent to Ff
COROLLARY 4. If a figure Fl is symmetric with a figure F2, and F^ is congruent to a figure FS, then Fl is symmetric with Ff
Since translations and point reflections leave the absolute invo- lution invariant, the definition of congruence given in this section includes the definitions in §§ 39 and 46 as special cases. Theorem 6 shows that the theory of congruence and symmetry in general belongs to the geometry of the parabolic metric group. It must be remem- bered, however, that the theory of congruence of point pairs on parallel lines belongs to the affine group. In other words, the part of the theory of congruence developed in Chap. Ill is independent of the choice of the absolute involution.
In case the absolute involution has double points, the theory of congruence of point pairs on the minimal lines (§ 56) is different from that on other lines. As will appear in the following sections the
§57] MINIMAL LINES 125
theory on any line which is not minimal is essentially the same as that developed in Chap. Ill on the basis afforded by the group of translations and point reflections. On a minimal line, however, the set of points [P] such that OP0 is congruent to OP consists of all points on this line except the point 0. For let II and /2 denote the double points of the absolute involution, 7t being the one on the line OP0. Let Q be a point of the line 0/2 distinct from 0 and from /2, and let P be any point of 0/t distinct from O and from If If AI be the orthogonal line reflection whose center is the point at infinity of the line P0Q and whose axis passes through 0, and A2 be the orthogonal line reflection whose center is the point at infinity of the line QP and whose axis passes through 0, we have A1(tf) = Q and A2(Q) = P. Hence the rotation A2A1 transforms P0 to P. Combining transformations oftheformA2A1 with transla- tions it is clear that we have
THEOREM 7. Any pair of points on a minimal line is congruent to any other pair of points on the same line.
For example, if a mid-point of a pair AB were defined to be a point C such that A C is congruent to CB, we should have that when- ever the line AB is minimal, the point C may be any point on this line different from A and B. The theorems on mid-points in Chap. Ill would in general have exceptional cases. It is to avoid this difficulty that we have adopted the definition of mid-point given in § 40, Chap. III. A similar remark applies to the definition of ratio of collinear point pairs in § 43, Chap. III.
DEFINITION. A parallelogram ABCD whose sides do not pass through double points of the absolute involution and in which the point pair AB is congruent to the point pair AD is called a rhombus. \ rhombus which is also a rectangle is called a square.
126
KTCLIDKAN IM.ANK < i K< >.M KTKY
[CHAI-. IV
EXERCISES
1. Prove tliat tin- x'r""l' "'" displacements and symmetries could be defined as the group of nil colliueations leaving invariant tin- set of all conies obtain- able bv translations from a fixed central conic.
2. Tin- parabolic metric jjroiip consists of all projective cullineations trans- forming the group of displacrni'-nN into itself.
3. Two ]>i)int pairs ou nouminimal lines are symmetric if and only if they are congruent.
4. The i*T|K'udicular bisector of a point pair AB contains all points/* such tliat .1 /' is congruent to 111'.
5. The. simple quadrangle .1 BCD is a rhombus if and only if the lines AC and BD are the perpendicular bisectors of the point pairs BD and A C respectively.
6. A parallelogram A BCD is a rectangle if and only if the ]K>int pair .(< ' is congruent to the point pair BD.
7. Socialize the quadrangle-quadrilateral configuration (§ 18, Vol. I) to the case where the vertices of the quadrangle are the vertices of a square.
58. Pairs of orthogonal line reflections. THEOREM 8. If A1? A.,, A8 are three orthogonal line reflections whose axes pass through a point 0 (ordinary or ideal), the product A8A0At is an orthogonal line reflection whose axis passes through 0.
Proof. In case the three axes are parallel, the product A8A2 is a translation, and so by Theorem 4 is expressible in the form A4At, where A4 is an orthogonal line reflection whose axis is parallel to the other axes. Hence
Q<
Fiu. 48
In case two of the axes are not parallel, the third axis must pass through their common point O. Let Pbe any point not collinear with 0 and a circular point. Let Ca be the conic, existent and unique according to the lemma of § 56, which passes through P, has O as center, and has the absolute involution as an involution of conjugate
§58] ORTHOGONAL LINE REFLECTIONS 127
points. If Ql be any point of C'2, let A1(^1) = ^, Af«?a)=0,,
According to this construction the line QlQt is parallel to QtQ& and Q2Qa to (?6<i>6, where in case Q{ = Qj, the line Q{Qj is taken to mean the tangent to C2 at $,.. Hence, by Pascal's theorem (Chap. V, Vol. I) or one of its degenerate cases, it follows that Q3Qt is parallel to Q9Q^ Hence
and (A.A.A.^Q,) = Qf
Since Ql is an arbitrary point of C2,
(AAA/-1
The transformation A3A2A1 is not the identity, because it cannot leave invariant a point, different from 0, of the axis of At unless A2 = Ag, and in the latter case the product is equal to At. Since A8A2A1 leaves invariant the line QvQt (or the tangent at Qv, if Ql = Qt), it leaves in- variant the point at infinity of this line and also the line through 0 perpendicular to it. As A3A2A1 is of period two, it follows that it is an orthogonal line reflection.
COROLLARY 1. If Alf A2, and A3 are any three orthogonal line reflections whose axes meet in a point or are parallel, there exists an orthogonal line reflection A4 such that A2At = A8A4, and an orthogonal line reflection A5 such that A2Aj = A5A3.
Proof. By the theorem, A4 exists such that
A8AA=A4. Hence A2Aj = A8A4.
In like manner, A6 exists such that
AAA»=A,
Hence A2At = A6A8-
COROLLARY 2. The product of any odd number of orthogonal line reflections whose axes meet in a point or are parallel is an orthogonal line reflection.
Proof. By the theorem, whenever n ^ 3, the product of n orthog- onal line reflections whose axes are concurrent reduces to a product of Ti—2. Thus, if n is odd, the number of line reflections can be reduced by successive steps to one.
128 KICLIDEAN PLANE GEOMETRY [< ,,A,-. iv
If ii is even, this process reduces the number of line reflections in the product t<» two. Thus we have
COROLLARY 3. Tht product of any even number of orthogonal line /•• rhrtions is a rotation in case their axes meet in a point, and is a translation in case the axes are parallel.
COROLLARY 4. An orthogonal line reflection is not a displacement.
( 'OROLLARY 5. The set of all rotations having a common center is a commutative group.
Proof. A rotation is defined as a product of two orthogonal line reflections whose axes meet in an ordinary point. So, by definition, the identity is a rotation, and the inverse of a rotation A.AJ is the rotation AjAa. The product of two rotations is a rotation by Cor. 3. Hence the rotations having a given point as center form a group. To show that any two of these rotations are commutative amounts to showing that
(i) AAAA = AAAA
whenever the A's are orthogonal line reflections whose axes concur. By the theorem we have
and hence
AAAA=AAAA
But since A, A A, = A, A A,,
O 4 1 1 * 3
A2A3AA = AAAA>
which combined with (2) gives (1).
THEOREM 9. Any displacement leaving a point 0 invariant is a rotation about 0.
Proof. The given displacement is a product of an even number, n, of orthogonal line reflections, An • • • A:. Let A,' be the line reflection whose axis is the line through 0 parallel to the axis of A,. Then the product T, = A,Aj is a translation (Theorem 3) and
A, = T,.A;.
Thus A,, • • • Ax = TX • • • T^;,
where each T, is a translation. But by Cor. 2, Theorem 11, Chap. 11 1, if 2 is any affine collineation, T,.£ = 2T/, where T/ is a translation or the identity. Hence
A • • • A = A' • • • A'T' • • -T'
iVn -ilj — ^VB JV, !„ 1 ,.
§§58,59] DISPLACEMENTS 129
But since An • • • A, and A'H • • • A[ leave O invariant, the product T,| • • • Tj leaves O invariant, and hence, by Theorem 3, Chap. Ill, is the identity. Hence
VV-At— Ai
where A{, • • •, A^ are orthogonal line reflections whose axes pass through 0. By Cor. 3, Theorem 8, A'n • • • A.[ is a rotation about O.
59. The group of displacements. THEOREM 10. Let 0 be an arbi- trary point. Any displacement can be expressed in the form PT, where P is a rotation about 0 and T a translation.
Proof. By precisely the argument used in the last theorem the given displacement can be expressed in the form
A' •••A'T' •••T'
**•• •'Vl12n *JJ
where A,'(t = l, • • •, 2?i) is an orthogonal line reflection whose axis passes through O, and T/(i = 1, • • •, 2 n) is a translation or the identity. The product TJ. • • • T{ is, by Theorem 6, Chap. Ill, a translation. By Cor. 3, Theorem 8, A.^B • • • A{ is a rotation or a translation. Since it leaves 0 invariant, it is a rotation.
COROLLARY 1. Any displacement can also be expressed in the form T'P', where T' is a translation and P' a rotation with 0 as center.
COROLLARY 2. Any symmetry is a product of a line reflection wJwse axis contains an arbitrary point and a translation.
THEOREM 11. Any displacement, except a translation having a double point of the absolute involution as center, is a product of two orthogonal line reflections.
Proof. Let 0 be an arbitrary point. By the last theorem the given displacement reduces to PT, where T is a translation and P a rotation about 0. If the center, L, of T is not a double point of the absolute involution, by Theorem 4,
where ^ and ln meet L in the conjugate of L relative to the absolute involution and where 12 passes through 0. By Cor. 1, Theorem 8, there exists an orthogonal line reflection {Mm} such that
P = {Mm} - {LIJ.
Hence PT = {Mm} • (Z/J • {LIJ
= {Mm} - {LIJ.
130 Kt ( LIDKAX PLANE GEOMETRY [( HA,-, iv
If P is net tin- identity, it ia clear that m and ^ cannot be parallel, ami lu-nce PT is a rotation.
In case T is a translation whose center is a double point of the absolute involution, it can be expressed (Theorem 8, Chap. Ill) as a product <>f two translations Tt, T2 whose centers are not double points of the absolute involution. Hence, if P is not the identity, PT0 is a rotation, and thus PTsTt is also a rotation. In case P'is the identity, we have the exceptional case noted in the theorem.
COROLLARY. A displacement is either a rotation or a translation.
The following two theorems have the same relation to the para- bolic metric group and the group of displacements, respectively, that the fundamental theorem of projective geometry (Assumption P) has to the projective group on a line.
THEOREM 12. A transformation of the parabolic metric group leaving invariant two ordinary points not collinear with a double point of the absolute involution is either an orthogonal line reflection or the identity.
Proof. Denote the given fixed points by 0 and P, and let Cz be the conic through P having 0 as center and the absolute involution as an involution of conjugate points. Since C2 is uniquely determined by these conditions (cf. the lemma in § 56), it is left invariant by the given transformation F. Now F leaves 0, P, and the point at infinity of the line OP invariant. Hence the line OP is point-wise invariant, and every line I perpendicular to it is transformed into itself. Since C2 is also invariant and each of the lines perpendicular to OP meets C* in at most two points, F is either the identity or of period two. If of period two, it is evidently an orthogonal line reflection.
THEOREM 13. A displacement leaving invariant a point 0 and a line I containing 0 but not containing a double point of the absolute involution is either the identity or a point reflection with 0 as center.
Proof. Let P be any ordinary point of I distinct from 0, and let C2 be the conic through P having 0 as center and the absolute invo- lution as an involution of conjugate points. A displacement leaving o invariant, being a product of two orthogonal line reflections whose axes meet in 0, must leave 6'2 invariant. Hence it either leaves P invariant or transforms it into the other point in which the line OP meets C'2. In the first case the transformation must, by Theorem 12
§§ :.«.», <*>J I » I SPLACEMENTS 131
and Cor. 4, Theorem 8, reduce to the identity. In the second case the given displacement, which we shall denote by A, multiplied by the orthogonal line reflection A whose axis is the line through 0 perpen- dicular to OP, leaves P invariant. Hence, by Theorem 12,
AA = A',
where A' is a line reflection having OP as axis or the identity. Hence
A = A'A.
Since A cannot be a line reflection, A' cannot be the identity. Since the axes of A and A' are perpendicular, A is a point reflection.
EXERCISES
1. A displacement which carries a point .4 to a point B and has a point 0 (ordinary or not) as center is, if the line OA is not minimal, the product of an orthogonal line reflection whose axis is OA followed by one whose axis is the line joining O to the mid-point of the pair .4/2.
2. If three of the perpendicular bisectors of the point pairs AB, BC, CD, DA of a simple quadrangle meet in a point, the fourth perpendicular bisector passes through this point.
*3. Any affine transformation which leaves a central conic invariant is a line reflection whose center and axis are pole and polar with regard to the conic or a product of two such line reflections.
*4. In case the absolute involution is without double points, the group of displacements can be denned as the group of transformations common to the parabolic metric group and the equiaffine group. Thus two ordered point triads are congruent if they are both equivalent and similar. Develop the theory of congruence on this basis, and show what difficulties arise in case the absolute involution has double points.
60. Circles. DEFINITION. A circle is the set of all points [P] such that the point pairs OP, where O is a fixed point, are all congruent to a fixed point pair OP0, provided that the line OP0 does not contain a double point of the absolute involution. The point 0 is called the center of the circle.
Since the displacements form a group, it is clear that PQ may be any one of the points P. It has already been proved (§ 57) that if the line OPQ contained an invariant point of the absolute involution, the set [P] would consist of all ordinary points, except O, of the line OPQ.
THEOREM 14. A circle consists of the ordinary points of a conic sec- tion having the pairs of the absolute involution as pairs of conjugate points. The center of the circle is the pole of /„ with respect to the circle.
EUCLIDEAN PLANE GEOMETRY [CHAP. IV
Proof. Let O be the center of the circle and P0 any point of the circle. The circle consists of all points obtainable from J% by displace- ments which leave 0 invariant. If one of the line reflections of which each of these displacements is a product be taken to have OP0 as axis (Cor. 1, Theorem 8), it follows that the circle consists of the points obtainable from P0 by orthogonal line reflections whose axes pass through 0. But the system of points so obtained is identical by construction with the ordinary points of the conic referred to in the lemma of § 56.
COROLLARY. In case the absolute involution has no double points, every circle is a conic section. In case the circular points exist, they and the points of any circle form a conic section.
THEOREM 15. The ordinary points of any proper conic, with regard to which the pairs of the absolute involution are pairs of conjugate points, form a circle.
Proof. A conic C2 with regard to which the pairs of the absolute involution are conjugate points cannot be a parabola, since all points of Z» are conjugate to the point of contact of a parabola. Hence Ca has an ordinary point 0 as center. Let P be any point of C*. By definition there is one and only one circle through P which has 0 as a center. By Theorem 14, this circle is a conic through P having 0 as center and the pairs of the absolute involution as pairs of conjugate points. By the lemma of § 56 there is only one such conic. Hence the circle through P with 0 as center contains the ordinary points of C*.
THEOREM 16. Three noncollinear points, no two of which are on a minimal line, are contained in one and only one circle.
Proof. Let the three points be P0, Pl} and P2. Let Z« be the point at infinity .of the line J£/J and I the perpendicular bisector of the point pair P0Pt. The polar of Z. with regard to any circle through P0 and Pl must, by Theorem 14, pass through the mid-point of P0P^ and the conjugate of L* in the absolute involution. Hence the polar of Lm with regard to any circle through P0 and Pl must be I. In like manner, the polar of the point at infinity M* of the line P^P2 with regard to any circle containing 7f and P2 must be the perpendicular bisector m of P^Py Since the points P0, Pv P^ are not collinear, I and m intersect in an ordinary point 0, which must be the pole of
§60]
CIRCLES
FIG. 49
L* J/«, = /„ with regard to any circle through PQ, Plt and ^. Since, by definition, there is one and only one circle through P with O as center, there cannot be more than one circle through PQ) Plt and Pv
Since the product of the orthogonal line re- flection with OP0 as axis by that with I as axis transforms the point pair OP0 into the point pair OP^ the circle through P0 with 0 as center con- tains Pr A like argument shows that it contains P2. Hence there is one circle containing P0, Pv and P2.
Observe that we do not prove at this stage that a circle has a point on every line through its center. This could not be done without further hypotheses on the nature of the plane than we are making at present.
EXERCISES
1. The locus of the points of intersection of the lines through a point A with the perpendicular lines through a point B, not on a minimal line through A, is a circle whose center is the mid-point of the pair AB.
2. A tangent to a circle is perpendicular to the diameter through the point of contact.
3. Any two conjugate diameters of a circle are orthogonal.
4. If the tangents at two points A and B of a circle meet in a point O, the pairs OA and OB are congruent.
5. If / is the perpendicular bisector of a point pair AB, then the circles through A and B meet / in pairs of an involution whose center (§ 43) is the mid-point of A B.
6. The system of all circles having a common center meet any line in the pairs of an involution.
7. A parallelogram which circumscribes a circle must be a rhombus.
8. A parallelogram inscribed in a circle is a rectangla
9. If two circles have two points in common, the pair of tangents at one common point is symmetric to the pair of tangents at the other.
10. The feet of the perpendiculars from any point of a circle to the sides of an inscribed triangle are collinear.
KICI.IDKAN 1'LANK (JKo.MF/rKY CHAP. IV
61. Congruent and similar triangles. Two of the three fundamental criteria for tin- congruence of triangles can be derived at tin- present stage. The third criterion, that in terms of " two sides and the included angle," essentially involves order relations and is given in § 63.
In the following theorems we shall restrict attention to triangles none of whose sides pass through double points of the absolute invo- lution. The sides of a triangle ABC which are opposite to the vertices . I. />', fare denoted by a, b, c respectively. It will be observed that instead of angles we refer to ordered line pairs.
THEOREM 17. Two triangles ABC and A'B'C' are congruent in such a way that A corresponds to A' and B to B' if the point pair AB is con- gruent to the point pair A'B' and the ordered line pairs ca and cb are congruent to the ordered line pairs c'a' and c'b' respectively.
Proof. By hypothesis, there is a displacement F carrying A and B to A' and // respectively. Let T («) = «", T(b) = b", and T(C) = C". If a" * a1, we should have the ordered line pair c'a' congruent to c'a", and hence there
would be a transformation leaving B' and c' invariant and carrying a' to a", but this transformation, by Theorem 13, would be the identity or a point reflection with B1 as center contrary to the assumption that a"=£ a'. In like manner it follows that b"= b', and hence that C"= C'.
THEOREM 18. If in two triangles ABC and A'B'C' the point pairs All, BC, CA are congruent, respectively, to A'B', B'C', C'A', the pair of lines be is congruent to the pair of lines b'c'. The two triangles are either congruent or symmetric.
Proof. By hypothesis, there is a displacement which carries A'B' to A B. Let C" be the point into which C is carried by this displacement. Let C'" be the point to which C" is carried by the orthogonal line reflection of which AB is axis. Now if C were not identical with C" or C'", we should have three congruent point pairs AC, AC", AC"' and
§§<•>!, 62] TRIANGLES
other congruent point pairs B( ', l>( '", BC'". That is, there would IK- two circles, one with A as center and one with // as center, having tluve points in common. If C, C", C'" were colliuear, or if two of them were on a minimal line, this would contradict Theorem 14 ; otherwise it would contradict Theorem 16.
The conclusions of the theorem are now obvious.
The theorems converse to the above are not difficult and are stated in the exercises below. The theorems on similar triangles (Exs. 3, 4, 5) are proved in an analogous way, using Theorem 12 instead of Theorem 13. For these theorems we used the following definition :
DEFINITION. Two figures are said to be directly similar if and only if one can be transformed into the other by a similarity trans- formation which effects on /„ the same transformation as some dis- placement. A transformation of this sort is called a direct similarity transformation.
EXERCISES
1. If two ordered point triads are congruent, the corresponding ordered point pairs and line pairs are congruent.
2. If two ordered point triads are symmetric, the corresponding point pairs are congruent and the corresponding ordered line pairs are symmetric.
3. If the ordered line pairs ab, bc,ca are congruent, respectively, to the ordered line pairs a'l', b'c', c'a', the ordered triad abc is directly similar to the ordered triad a' b'c'.
4. If the ordered line pairs ab, be, ca are symmetric, respectively, to the ordered line pairs a'b', b'c', c'a', the ordered triad abc is similar to the ordered triad a' b'c'.
5. If two ordered triads abc and a'b'c' are directly similar, the ordered pairs ab, be, ca are congruent to a'b', b'c', c'a' respectively.- If the ordered triads are similar but not directly similar, the ordered pairs ab, be, ca are symmetric to a'b', b'c', c'a' respectively.
6. A direct similarity transformation is a product of a displacement and a homology.
62. Algebraic formulas for certain parabolic metric groups. Adopt- ing a system of non homogeneous coordinates (x, y) for which /„ is the singular line, and a system of homogeneous coordinates for which
the line /„ has the equation x0 = 0, and any involution on it can be written in the form (§§ 54, 58, Vol. I),
a? = 0, axx + bxx + bx + cx = 0.
180 EUCLIDEAN PLANE GEOMETRY [CHA, iv
If the coordinate system be chosen so that (0, 1, 0) and (0, 0, 1) are conjugate points in this involution, the bilinear equation reduces to
(3) cu^-f cx9xa= 0.
Here the point (0, 1, 1) is paired with the point (0, c, — a). In case the involution contains two pairs of points which are harmoni- cally conjugate, one pair may be chosen as (0, 1, 0) and (0, 0, 1) and the other pair as (0, 1, 1) and (0, 1, — 1). In that case (3) reduces to
(4) 3^+^=0.
For the rest of this section we assume that the absolute involution contains two pairs of points which are harmonically conjugate with respect to each other. Such involutions exist in every plane satis- fying Assumption HQ, since any two distinct collinear pairs of points determine an involution. Hence this assumption is no restriction on the nature of the plane in which we are working. It is, moreover, easy to replace the formulas which we shall obtain from (4) by the more general but more cumbersome formulas based on (3).
The equations of the transformation required to change (3) into (4) are
*o = aro» x\ = Vcxv Xt = Vaxr
Hence it is clear that in the complex geometry (§ 5) every involution may be reduced to the form (4), and in the real geometry only those involutions can be reduced to this form which are such that a/c > 0. The involutions of the latter type are direct (§ 18).
The equations of the affine group are
(5)
and if the involution (4) is to be transformed into itself, all pairs xlt xa and xlt xa which satisfy
• ^+^3=°
must also satisfy
K*, + &!«,) («!«! + ^xj + («,*! + \x} K*j + &,*,) = 0, which is the same as
(a « + a,*) xft + (a161 + afc} (xfr + x^) + (6? + 622) x& = 0.
Hence
.aA+aA=°>
§<•--'] FORMULAS 137
are the necessary and sufficient conditions that (5) leave (4) invariant. Combining these two equations, we obtain
or
Thus we infer aa = ± bl and al=^ ba. Hence
THEOREM 19. The equations of the parabolic metric group are
where e2 = 1.
Any conic section has an equation of the form (§66, Vol. I) (7) Vo2 + anx? + a^xl + 2 anx^ + 2 a^x^ + 2 a12a^c8 = 0,
which determines on the line XQ = 0 an involution whose double elements satisfy
Comparing with (4), we have that a circle must satisfy the condition
If this circle is to have (1, 0, 0) as center, i.e. as pole of x0= 0, the equation (7) must also satisfy the condition
Thus, returning to nonhomogeneous coordinates, the equation of a circle with the origin as center must be of the form*
(8) x*+f=k.
According to § 59, the transformations of the parabolic metric group leaving such a circle invariant are all displacements or sym- metries, and, moreover, all displacements and symmetries leaving the origin invariant leave this circle invariant. Substituting (6) in (8), we see that a displacement or symmetry leaving the origin invariant is of the form i_ , />
y'= e(-#£ + ay),
* This argument does not prove that every equation of this form represents a circle. The answer to this question depends on the value of k.
KM l.IhKAN PLANE GEOMETRY [CM.U-. iv
Simv any displacement or syniim'try is expressible as the resultant of one leaving the origin invariant and a translation ('riieorein 10, Cor. 1 >. wr have
TllKOKEM 20. The »->/imfiinni of the group of displacements and symmetries are
.••' = <™ + 0y '
a' + ff- = 1 and e'—l.
By $ .~>4, Vi»l. I, a transformation of the form (0) effects an involu- tion on /, if and only if c = — 1. By Theorem 10, Cor. 2, any symmetry leaving the origin invariant is a line reflection. Hence
THEOREM 21. The displacements are the transformations of the type (9) for which € = 1 and the symmetries those for which e = — 1.
EXERCISES
1. The equation of a circle containing the point («2k>) an(^ having the point (a^j) as center is
(x - atf + (y - M2 = (a, - fll)S + (/,, - bj*.
2. Two lines ax + by + c = 0 and a'x + l/y + c' = 0 are orthogonal if and only if aa' + W = 0.
3. In case the absolute involution has double points, the equiaffine trans- formations of the parabolic metric group are of the form (9), where a2 + f& = e and e = ± 1.
63. Introduction of order relations. Let us now assume that the plane which we are considering is an ordered plane in the sense of § 15. We may therefore apply the results of Chap. II, particu- larly of §§ 28-30. Let us also assume that the absolute involution satisfies the condition referred to in § 62, that there exist two pairs of points conjugate with regard to the absolute involution which separate each other harmonically. By Theorem 9, Chap. II, and its corollaries, it follows that any two pairs of the absolute involution separate each other, and that the absolute involution has no double jM»ints.* This result may conveniently be put in the following form :
TMK.DKKM 22. Two pairs of perpendicular lines intersecting in the same point separate each other. No line is perpendicular to itself.
• The geometry arising from the hyperbolic case has been studied by Wilson and Lewix in the article referred to in § 48.
§63] ORDER RELATIONS 139
The restrictions which we have just introduced enable us to state the fundanu-ntal tlit'on-m (Thruivtn 1.'!) about the group of displace- ments in the following more precise form :
THEOREM 23. The only displacement leaving a ray invariant is the identity.
Proof. Let A be the origin and B any point of the ray. Since any col- lineation preserves order relations,^ is transformed into itself. Since the line A B is invariant, the displacement is a point reflection or the identity (Theorem 13). But a point reflection would change B into a point of the ray opposite to the ray AB, and thus not leave the ray AB invariant.
With the aid of this theorem we can complete the set of funda- mental theorems on congruent triangles, the first two of which were given in § 61.
THEOREM 24. Two triangles ABC andA'B'C' are congruent if the point pairs AB, AC and the angle 4 CAB are congruent respectively to the point pairs A' B', A'C' and the angle 4 C'A'B'.
Proof. Since the angle* 4 CAB is congruent to the angle 4 C'A'B', there exists a displacement At carrying A to A' and the rays AC and AB to A'C' and A'B' respectively. Since the point pair AB is con- gruent to A'B', there is also a displacement A2 carrying A to A' and B to B', and since AC is congruent to A'C', there is a displacement A3 carrying A to A' and C to C'. By Theorem 23, \ = A2 and \ = A3. Hence the displacement At carries the triangle ABC to A'B'C'.
EXERCISES
1. Two triangles ABC and A'B'C' are congruent if the point pair AB is congruent to the point pair A'B' and the angles 4 CAB and 4 CBA are con- gruent respectively to the angles 4 C'A'B' and 4 C'B'A'.
2. If two triangles ABC and A'B'C' are congruent in such a way that A correspond^ to A' and B to B', the angles &ABC, &BCA, &CAB are con- gruent to the angles & A'B'C', &B'C"A', 4 C'A'B' respectively.
3. If two triangles ABC and A'B'C' are symmetric in such a way that .1 corresponds to A' and fi to 11', the angles &ABC, &BCA, &CAB are con- gruent to the angles 4 C'B'A', 4 A '( "B', 4 B'A'C' respectively.
4. Let A, B, C be three collinear ]>oints and P*, the point at infinity of the line joining them ; B is between A and C if and only if
5. An orthogonal line reflection interchanges the two sides of its axis. * Note that an angle is an ordered pair of rays (§ 28).
L40
l-l t I.IDKAN IM.ANK (IKn.METUY
[CHAP. IV
64. The real plane. Let us finally assume that we are dealing
with tlu- gouiiH-iry of reals. In consequence, we have the theorem
that any one-dimensional projectivity which alters sense (i.e. for
which A < 0) has two douhle elements. This may be put into the
following form as a theorem of the affine geometry.
TIFEOKKM 25. If Al and At are any two points of an ellipse, any
in a point between
and A3, meets the
line I, meeting the line tlli /me in two points.
Proof.* Let us de- note the given ellipse by E*, and let A be a variable point on it. Let Ll and Ly be the points in which I is met by A^A and A^A respectively, and let Ql and <?2 be the points in which /„ is met by A^A and A2A respectively. Also let Qs be .the point in which A^L^ meets/.. By construc- tion, and by the defi- nition of a conic,
(10)
The projectivity [^Jx [(?2] is direct, because, by the remark at the beginning of this section, if the projectivity altered sense it would ha»e two double points, and these, by the definition of the projec- tivity, would be points of intersection of /„ with E2, contrary to the hypothesis that E* is an ellipse.
Let C and C» be the points of intersection of A^A2 with I and /. respectively. Also let L* be the point at infinity of I. Then, by the hypothesis that C is between Al and A^
\
FIG. 61
•A simpler proof of this theorem, which, however, involves more preliminary theorems, in given in the next chapter (§ 76).
§«4] THE REAL PLANE 141
I '.ut, by construction, A^CA^C* =1 Q3LX QaC*. Hence, by Theorem 6, Chap. II,
But the points (7., Z., #2 are carried to <7«, £., #8, respectively, by the projectivity [#2] x[#8], indicated in (10). Hence the projectivity
[#J X t@J is °PPosite- Since [#,] Al#J is direct, [QJxt^J is °PP°- site. From this, since ^1 and Qa are carried by a perspectivity with Al as center to Ll and Z2 respectively, it follows (Theorem 6, Chap. II)
that the projectivity r T -\_rT -,
L^J A L^J
is opposite. By the remark at the beginning of the section this pro- jectivity must therefore have two double points, and by the definition of the projectivity these double points must be points of intersection of I with E\
COROLLARY 1. The points in which I meets the ellipse are separated l>y A^ and An relative to the order relations on the ellipse.
Proof. Let-Dj and Z>2 (fig. 51) be the two points in which I meets the ellipse, and let A, A^ A2, etc. have the meanings given them in the proof of the theorem. Then since the projectivity [ZJ -^ [ij is opposite, S(D1D2L1)^S(D1D2L2).
Hence the lines ADl and AD2 separate the lines AA1 and AAZ, which, according to the definition in § 20, implies that the pair of points DJ)^ separates the pair A^42 on the ellipse.
COROLLARY 2. The points in which I meets the ellipse are on opposite sides of the line A^A^
^ Proof. Let a be the tangent at A^ By the first corollary the lines a and A^A^ separate the lines A1Dl and A^D^ Hence, if A' denote the point in which a meets D^D^, I>1 and Z>2 separate A' and C. Now A' is not between Dl and Z>2, because if it were, the line a would meet the ellipse in two points instead of only in one. Hence C is between Dj and Z>2, and hence Dl and D2 are on opposite sides of I.
THEOREM 26. A rotation which transforms a given circle into itself transforms any triad of points on the circle into a triad of points in the same sense relatively to the order relations on the circle.
U-J EUCLIDEAN PLANE GEOMETRY [CHAF.H
Proof. \j&t the given triad of points be A, B, C, let 0 be any other point <>f tin- fiivlf. and let .4., #., Cm be the points at infinity of the linrs ". I. "/•', OC respectively ; let (X,4',#', C', A'X,B'X>CL be the points to which O, A, B, C, A., B,, C., respectively, are carried by the given rotation ; Irt A", B", C" be the points at infinity of the lines OA', OB', OC' respectively.
The given rotation effects on L a transformation which is the prod- uct of two hyperbolic involutions. Hence S(AaDBa>Cx) = S(A'a,B'a>C'ae). As in the proof of Theorem 25, the projectivity A^B^CL-^A'lB'lC'l is direct because otherwise it would have double points and these would be common to the circle and L. Hence S(ALB'nCL}=S(A'lB'lC'l} and, therefore, S(A*B» C«) = S(A'lB'l C'l). Projecting from 0, we have, by the definition of sense on a conic (§ 20), that
S(ABC) = S(A'B'C').
Theorem 2fi, which is here proved only for a real space, can be proved for any ordered space by the methods of the next chapter. This theorem states one of the most intuitionally immediate properties of a rotation. In fact, most of the older discussions of the notions of sense describe sense, without further explanation, as " sense of rotation."
EXERCISES
1. If 4 A OB is any angle, and PQ, any ray, there is one and only one ray PR on a given side of the line PQ such that 4 A OB is congruent or symmetric to 4 QPR.
*2. Prove that Theorem 25 is not true in a space satisfying Assumptions A, E, H, Q.
65. Intersectional properties of circles. THEOREM 27. If A and B are any two distinct points, then on any ray having a point 0 as origin there is one and only one point P such that the pair AB is congruent to the pair OP,
Proof. Let Bl be the point to which B is carried by the translation which carries A to O. The circle through Bl with 0 as center contains all points Q such that OQ is congruent to AB. Let 7?2 be the point to wlu'ch Bl is transformed by a point reflection with 0 as center. Then since O is between B^ and 7?2, any line I through 0 (and distinct from 0/fj) must meet the circle in two points, according to Theorem 25. But by Theorem 23 neither of the rays on / whicli have 0 as origin can contain more than one point of the circle. Hence each of these
§05]
INTKKSKCTION OF CIKCLKS
143
rays contains just one point of the circle. Hence each ray with O as origin contains a single point P such that^/? is congruent to OP. Combining this theorem with Theorem 23, we have
Til KOI; KM 28. There is one and only one displacement carrying a given ray to a given ray.
This result characterizes the group of displacements in the same way that the proposition that there is a unique projectivity of a one-dimensional form carrying any ordered triad of elements to any ordered triad characterizes the one-dimensional protective group.
THEOREM 29. If two circles are such that the line joining their centers meets them in two point pairs which separate each other, the circles have two points in common, one on each side of the line joining the centers.
Proof. Let the two circles be C* and Of, and let them meet the line joining the centers in the pairs PlQl and P^Q^ respectively. Let A be the center (§ 43) of the involution F in which T^j and P2Q.t are pairs, and let a be the perpendicular to the line P& at A.
SinceTJ and Q1 separate PZ and Q.2, the ordered triads P&H and Q^Q^ are in the same sense. The involution F inter- changes these two triads and hence transforms any triad into a triad in the same sense. Hence A is between Pl and Qr Hence, by Theorem 25, the line meets the circle C? in two points Al and A.t ; and by the second corollary of this theorem, Al and A^ are on opposite sides of the line P^QV
The lines AJ\ and AlQl are orthogonal since 7J and <?t are the ends of the diameter of a circle through A^. The line A^A is orthogonal to the line through Al parallel to PlQr Hence the involution F is per- spective with the involution of pairs of orthogonal lines through A^ Hence Av P2, and <?2 are on a circle whose center is on the line P^Qr By Theorem 16 this circle must be C%. Hence C\ and (7,a have Al in common. A similar argument shows that At is on f* and T*.
FIG. 52
144 EUCLIDEAN PLANK (JKo.MKTRY [CHAP. IV
66. The Euclidean geometry. A set of assumptions. In the geom- t-tiv of r.-als i In- rot-llicients of the formulas derived in § 62 are real numbers. The formulas given for displacements in that section are tlu- \vi-ll-kno\\ii equations for the "rigid motions" of elementary geometry. Hence the geometry of the parabolic metric in a real plane is the Euclidean geometry.
This result can also be established by considering a set of postu- lates from which the theorems of Euclidean geometry are deducible and proving that these postulates are theorems of the parabolic metric geometry. It then follows that all the theorems of Euclidean geometry are true in the parabolic metric geometry.
As a set of assumptions for Euclidean geometry of three dimensions we may choose the ordinal assumptions I-IX which are stated in § 29, together with the assumptions of congruence (X-XVI) stated below. For our immediate purpose, however, a set of assumptions for Euclid- ean plane geometry is needed. To obtain such a set we merely replace VII and VIII by the following :
VII. All points are in the same plane.
Thus our set of postulates for Euclidean plane geometry is I- VI,
vn, ix-xvi.
Assumptions X-XVI make use of a new undefined relation between ordered point pairs which is indicated by saying "AB is congruent to CD" It must be verified that the new assumptions are valid when this relation is identified with the relation of congruence defined above.
X. If A =£ B, then on any ray whose origin is a point C there is one and only one point D such that AB is congruent to CD. Proof. This is the same as Theorem 27.
XL If AB is congruent to CD and CD is congruent to EF, thenAB is congruent to EF.
Proof. This is a consequence of the fact that the displacements form a group.
XII. If AB is congruent to A'B', and BC is congruent to B'C' and {ABC} and {A'B'C1}, then AC is congruent to A'C'.
Proof. By Theorem 28, there is a unique displacement which carries A and B to A' and B' respectively. This displacement carries
§<*;] ASSUMPTIONS 145
C to a point C' such that {A'B'C1}, because a collineation preserves order relations. Moreover, the point C' so obtained is such that BC is congruent to B'C' and AC to A'C'; and, by Theorem 27, there is only one point C' in the order {A'B'C'} such that BC is congruent to B'C'.
XIII. AB is congruent to BA.
Proof. AB is transformed into BA by the point reflection whose center is the mid-point of AB.
XIV. If A, B, C are three noncollinear points and D is a point in the order {BCD}, and if A'B'C' are three noncollinear points and D' is a point in the order {B'C'D'} such that the point pairs AB, BC, CA, BD are respectively congruent to A'B', B'C', C'A', B'D', then AD is congruent to A'D'.
Proof. Since AB is congruent to A'B'y there exists a displacement A which carries AB to A'B'. Let A(C') = C1, A (D) = Dv Also let C2 and Z>2 be the points to which Cl and Dl are trans- formed by the orthogonal line reflection having A'B1 as axis.
According to § 57, the pair BC is con- gruent to BtCl and ioB'C^ CA to C^A' and CyA'; BD to B'Dl and B'D^ and AD to A'Dl and A'D2. It follows that C' must coincide with Cl or C2, for
otherwise there would be two circles, one with A' as center and the other with B' as center, containing the three points C^ C9, C'.
If C' = Cj, it follows, by Theorem 23, that D1 = Dv and hence that AD is congruent to A'D'. If C'= C2, it follows, similarly, that D'=Dt, and hence that AD is congruent to A'D'.
DEFINITION. If 0 and XQ are two points of a plane a, then the set of points [A'] of a such that OX is congruent to OA'Q is called a circle.
XV. If the line joining the centers of two coplanar circles meets them 'In pairs of points, I^Ql and P2Q.2 respectively, such that {I^I^Q^ and {P1Q1Q.2}, the circles have two points in common, one on each side of the line joining the centers.
Proof. This is the same as Theorem 29.
EUCLID KAN PLANE GEOMETRY [CHAI-. iv
XVI. // .1, 1'-. C art tlurc points in the order {ABC} and /.' . /.'..,
Jit, • - • tire jwints in tin- <>,•</>/• [.l/:/!^, •;.-///l/>a}, • • • such that A I- /.s
• , • • . - ,-.. point pain /•'/;.. /•',/•'.,, • • -, //"/<• ///»•/•<• <(/•»• nut
more than <i jinitc number of the points B^ B^ • • • between A and C.
Proof. Let B* be the point at infinity of the line Ail. Then J$l is the harmonic conjugate of A with respect to B and B*,, B^ is the har- monic conjugate of B with respect to Bl and B<* ; and so on. Thus J. /•', /.',, /*,••• form a harmonic sequence of which J5. is the limit-point. Since C has a finite coordinate, the result follows from § 8, Chap. I.
The set of assumptions I-XVI is not categorical It provides merely for the existence of such irrational points as are needed in constructions involving circles and lines (see § 77, below). It can be made categorical by adding Assumption XVII, § 29. It must be noted, however, that when XVII is added, X-XVI become redundant in the sense that it is possible to introduce ideal elements and then bring in the congruence relations by means of the definitions in tlu's and the preceding chapters.
In order to convince himself that the assumptions given above are a sufficient basis for the theorems of Euclid, the reader should carry out the deduction from these assumptions of some of the fundamental theorems in Euclid's Elements. An outline of this process will be found in the monograph on the subject from which the assumptions have been quoted.*
In making a rigorous deduction of the theorems of elementary geometry, either from the assumptions above or from the general projective basis, it is necessary to derive a number of theorems which are not mentioned in Euclid or in most elementary texts. These are mainly theorems on order and continuity. They involve such matters as the subdivision of the plane into regions by means of curves, the areas of curvilinear figures, etc., all of which are fundamental in the applications of geometry to analysis, and vice versa. In so far as these theorems relate to circles, they have been partially treated in §§ 64-65 and will be further discussed in the next chapter. The methods used for the more general theorems on order and continuity, however, are less closely related to the elementary part of projective geometry and will therefore be postponed to a later chapter.
• Foundations of Geometry, by Oswald Veblen, in Monographs on Modern Mathematics, edited by J. W. A. Young, New York, 1911.
§§ tw, «,7J DISTANCE 147
67. Distance. In § 43 we have defined the magnitude of a vector OB as its ratio to a unit vector OA collinear with it ; but in the affiue geometry the magnitudes of noncolliuear vectors are abso- lutely unrelated. In the parabolic metric geometry we introduce the additional requirement that any two unit vectors OA and O'A' shall be such that the point pair OA is congruent to the point pair O'A'.
Thus, if a given unit vector OA is fixed and C'2 is the circle through A with 0 as center, any other unit vector must be expressible in the form Vect (OP), where P is a point of the circle. This gives two choices for the unit vector of any system of collinear vectors, and each of the two possible unit vectors is the negative of the other. Therefore, while it is possible under our convention to compare the absolute values of the magnitudes of noncolliuear vectors, there is no relation at all between their algebraic signs. This corresponds to the fact that there is no unique relation between particular sense classes on two nonparallel lines.
Formulas in which the magnitudes of noncollinear vectors appear must, if they state theorems of the Euclidean geometry, be such that their meaning is unchanged when the unit vector on any line is re- placed by its negative. This condition is satisfied, for example, in Exs. 2 and 4, § 71.
The ratio of two collinear vectors is invariant under the affine group; the magnitude of a vector is invariant under the group of translations; but the absolute value of the magnitude of a vector, according to- our last convention, is invariant under the group of il is placements. The last invariant may be defined directly in terms of point pairs as follows :
DEFINITION. Let AB be an arbitrary pair of distinct points which shall be referred to as the unit of distance. If P and Q are any two points, let C be a point of the ray AB such that the pair AC is congruent to the pair PQ. The ratio
AB
is called the distance from P to Q, and denoted by Dist (PQ). If L is any point and / any line, the distance from L to the foot of the perpendicular to I through L is called the distance from L to I.
Us EUCLIDEAN PLANE GEOMETRY [CHAP, iv
It follows directly from the theorem above that Dist i /'',» is uniquely defined and positive whenever P =£ Q, and zero whenever P=Q. From the corresponding theorems on the magnitudes of vectors there follows the theorem that if {ABC}, then
I >ist (AB) + Dist (BC) = Dist (AC). Other properties of the distance-function are stated in the exercises.
The notion of the length (or circumference) of a circle may be defined as follows : Let Pv Pt, •••, Pn be n points in the order {P^P^ • • • Pn} on a cin-lr, and let
^ + . . . + Digt (P^).
It can easily be proved that for a given circle C'2, the numbers j> obtained from all jKxssible ordered sets of points Pv !'„, • • •, Pn, for all values of n, do not exceed a certain number.
DKKINITION. The number c, which is the smallest number larger than all values of p, is called the length or circumference of the circle C2.
The proof of the existence of the number c will be omitted for the reasons explained below. The existence of c having been established, it follows with- out difficulty that if c and c' are the lengths of two circles with centers O and (/, respectively, and passing through points P and P', respectively,
c _ Dist (OP) ~
Choosing the point pair O'P' as the unit of distance and denoting the con- stant c' by 2 ir, this gives the formula
(11) c = 2ir-Dist(0P).
The theory of the lengths of curves in general could be developed at the present stage without any essential difficulty. This subject, however, is very different (in respect to method, at least) from the other matters which we are considering, and therefore will be passed over with the remark that, starting with the theory of distance here developed, all the results of this branch of geometry may be obtained as applications of the integral calculus. Even the theory of the length of circles which we have summarized in the paragraphs above involves the ideas, if not the methods, of the calculus.
EXERCISES
1. Two point pairs AB and CD are congruent if and only if Dist (AB) =s l)'ist(CD).
2. If A, B, C are noncollinear points, Dist (AB) + Dist (BC) >Dist (AC).
3. Two triangles ABC andA'B'C' are similar in such a way that A corre- sponds to A', B to B', and C to C' if and only if
Dist (AB) _ Dist (A C) _ Dist (BC) Dist (A'B') ~ Dist (A'C') ~ Dist (B'C) '
§§«7,68] ARK A 149
4. Relative to a coordinate system in which the axes are at right angles, the distance between two points (xv #,), (x2, y2) is
the jwsitive determination of the radical being taken. The distance from a point (#!#!) to a line ax + by + c = 0 is the numerical value of
68. Area. The area of a triangle, as distinguished from the measure of an ordered point triad, may be defined as follows :
DEFINITION. Relative to a unit triad OPQ (§ 49) such that the lines OP and OQ are orthogonal and the point pairs OP and OQ are congruent to the unit of distance, the positive number
is called the area of the triangle ABC, and denoted by a (ABC).
As was brought out in Chap. Ill, the theory of measure of polygons belongs properly to the affine geometry. But the standard formula for the area of a triangle in terms of base and altitude (Ex. 1, below) involves the ideas of distance and perpendicularity and hence belongs to the parabolic metric geometry. It should be noticed that this formula assumes that the side of the triangle which is regarded as the base does not pass through a double point of the absolute invo- lution. This condition is satisfied under the hypotheses of §§ 63, 64, but is not always satisfied in a complex plane ; whereas the definitions of equivalence and measure as given in Chap. Ill are entirely free of such restrictions.
The theory of areas in general depends on considerations of order and continuity which we have not yet developed, and which, like the theory of lengths of curves, belongs essentially to another branch of geometry than that with which we are concerned in this chapter. We shall, however, outline the definition, of the area of an ellipse from the point of view of elementary geometry, because the derivation of the area of an ellipse from that of the circle affords rather an interesting application of one of the theorems about the affine group.
Let Pv P2, • • •, Pn be any finite number of points in the order { P1P2 • • Pn] on an ellipse E* with a point 0 as center, and let
A =
It can easily be proved that there exists a finite numl>er, a(E*), which is the smallest number which is greater than all values of A formed according to the rule above.
L50
KIVUDKAN PLANK ( i K» ».M KTRY
ICHAP. IV
|»i t IN ..... N. Tin- nmnlHjr a(E*)ia called the urea of the ellipse. In case E* is a cirri.-, < "-, it is easy to prove that
\v li.-re IT is the constant denned above and r = Dist (OPj).
Now suppose E* is an ellipse with two }>eri>endicular conjugate diameters <>A ami ui: wliidi inert K* in -I and .6 respectively, and let C'2 be the circle through A with f> as center, and let C be the point in which the ray OB (**. The hoinology F with OA as axis and the point at infinity of OB
center, which transform* B to C, is an affine transformation carrying the E* to the circle C*. This hoinology transforms the triangle OAB to the triangle OAC\ and the areas of these triangles satisfy the relation
a(OAC) _ Dist (PC) _ ^ a (04 fl) ~ Dist (OBJ ~
It follows, by § 50, that the hoinology transforms any tri- angle into one whose area is Z- times as large. By the definition of the area of an ellipse, therefore,
rt(C') _ Dist (PC) a (E2) ~ Dist (OB) '
Denoting Dist (OA ) by a and Dist (OB) by b, this gives
_, _. FIG. 64
/ »~> a (E*) = - — trah.
EXERCISES
1. The numerical value of the measure of a point triad ABC is equal to I)ist(/l/f) • Dist(CC'), where C' is the foot of the perpendicular from C to the line A B.
2. If afx-il is a simple quadrilateral whose vertices are on a conic and P is a
variable jniint of the conic,
• Dist(7>c)
Dist (Pb)- Dist (Pd) i» a constant (cf. Ex. 2, § 51).
3. If a project! ve collineation carries a variable point M and two fixed lines a, b to „!/', fi', // re8j>ectively, the number
is a constant.
§§<«,«] ANGULAR MEASURE 151
4. Let Fbe the center of a homology T ami / tin; vanishing line, P"1 (/«,). If /' is a variable point and Q = r(/>),
where £ is a constant.
5. The area of an ellipse is ir«/2, where a is the area of any inscribed parallelogram whose diagonals are conjugate diameters.
6. Among all simple quadrilaterals circumscribed to an ellipse, the ones whose sides are tangent at the ends* of conjugate diameters have the least area.
7. Among all simple quadrilaterals inscribed in an ellipse, the ones whose vertices are the ends of conjugate diameters have the greatest area.
8. Of all ellipses inscribed in a parallelogram, the one which has the lines joining the mid-points of opposite sides as a pair of conjugate diameters has the greatest area.
9. Of all ellipses circumscribed to a parallelogram, the smallest is the one having the diagonals as conjugate diameters.
69. The measure of angles. The unit of distance may be chosen arbitrarily, because any point pair can be transformed under the par- abolic metric group into any other point pair. It is otherwise with angles or line pairs, because, for example, an orthogonal line pair can- not be transformed into a nonorthogonal pair. Therefore the systems of measurement for angles obtained by choosing different units are, in general, essentially different. We shall give an outline of the generally adopted system of measurement, basing it upon properties of the group of rotations leaving a point O invariant.
Let P0 be an arbitrary point different from 0, and C2 the circle through P0 with 0 as center. Let Pl (fig. 55) be the point different from P0 in which the line PQO meets C2, and let P, and P. be the points in wliich
2 5
the perpendicular to P00 at 0 meets C2. By Cor. 1, Theorem 25, these points are in the order {P^PPP on the circle. Let <r denote the
segment P0P,Pr Any line through 0 meets C2 in two points which are separated by P0 and Plt and hence meets a in a unique point. Let P. be the point in which the line through 0 perpendicular to PQP, meets <r. And, in general, let [/J], n = I, 2, • • • be the set such
2»
that PI is the point in which the line through 0 perpendicular to
2"
meets <r.
•The ends of a diameter are the points In which it meets the conic.
l.VJ
Kl CLIDKAN 1M.ANK (JKo.MKTRY
[CHAI-. IV
The line OP obviously meets the line P0P. in the mid-point of the pair /„'/*, and the mid-point is between P0 and P.. Hence, by Cor. 1 , Theorem 25, we have the order re- lation {PP^Pj, where P1 denotes, for the moment, the point not on a in which the line OP. meets the circle. Since O is between J? and P', the same corollary gives {P^P.P^P1}.
Since P. is on the segment <r, we have either {P0P,P.P^ or {J^PP/$. The second of these alternatives, how- ever, when combined with {P'P0P.P^, would imply {P'P^P.P.}, con- trary to {JfjJP/^P'}. Hence {PQP.P,P^ is impossible, and we must have {P0P.P.P^. In like manner it is proved that {PQP.P.P^ and, in general, that (P • • • PP PP\
2" y-i
Let II denote the rotation (a point reflection in this case) which
.1
leaves 0 fixed and transforms P0 to Pv and let IT2" denote the rotation transforming P0 to Pr The rotation II*, being the product of the
3"
orthogonal line reflection with OP. as axis followed by that with OP. as axis, carries the point pair OP. to the point pair OPr Hence*
FIG. 55
In like manner it follows that
• The symbol A", where A is any transformation and n a positive integer, has been defined in § 24, Vol. I.
§<*'] ANGULAR MEASURE 153
1 IH
Let us denote (II2")"1 by II2", where m is any positive or negative integer,
Now all rotations are direct (Theorem 26). Hence S(P0P,P) =
Combining these relations with {P^PPJ , we have the order relation {P0P.PP.Pi}, and in general, by a like argument,
Hence we have {P0PmPm,P^, whenever 0 < — < — < 1, as can easily
& & -" 2"
be seen on reducing the two fractions to a common denominator.
Since II2 = 1, it follows that whenever ra/2" is expressible in the form 2 k + a, k being an integer,
(12) n"+"=ir and P2k+a = Pa.
DEFINITION. Let TT be the constant defined in § 67, (11). The number a • TT, where a = m/2", is called the measure of any angle congruent to 4-PQOPa. An angle whose measure is air is also said to be equal to 2 a right angles.
The measure of an angle is indeterminate according to this defi- nition. In fact, according to (12), whenever the measure of an angle is y8, it is also Ikir + ft, where k is any positive or negative integer. This indetermination can be removed by requiring that the measure /8 chosen for any angle shall always satisfy a condition of the form
OS#<27T, Or -7T</3^7T.
Since the rays OPm do not include all rays with 0 as center, the
2»
definition just given does not determine the measures of all angles. The required extension may be made by means of elementary con- tinuity considerations, the details of which we shall omit. The essential steps required are: (1) to prove that if P be any point in the order
{P0PPPj}, there exists a positive integral value of n such that {J^PP.P^ ;
•F (2) hence to prove that if P be any point on the circle not of the form
/'. the points of the form Pm fall into two classes, [/*] and [Pft], such
that {P0PaPPft}, and there is no point, except P, on every segment PaPPft of the circle ; (3) having required that 0 < a < ft < 2, to define II2*"1"* (where k is an integer, positive, negative, or zero, and x is the number
i:,l EUCLIDEAN PLANE GEOMETRY [CHAP, iv
such that a<.c<@ for all a's and {?») as the rotation about 0 carry- ing /' t" /' : (4) tt> show that if u* is a rational number m/n, (IIX)"= II"1 ; 10 define measure of angle as above, but with the restriction that a = //i/2" removed ; (6) to prove that the measure of the sum of two angles differs from the sum of the measures by 2 &TT, the sum being ik-tined as below.
DKKIMTIUN. If a, b, c are any three rays having a common origin, but not necessarily distinct, any angle 4«1c1 congruent to 4ac is said to be the sum- of any two angles 4«a&a and 4igcg such that 4 "./>., is congruent to 4 ab and 4&ges is congruent to 4£>c. The sum 4.a]cl is denoted by 4 <tj>., + 4 62ca.
For some purposes it is desirable to have a conception of angle according to which any two numbers are the measures of distinct angles. This may l>e obtained as follows :
DKKINITION. A ray associated with an integer, positive, negative, or /ero, is called a iitiinheretl ray. An ordered pair of numbered rays having the same origin is called a numbered angle. If the measure of an angle 4 hk in tin- curlier sense is a, where o^a<2ir, the measure of a numbered angle in which h is associated with in, and /• with n, is
2(« — m)ir + a.
Denning the sum of two numbered angles in an obvious way, it is clear that the sum of two numbered angles has a measure which is the sum of their measures.
The trigonometric functions can now be defined, following the elementary textbooks, as the ratios of certain distances multiplied by ± 1 according to appropriate conventions. This we shall take for granted in the future as having been carried out.
70. The complex plane. Instead of the assumption in § 64, we could assume that the Euclidean plane is obtained by leaving out one line from the complex projective plane (A, E, J, or A, E, H, C, R, I). All the results of Chap. Ill and of the present chapter up to § 63 are applicable to this case. The rest of the theory, however, is essentially different from that of the real plane, because the absolute involution necessarily has two double points and because a line does not satisfy the one-dimensional order relations. Thus the minimal lines play a principal role and must be regarded as exceptional in the statement of a large class of theorems; and another large class of theorems of elementary geometry (those involving order relations) disappears entirely.
§70] THK COMI'LKX I'LANH 155
For the present, therefore, we shall confine attention to the geometry of reals, but shall make use, whenever we tind it convenient to do so, of the fact (§ 6) that a real space S inay.be regarded as immersed in a complex space, S', in such a way that every line / of S is contained in a unique line I' of S'. As a direct consequence it follows that any conic C2 of S is a subset of the points of a unique conic of S'. For any five points of C'2, regarded as points of S', determine a unique conic of S' which, by construction (§ 41, Vol. I), contains all points of C*2 and is uniquely determined by any five of its points. Similar reasoning will show that any plane TT of S is contained in a unique plane IT' of S'; and like remarks may be made with regard to any one-, two-, or three-dimensional form.
A like situation arises with respect to transformations. A protective transformation II of a form in S is fully determined, according to the fundamental theorem of projective geometry, by its effect on a finite set * of elements of S. Since the fundamental theorem is also valid in S', there is a unique projective transformation II' which has the same effect on this set of elements as II.
Specializing these remarks somewhat we have : A Euclidean plane IT of S is a subset of the points of a certain Euclidean plane TT' of S'. The line at infinity L associated with TT is a subset of the line at infinity /', associated with TT'. The absolute involution I on 1M deter- mines an involution I' on ll in which all the pairs of I are paired. The involution I' has two imaginary double points, the circular points (§ 56), which shall be denoted by I1 and /2. Since a circle in TT is a conic having I as an involution of conjugate points, every circle in TT is a subset of the points on a conic in TT' which passes through II and /2.
The problem of the intersection of a line and a circle, or indeed of a line and any ellipse, can now be discussed completely. In the proof of Theorem 25 the intersection of a line / and an ellipse E* was seen to depend on finding the double points of a certain projectivity [£J -^ [LJ on 1. Any three points L{, L", L"', and their correspondents Z/2, 7/2', L"', determine a projectivity on the complex line I' containing /, and, by the fundamental theorem of projective geometry, this pro- jectivity is identical with [^J^^,] so far as real points are concerned. The double points of this projectivity are common to the complex
*For example, in case of a one-dimensional form any three elements of the form are such a set.
i;i ( 1.1DKAN I'LANK (JKO.MKTKY [CHAI-. l\
line containing / and tin- complex conic containing K~. Tlii-M- points are real if the hyi>otliesi8 of Theorem 25 is satisfied; they are n-al and coincident if / is tangent to E* ; otherwise they are imaginary.
A similar discussion will be made in the next section of the prol>- lem of the intersection of two circles, but first let us make certain definitions and conventions which will simplify our terminology.
According to the definitions in § 6, any point of S' is said to be complex, and a complex point is real or imaginary according as it is contained in S or not. In the case of lines, however, we have three things to distinguish: a line of the space S, a line of S' which contains a line of S as a subset, and a line of S' which contains no such subset. In current usage a line of the last sort is called imnginary, a line of either of the first two sorts is called real, and a line of either of the last two sorts is called complex. The current terminology therefore permits a confusion between a real line as a locus in S and a real line as a particular kind of a complex line.
In most cases, however, no misunderstanding need be caused by this ambiguity of language, and we shall in future usually employ the same notation for the real line I of S and the line I' of S' which contains /. The same remarks apply to conic sections and, indeed, to all one-dimensional forms.
DEFINITION. Any element (point, line, or plane) or set of elements of S' is said to be complex. Any element or set of elements of S is said to be real. A line or plane of S' which contains a line or plane, respectively, of S is said to be a real line or real plane of S'. A one- dimensional form of S', a subset of whose elements are real elements of S' and contain all the elements of a one-dimensional form of S, is called a real one-dimensional form of S'. An element or one-dimen- sional form of S' which is not a real element or real one-dimensional form of S' is said to be imaginary.
DEFINITION. A projective transformation of a real form of S' is said to be real if it transforms each real element of S' into a real element of S'.
Strictly speaking, these definitions distinguish between the two senses of the word " real " by phrases such as " real line of S'." But in practice we shall drop the " of S'." The one-dimensional forms as thus far defined are all of the first or second degrees, but the defini- tion can be extended without essential modification to forms of higher
§§70,71] THE COMPLEX PLANE 157
degree ami also to forms of more than one dimension. We shall take this extension for granted whenever we have occasion to use it.
In accordance with these conventions, the points /t and /2 which are really the double points of I' will be referred to in future as the double points of the absolute involution I. In like manner, any line I and circle C'2 which have no real points in common will be said to have in common the two points common to the complex line and the complex conic which contain I and (72 respectively.
The utility of these conventions will be understood by the reader if he will write out in full the discussion of pencils of circles in the following section, putting in explicitly, in notation and language, the distinction between elements of S and S'.
It is also convenient in many cases to extend the formulas for distance, area, etc. given in §§ 67~69 to imaginary elements. Thus, for example, in case (x^ y^) and (x^ y^) are imaginary points such that (x\~ aa)2 + (#1- #2)2 is a positive real number, V(a?1— a;s)a + (y1— ya)a will be referred to as the distance from (x^, y^) to (#2, y2). Extensions of terminology of this self-evident sort will be made when needed, without further explanation.
71. Pencils of circles. Consider two circles Cf and C2 in a real Euclidean plane. Let their centers be denoted by Cl and C2, and in case Cl =£ C2, let b denote the line C^Cf By Theorem 25, & meets each circle in a pair of real points which we shall denote by PlQl and I^Q^ respectively. The two pairs may be entirely distinct, in which case let F denote the involution on b transforming each pair into itself ; or they may have one point in common, in which case the line through this point perpendicular to & is a common tangent of the two circles. The two pairs cannot coincide, because the circles would then coincide. Thus four cases may be distinguished :
(1) The circles have the same center.
(2) The circles have a common tangent and point of contact.
(3) The involution F is direct.
(4) The involution F is opposite.
A circle is, by § 60, a real conic which, according to the terminology of the last section, contains the double points of the absolute invo- lution. Let us denote these points (the circular points) by Ii and It and apply the results of § 47, Vol. I, on pencils of conies.
KITLIDKAN 1M.ANK ( i K< >.M KTRY
[CHAI-. IV
In the tirst case let <> denote the common <vntn- «>f the t\vi» < ii The lines o/f and <>/., are tlu-n tangent to both circles at /t ami /, n-sjK-ctively. Ilrnrr. l.y ivtVivmv to £ 47, Vol. I, it is evident that the two rhvli's briong to a pencil of circles of Type IV.
In the second ease ('{ and C* have in common the points II and /2 as well as a common tangent and point of contact. Hence they belong to a pencil of Type II which contains all circles touching* the given line at the given point.
In the third case, since the involution F is direct, the pairs /'/,', and 1\(}^ separate each other. Hence, by Theorem 29, the circles have two real points, Al and A2, in common. Hence they belong to a pencil of Type I consisting of all conies through Alt A2, Ilf and /2. This may also l>e seen as follows :
Since the involution F has no double points (§ 21), it has a center (§ 43) which we shall call 0. Let a be the line perpendicular to b at O. Then by the argument used in the proof of Theorem 29, 0 is between Pl and Qj. Hence a meets Cf in two real points Al and Az (fig. 52). The pencil of conies through AJ} A3, Ilt /2 meets b in the pairs of an involution among which are PlQl and O and the point at infinity of b. Hence C* is a conic of the pencil, and hence a meets C* in Al and A2. In this case, therefore, the two circles belong to a pencil of Type I.
In the fourth case the involution F cannot have a double point at infinity, because then the other double point would have to l>e the mid-point of /,'V, and also of I*Va, and
thus C'f and ('.; would have a common center. Hence in this case also the center O of the involution F is an ordinary point. Let a denote
FlCf
• A conic and one of its tangent lines are said to touch each other at the point of contact. Two conies touching a line at the same point are said to touch each other.
§71] PENCILS OF CIRCLES 159
the perpendicular to b at 0, and let Al and A2 be the points in which " meets (7*. These points are imaginary ; for otherwise, since they are interchanged by the orthogonal line reflection with b as axis, O would be between them, and hence, by Cor. 1, Theorem 25, 0 would be between P^ and Qv contrary to the hypothesis that F is opposite. Precisely as in the third case it follows that A^ and A^ are also on r.;. Hence in this case also C* and C£ belong to a pencil of Type I.
In each case the facts established make it clear that the two circles could not both be members of more than one pencil of conies. Since any two circles fall under one of the four cases, we have
THEOREM 30. DEFINITION. Any circle contains the real points of a certain conic in the complex plane. Two conies determined by circles are contained in a unique pencil of conies, which is of Type I, II, or IV. The set of circles which the conies of such a pencil have in common with the real plane is called a pencil of circles. If the pencil of conies is of Type IV, the pencil of circles is the set of all circles having a fixed point as center ; if the pencil of conies is of Type II, the pencil of circles is the set of all circles tangent to a given line at a given point ; if the pencil of conies is of Type I, the pencil of circles is the set of all circles having a given pair of distinct real points in common, or else the set of all circles with centers on a given line and meeting this line in the pairs of an involution with two ordinary double points.
DEFINITION. The line a joining the centers of two nonconcentric circles is called the line of centers of the two circles or of the pencil of circles which contains them. If the circles have a common tangent and point of contact, this tangent is called the radical axis of the two circles or of the pencil of circles ; if not, the line perpendicular to a at the center of the involution in which the circles of the pencil meet a is called the radical axis. The double points of this involution are called the limiting points of the pencil of circles. Any circle of the pencil is said to be about either one, or both, of the limiting points.
The discussion above has established
THEOREM 31. Tlie radical axis of two circles passes through all points common to them which are not on the line at infinity. TJie limiting points of the pencil ivhich they determine are real if the cir- cles meet only in imaginary points and imaginary if they meet in lira real points.
160
i;i ( I.IDKAN IM.ANK ( I K< >.M KTK V
[CHAI-. IV
TiiK"KKM 32. The rir<- ti I <t i- jioiiit*, t)«' limiting points of a pencil of circles of Type /, <ni</ (In- two points not at infinity in which the circles of the pencil intersect are the pairs of opposite vertices of a complete quadrilateral. The sides of the diagonal triangle of this quadrilateral are /., the radical axis, and tJie line of centers of the pencil.
Proof. Let Av and At (fig. 57*) be the points other than II and It common to the circles of the pencil, and let B^ and 7?2 be the points of intersection of the pairs of lines I^Ayl^A^ and /^42, I2Al respectively. Whether Al and A3 are real or imaginary, the line A^4f = a, which is the radi- cal axis, is real Hence its point at infinity -4. is real ; and hence the line BJ$^ the polar of A,, with regard to any circle of the pencil, is real.
Since the line b = B^B^ is the polar of A^, it con- tains the centers of all conies through Ay AZ, Iy /2. Hence b is the line of centers of the pencil of circles through A^ and A2. The points B^ and 7?2 being diagonal points of the complete quadrangle A^AJJ., are evidently the double points of the involution in which the pencil of circles meets b, and hence are the limiting points of the pencil.
Taking Theorems 31 and 32 together, we see that any pair of real points Ay Ay determines a pair of imaginary points By /?2 such that either pair is the pair of limiting points of the pencil of circles through the other pair; that, conversely, any pair of imaginary points By Bt, which are common to two circles, determines two real points Ay A^ which are in the above relation to By 7?2; and that the three pairs AyA.^ ^,^,> IJi are pairs of opposite vertices of a complete quadri- lateral. The relation between the two pencils of circles, the one
• Fig. 57 is, of course, a diagram in which certain imaginary elements are repre- sented by real ones. On the use of figures in general, cf. p. 16, Vol. I.
FIG. 67
§71]
1'KNCILS OF CIRCLES
161
through Av and A^ and the other about A^ and AZ, is thus extremely symmetrical. It can be described in purely real terms by means of the following theorems and definition :
THEOREM 33. DEFINITION. If two circles have a point in common such that the tangents to the two circles at this point are orthogonal, the two circles have another such point in common. Two circles so related are said to be orthogonal to each other.
Proof. An orthogonal line reflection whose axis is the line of cen- ters transforms each circle into itself and transforms the given point of intersection into another point of intersection. Since orthogonal lines are transformed to orthogonal lines, the tangents at the second point are also orthogonal.
THEOREM 34. If a line through the center of a circle C2 meets the circle in a pair of points P1Ql and meets any orthogonal circle K2 in a pair of points P^Q^ the pairs PlQl and P^Q^ separate each other harmonically. Conversely, if PlQl and P^Q^ separate each other har- monically, any circle through P^ and Q2 is orthogonal to C2.
Proof. Let T be one of the points common to the two circles, and let t be the tangent to the circle TPtQ.£ at T. The pencil of circles tangent to t at T meets
the line P^ in the pairs
of an involution F, and
hence the first statement
of the theorem will follow
if we can prove that Pv
and #1 are the double
points of this involution.
The line perpendicular
to t at T and the line
perpendicular to P^ at Pl
are tangents to the circle TP1Q1 at T and P1 respectively, and hence
(Ex. 4, § 60) meet in a point M such that the pairs MPl and MT are
congruent. Hence the circle through T with M as center is tangent
to t at T and to P^ at Pr Hence Pv is a double point of F. A similar
argument shows that Ql is also a double point.
To prove the converse proposition we observe that there is only one circle through Pt and T and orthogonal to C*. One such circle, by
FIG. 58
EUCLIDEAN PLANE GEQMETBY [CHAI-. iv
tlu- argument above, passes through the point (/., which is liunimn- irally serrated from 7J by Pl and Qv Hence the circle J^Q.2T is orthogonal to (7s.
As a corollary we have
COROLLARY 1. The set of all circles orthogonal to a pencil of Type I is the pencil of circles through the limiting points of the first pencil.
Another form in which this result may be stated is the following :
COROLLARY 2. Let C* be a circle, Al any point not its center, and A^
the point on the line joining A to the center of C* which is conjugate
to A icith regard to the conic C*. TJien all circles through Al and
orthogonal to C* meet in Af
DEFINITION. Two points are said to be inverse with respect to a circle if and only if they are conjugate with regard to the circle and collinear with its center. The transformation by which ever}7 point corresponds to its inverse is called an inversion or a transformation by reciprocal radii.
Thus the center of the circle is inverse to ever}' real point at infinity. We shall return to the study of inversions in a later chapter.
EXERCISES
1. In case the limiting points of a pencil of circles are real, the radical axis is their perpendicular bisector.
2. If O is any point of the plane of a circle, and a variable line through 0 meets the circle in two points X, Y, the product OX • OF is constant, and equal to (OT)* in case there is a line OT tangent to the circle at T. The product (>.\ • OY is called the power of 0 with respect to the circle.
3. The power of any point of the radical axis of a pencil of circles with resjiect to all circles of the pencil is a constant, and this constant is the same for all points of the radical axis.
4. If O is the center of a circle, C any point of the circle, and Av and .4, any two points inverse with respect to it,
5. Through two points not inverse relative to a given circle, there is one and hut one circle orthogonal to it.
6. By a center of similitude of two circles is meant the center of a dilation (§ 47) or translation which transforms one of the circles into the other. If the circles are concentric, they have one center of similitude ; if they are not con- ri-ntric, they have two. The centers of similitude harmonically separate the rt-Mt«T8 of the two circles. The one which is between the centers of th«; two
§§:i,72] AN<;tLAR MEASURE 163
circles is called the interior, and the other is called the exterior, center of similitude. The common tangents of two circles meet in the centers of similitude.
7. Three circles whose centers are not collinear determine by pairs six centers of similitude which are the vertices of a complete quadrilateral having the centers of the circles as vertices of its diagonal triangle. Generalize to the case of n circles.
8. If a circle A"2 meets two circles Cj2 and C.| in four points at which the pairs of tangents are congruent or symmetric, the four points are collinear by pairs with the centers of similitude of Cj2 and C'|. Prove the converse proposition.
72. Measure of line pairs. The circular points Ilt 72 figure in a very important formula for the measure of a pair of lines.* With the exception of these two points, and two lines ilf iz which pass through them, all the points and lines to which we shall refer in this section are real.
The center and the point at infinity of the axis of an orthogonal line reflection are harmonically conjugate with regard to II and /2. Hence any orthogonal line reflection, regarded as a transformation of the complex space, interchanges Iy and 72, and any displacement leaves 7j and /2 separately invariant. Moreover, there exists a displacement transforming any (real) point of l^to any other (real) point of fc. Hence a necessary and sufficient condition that a pair of points P, P' of lx be transformable by a displacement to a pair Q, Q' of I* is
(13)
Now any pair of lines meeting I*, in P and P' can be transformed by a translation into any other pair of lines meeting it in P and P', and any pair of lines meeting 1& in Q and Q' can be transformed by a translation into any other pair of lines meeting it in Q and Q'. Hence the necessary and sufficient condition that a pair of lines meeting J. in P and P' be congruent to a pair of lines meeting it in Q and Q1 is (13).
This suggests as a possible definition of the measure of a pair of nonparallel lines l^, Bft^v,),
where il and i_2 are the lines joining the point of intersection of /t ami l^ to 7t and /2 respectively. It would satisfy the requirement of
•This formula is due to A. Cayley. Cf. Encyclopadie der Math. Wiss III AB 9, p. 901, footnotes 98 and 99.
Kit I.1DKAN PLANK UKo.MKTKY [CHAP. IV
being unaltered by displacements. In the case of measure »>f point pairs, however, we have
Dist(AB) + Dist(£C) = Dist(^C)
whenever {ABC}, and this condition is not satisfied by the cross ratio given above. We have, in fact,
(14) B (V,, v.) • B (V,, v.) = B ft/,, V, )
whenever / , /,, ls are concurrent. This is easily verified by substituting in the formula for cross ratio (§ 56, Vol. I). From (14) it is obvious that if we define
(15) »(Vt)=elogB(JA,Vt), the measure of line pairs will satisfy the condition
m(/1/2)-fm(y8)=m(y8)
whenever /lf /2, 13 are concurrent. Since the logarithm is a multiple- valued function, we must specify which value is chosen; and we must also determine the constant c conveniently.
Making use of the same coordinate system as in § 62, any point on /. may be denoted by (0, a, J3). In case a/ ft is real, (a/ ft}* > 0, and hence a and ft may be multiplied by a factor of proportionality so that
(16) «'+/32=l.
Throughout the rest of this section we shall suppose a and ft subjected to this condition. This is equivalent to supposing that
a = cos (0 + 2 mr), ft = sin (6 + 2 mr),
where 0 ^ 6 ^ 2 tr, and n is an integer, positive, negative, or zero. The double points of the absolute involution satisfy the condition
and so may be written
/, = (0, 1, i) and 72=(0, 1, - t),
where i = V^Ti. Now if ^ and 13 meet L in (0, alt 0J and (0, «3, respectively, it follows that (§ 58, Vol. I)
§72] ANGULAR MEASURE 165
The numbers a = ttlay+ fifi^ and fi = a1@2—a.l@l satisfy the condition «2+ /9s = 1. In fact, if al= cos^t and aa = cos#2, then a = cos0 and £ = sin 6, where 0 = 0^-0^+2 nir. Hence
B (*A, *\v) = «'-£*+ 2 «*£
Here again, a = a2 — /3s and # = 2 a;8 satisfy the condition
In fact, a = cos 2 0. Thus
(17) B(V2, t>2)=« + i£
= cos 2 0 + 1 sin 2 0
= e2" Hence
(18) logB(/t/2, V'2)=2*0,
where 2 0 is real and may be chosen so that 0 ^ 2 0 < 2 TT. Hence,
___ 4
choosing the constant c in (15) as — - > we have
A
(19) m (V2) = ~ log B (V2, v2) = 0,
where 0 may be chosen so that 0 S 0 < TT.
The formula (19) is interesting in connection with the theorem that the sum of the angles of a triangle is equal to two right angles. This proposition can easily be established without the consideration of imaginaries, on the basis of the definitions in the last section. From our present point of view, however, it appears as follows : Let the three sides of a triangle be a, b, c, and let them meet the line at infinity in A*, Bx, Cx respectively. It is easily verifiable that
B (A.B., V2) • B (B.C., //2) • B (C.A., //,) = 1, from which it follows by (19) that
m (ab) + m (be) + m (ca) = IT.
Here we have a theorem on the line pairs rather than on the angles of a triangle. Indeed, (19) is necessarily a formula for the measure of a pair of lines and not of an angle, because of the fact that two opposite rays determine the same point at infinity.
The number m(ab) may also be defined as the smallest value between 0 and 2 TT, inclusive, of the measures of the four angles ^aA wnicn mav be formed by a ray al of a and a ray bl of b.
Following the common usage, we shall say that two pairs of lines which are congruent make equal angles, etc.
1GC l.l CI.IDKAN 1'LANK <;K<>MKTI;Y [CHA...V
EXERCISES
1. If .1 ami />' are auy two jH>ints, the locus of a point P such that tin- rays /'.I an. I /'/; make a constant angle is a circle.
2. If in two [.n-jrctivr flat (H-ncils three lines of one make equal angles wit It tin- corn-simiiding three lines of the other, the angle between any two lines of the one is the same as the angle between the corresponding lines of the other.
3. l! <>.l, OB, OC, OD are four lines of a flat pencil,
B (OA,OB; OC,OD) = ™*AOC + »in*BOC. sinAAOD sin 4 BOD
In case the four lines form a harmonic set,
4. If Av .12, Aa, A4 are four points of a circle,
where AfAj represents Dist (A{Aj) or — Dist (AfAj) according as 5 (OAfAj) = >'(f>.lr42) or not, 0 being an arbitrary point of the circle and S(OA;Aj) being a sense-class on the circle.
5. If a, b, c are the sides of a triangle and fl^, fcj62, c,r2 are pairs of lines through the vertices 6c, ca, ab respectively, the six lines av a2, bv b2, cv c2 are tangents of a conic if and only if
jfl) sin (c2a) _ . sin (c2/>)
6. The points of a ray having (x, y) as origin may be represented in
the form .. . » . , \o\
(x + Au, y + A/J),
where a and /J are fixed and A > 0. There is a one-to-one reciprocal corre- spondence between the rays having (x, y) as origin and the ordered pairs of values of a and /J which satisfy the condition
a2 + 02 = 1.
U'ht-n a and /3 satisfy this condition, the numerical value of X is the distance between (x, y) and (x + Xa, y + A/?).
7. Two angles formed by the pairs of rays
(*o + Ao, y0 + A/8) and (x0 + Aa', y0 + A/3'), Q
(70 + AS, y0 + A/8) and (x0 + AS', ^ + A/3') respectively are congruent if and only if
oaf -I- ftp = Ed' + ftp.
8. Relative to the homogeneous coordinates employed above, the formula for the distance between (x0, xt, .r2) and (?/0, ylt y2) may be written
o i
Z
.Vo .Vi .'/2 01-t
§7-0 GENERALIZATION MY PROJECTION 167
73. Generalization by projection. The relation established in § 66 between Euclidean and protective geometry furnishes a source of new theorems in each. A theorem which has been proved for protective geometry can be specialized into a theorem of Euclidean geometry, or a theorem of Euclidean geometry may be generalized so as to furnish a theorem of projective geometry.
The two processes, of generalization and of specialization, may often be combined in a happy way with the principle of duality or with other general methods of projective geometry. Thus a theorem proved for Euclidean geometry can be generalized into a theorem of projective geometry and the dual of the general theorem specialized into a new theorem of Euclidean geometry. As an example, let us take the theorem of Euclid :
A. The perpendiculars from the vertices of a triangle to the opposite sides meet in a point (the orthocenter).
The sides of the triangle meet the line at infinity in three points, and the three perpendiculars are lines from the vertices to the conjugates of these three points in the absolute involution. The Euclidean theorem is therefore a special case of the following projective theorem:
B. The lines joining tlie vertices of a triangle to the conjugates, with respect to an arbitrary elliptic involution on a line I, of the points in which the opposite sides meet I, are concurrent.
This is a portion of Theorem 27, Chap. IV, Vol. I, the orthocenter and the three vertices of the triangle being the vertices of a complete quadrangle. But though the Euclidean theorem is a special case, yet the general theorem for elliptic involutions in real geometry may easily be proved by means of it. For, given any elliptic involution whatever and any triangle, the involution can be projected into the absolute involution and the given triangle will go into a triangle of the Euclid- ean plane. Hence the general theorem, B, that certain three lines meet in a point could fail to be true only if the Euclidean theorem, A, failed.
It is to be noted that this proves the theorem only for a real space and an elliptic involution. In a complex space (§ 5) it might happen that any
i:r« l.IDKAN PLANE GEOMKTKY [CHAP. IV
whirh carried tin- involution into the absolute involution would furry the triimtflf into one whose sides are not all real.
Now consider the plane dual of the projective theorem, B.
B'. The points of intersection of the sides of a triangle with the conjugates in an arbitrary involution at a point L, of the lines joining the vertices to L, are collinear.
If the involution at L is taken as the orthogonal involution we have the Euclidean theorem:
A'. The three sides of a triangle are met in three collinear points by the perpendiculars from a fixed point to the lines joining this point to the opposite vertices.
The second of the two processes which we are here emphasizing, namely the discovery of Euclidean theorems by specializing projective ones, is bril- liantly illustrated in many of the textbooks on projective geometry. We may mention the following :
L. Cremona, Elements of Projective Geometry, Oxford, 1894.
T. Reye, Geometric der Lage, Leipzig, 1907-1910.
R. Sturm, Die Lehre von den GeometrischenVerwandtschaften, Leipzig, 1909.
R. Boger, Geometric der Lage, Leipzig, 1900.
II. Grassmau, Projective Geometric der Ebene, Leipzig, 1909.
J. J. Milne, Cross-Ratio Geometry, Cambridge, 1911.
J. L. S. Hatton, Principles of Projective Geometry, Cambridge, 1913.
The reader will find material for the illustration of the second process, namely the discovery of projective theorems by generalizing metric ones, in Euclid's Elements, and even more in such books as the following:
J. Casey, A Sequel to the First Six Books of the Elements of Euclid, Dublin, 1888.
C. Taylor, Ancient and Modern Geometry of Conies, Cambridge, 1881.
J. W. Russell, Elementary Treatise on Pure Geometry, Oxford, 190.').
The class of theorems which are here in question will be dealt with to some extent in the following chapter, and the methods available will be extended in Chap. VI by the study of inversions. But on account of the magnitude of the subject many important theorems will be found relegated to the exercises and many others omitted entirely. In nearly every such case, however, a good treatment can be found in one or another of the books on projective geometry referred to above.
The current textbooks do not often classify theorems on the basis of the geometries to which they belong (§ 34) and the assumptions which are neces- sary for their proof (§17). Some progress has been made on such a classifi- cation in the present book (cf. § 83 below), but more remains to be done.
GENERALIZATION BY PROJECTION 169
Another criticism on current books is that they employ imaginary points in a rather shy and awkward manner. This is doubtless due to the fact that, previous to a logical treatment of the subject based on definite assumptions, the geometry of reals was regarded as having, somehow, a higher degree of validity than the complex geometry. The reader will often find it easy to abbreviate the proofs of theorems in the literature by a free use of imaginary elements (cf. § 78).
EXERCISES
1. Generalize projectively the following theorems : (a) The medians of a triangle meet in a point.
(ft) The perpendiculars at the mid-points of the sides of a triangle meet in a point.
(c) The diagonals of a parallelogram bisect each other.
2. Let A v Bv Cj be the points in which the lines joining the vertices A,B,C, respectively, of a triangle to the orthocenter, O, meet the opposite sides. The circle through Ar Bl and Cl contains the mid-points of the pairs AB, BC, CA and of the pairs OA, OB, OC. This circle is called the nine point or Feuerbach circle of the triangle. Cf. Ex. 7, § 41.
3. A hyperbola whose asymptotes are orthogonal is said to be equilateral or rectangular. Every hyperbola passing through four points of intersection of two equilateral hyperbolas is an equilateral hyperbola.
4. All equilateral hyperbolas circumscribed to a triangle pass through its orthocenter.
5. The centers of the equilateral hyperbolas circumscribed to a triangle lie on the nine-point circle.
CHAPTER V*
ORDINAL AND METRIC PROPERTIES OF CONICS
74. One-dimensional projectivities. The general discussion of one- (liim-Msional projectivities in Chap. VIII, Vol. I, has a great many I* >ints of contact with the ordinal and metric theorems of the last three chapters. For example, a rotation leaving a point 0 invariant transforms into itself any circle C* with 0 as a center. The transfor- mation effected on the circle by the rotation is a one-dimensional projectivity having the point 0 as center and the line at infinity as axis. The defining property of the axis of the projectivity in this case is that if a pair of points AB of the circle be rotated into a pair A' I? (i.e. if ^.AOB be congruent to ^.A'OB1), then the line AB1 is parallel to the line A'B, which is a well-known Euclidean theorem.
The proposition that any rotation is a product of two line reflec- tions corresponds to the proposition that any projectivity is a product of two involutions. The point reflection with 0 as center is commut- ative with all the other rotations about O and hence effects on C* an involution which (§ 79, VoL I) belongs to all the projectivities effected on C* by the rotations of this group. This involution is harmonic (§ 78, VoL I) to the involution effected on C* by any orthogonal line reflection whose axis contains O, and hence all the involutions of the latter sort form a pencil. Tims all the theorems of § 7-9, Vol. I, can be specialized so as to yield theorems about the group of rotations with 0 as center.
There are many other applications of the theorems in Chap. VI, Vol. I, to affine and Euclidean geometry (a few of them are indicated in the exercises below), but the main application which we are to consider at present is to the theory of order relations. Let us first recall some of the ordinal theorems which have already been estab- lished, and interpret them on the conic sections. Extending the definition of § 4, we shall say :
• In the earlier chapters of this volume we have Hsed only the first seven chap- ters of Vol. 1. The present chapter may advantageously be read in mimertion with Chaps. VIII-X. Vol. I. Chap. IX is first used in § 77 and Chap. X in § 86.
170
§74] ONE-DIMENSIONAL PROJECTIVITIES 171
DEFINITION. A projectivity of a one-dimensional form in any onlt'ivd space is hyperbolic, parabolic, or elliptic according as it has two, one, or no double points.
With regard to involutions, we have already established the follow- ing propositions (§21): If an involution preserves sense, each pair separates every other pair. If an involution alters sense, no- pair *fl><u'ntf& any other pair. An involution which does not alter sense in t'l/ijttic; that is to say, the pairs of a hyperbolic involution do not separate each other. The double points of a hyperbolic involution separate every pair of the involution.
DEFINITION. If A, B, C, D are four distinct points of a conic, the point 0 of intersection of the lines AB and CD is called an interior point in case the pairs AB and CD separate each other* and an exterior point in case these pairs do not separate each other. The set of all interior points is called the interior or inside of the conic, and the set of all exterior points is called the exterior or outside of the conic.
The pairs AB and CD are conjugate in the involution with 0 as center. Hence, if these two pairs separate each other, this involution preserves sense and is such that any two of its pairs separate each other. Hence any two lines through 0 which meet the conic meet it in pairs of points which separate each other. That is to say, the def- inition of an interior point is independent of the particular choice of the points A, B, C, D. A like argument applies in case 0 is exterior. In case the involution with 0 as center has double points, the lines joining 0 t# these points are tangent to the conic. Hence the next to the last of the propositions about involutions stated above implies that there are no tangents through an interior point. These results may be stated as follows :
THEOREM 1. The points coplanar with a conic fall into three in a tn a II i/ exclusive classes: the conic itself, its interior and its exterior. Each interior point is the center of an involution on the conic which preserves sense, and each exterior point of oiw which alters sense. All points of a tangent, except the point of contact, are exterior points of the conic.
* Cf . § 20, particularly the footnote.
17 -2 coMC SKCTIONS [CHAI-.V
Now let O be any interior point. If (S is any point conjugate to O with regard to the conic, there exists (cf. fig. 59) a complete quad- rtngU- AUCD wtioM rorticCII :in- \<«\n{* »>n tin- conic such that J/> and C7> meet in 0 and AD and C# meet in O1. But by Theorem 7,
FIG. 59
Chap. II, if AB separates CD, then AD does not separate 1?C; and hence O1 is an exterior point. Hence the polar line of any interior point consists entirely of exterior points. Hence
THEOREM 2. All points conjugate to an interior point are exterior.
Suppose, further, that the tangent to the conic at B meets the line 0& in a point P and the line BD meets 00* in a point P' (fig. 59). Then P and P' are conjugate points with regard to the conic. Moreover,
ABCDj^OPO'P'.
Since A and C do not separate B and D, it follows that the pair OO1 does not separate the pair PP'. That is,
THEOREM 3. On a line containing an interior point of a conic the pairs of conjugate points with regard to the conic do not separate one another.
By elementary propositions about poles and polar there follow at once :
COROLLARY 1. The pole of a line which contains an interior point is an exterior point.
COROLLARY 2. The polar of an exterior point contains some in- terior points.
$74] ONE-DIMENSIONAL PROJECTIVITIES 173
In § 78, Vol. I, it was established that any projeetivity is a product of two involutions one of which is hyperbolic. Since a hyperbolic involution is opposite, it follows that if the given projeetivity is direct, it is a product of two opposite involutions ; and if the given projee- tivity is opposite, it is a product of a direct and an opposite involution. But in the second case the direct involution is, by the argument just made, a product of two opposite involutions. Hence
THEOREM 4. A direct projeetivity is a product of two opposite involutions, and an opposite projeetivity is a product of three opposite involution*. An opposite projeetivity is also expressible as a product of a direct and an opposite involution.
In the case of projectivities on a conic, the axis of the product of two involutions is the line joining their centers. Hence we have, as consequences of this theorem,
COROLLARY 1. Any line in the plane of a conic contains points exterior to the conic.
COROLLARY 2. A projeetivity whose center is an interior point, and whose axis therefore consists entirely of exterior points, is direct.
In the fourth exercise, below, we need the following definition :
DEFINITION. The line perpendicular to a tangent to a conic and passing through its point of contact is called the normal to the conic at this point.
EXERCISES
1. What transformations of the Euclidean group effect projectivities on /«, to which the absolute involution belongs? How are these distinguished from the remaining similarity transformations by their relation to the cir- cular points? What transformations of the Euclidean group are harmonic on /» to the absolute involution ?
2. Show that the measure of a line pair as defined in § 72 is the logarithm of the characteristic cross ratio of a certain projeetivity on /.. Obtain an analogous formula for the measure of an angle in terms of the characteristic cn.ss ratio of a projeetivity on a circle.
3. Any noninvolutoric planar collineation which leaves invariant a conic and a line transforms the points of the line by a projectivity to which belongs the involution of conjugate points with regard to the conic.
17 1 CONIC SKCTIONS ,<HM-. v
4. If /' is any fixed point of ;i conic ami !!<> a variable point pair Midi that 4 /«'/'(,.' is a right angle, tin- lines !!({ meet in a fixed point «\\ the norniiit at /'.
5. Tin- liiii-s joining homologous points in a uoninvolutoric project ivity on a conic are the tangents of a second conic.
6. If /' i> any ti\ed point of a conic and !!<} a variable pair of points sndi that 4 /.'/'^ has constant measure, the lines RQ. are the tangents to a second conic.
7. If a projectivity F on a line is a product of an involution having doulil" points, At and Hv followed by another involution, and if F-^-tj) =A0^ Al and F(J1) = A.,, then .lj and /^ an- liarmonically conjugate with regard to .!„ and .L whenever .•!„ ^ A2; and Bl = A0 whenever .10 = J.,.
8. If .1, and HI are a pair of an involution I which is left invariant \>y a projectivity F, and if F-^.lj) = ,40 ^ Al and F(.lj) = .!„ ^ -I0, then .!„ and .•lj are harmonically conjugate with regard to Al and Jir
9. Let .! and .1' l>e any pair of an involution I. If .1 ^ .1', any projec- tivity II which transforms I into itself and leaves .1 invariant is either tin involution, with .1 and A' as double points, or the identity.
10. Generali/e jj so, Vol. I, so as to apply to the group of translations and the equiaffine group, using the fact that the transformations in each of these groups are products of pairs of involutoric projectivities.
75. Interior and exterior of a conic.
THEOREM 5. Any two points of a conic are the ends of two linear segments one consisting entirely of interior points and the other entirely of exterior points.
Proof. Let the given points be denoted by A and B, let C and D be any two other points of the conic which separate A and B, and let <r and ff represent the segments A CB and ADB on the conic. By the definition of the order relations on the conic, the lines joining C to the points of er meet the line AB in the points of a segment a' whose ends are A and B, and these points satisfy the definition of interior In like manner the lines joining C to points of <r meet the A/> in a segment a-' which is complementary to 0' and consists entirely of exterior points.
In a real plane the following theorem is a consequence of what we have just proved, but in order to have the result for any ordered plane we give a proof which is entirely general.
I'u KIII; KM 6. Any two interior points of a conic are the ends of a segment consisting entirely of interior points.
§75]
INTERIOR OF A CONIC
175
Proof (fig. 60). Let A and C be two interior points. Let Al be any point of the conic not on the line AC. The lines A^C and A^A are not, tangent to the conic, since (Theorem 1) the involutions at A and C'are both elliptic. Let AO and Ba respectively be the points, distinct from Alt in which the lines A^A and A^C meet the conic. The two segments of the conic whose ends are A0 and B2 are projected by the lines through Al into the two segments of the line A C which have A and C as their ends. We shall prove that the segment <r of the line A C which is the projection of the segment complementary to A^AJi^ con- sists entirely of ulterior points.
Let B be any point of a. The line AJZ then meets the conic in a point C2 which is separated from Al by A0 and J?2. Let B^A meet the conic in Clt let C^B meet it in AZ, and let A2C meet it in BI} so that A^C^B^ form a Pascal hexagon whose pairs of opposite sides meet in A, B, C. Since A is an interior point, we have the order {C^A^B^A t}. Since B was chosen so that 6'2 and Al are separated by B2 and A0, we have {B^C^AJ. From these there follows {B^A^A^. Transforming this by the involution at A we have {C^^A^C^A^. Hence we have {B^C^A^C^B^A^. Since the involution with center at C is elliptic, we have {B^B^A^A^. Hence we have (B^C^A^CJS^A^^. Hence Cz and AI separate A2 and Clt and hence B is interior to the conic.
THEOREM 7. Any two exterior points are ends of a segment consist- ing entirely of exterior points.
Proof. Let the two exterior points be El and EZ. If the line E^E^ is tangent, all points on it except the points of contact are exterior, since each of these points is the center of a hyperbolic involution on the conic. In this case the theorem is obvious. If the line E^E% meets the conic in two points, the theorem reduces to Theorem 5. If the line E^E^ does not meet the conic, and both the segments with l-\ and JS2 as ends should contain interior points, /t and /2 respectively, then neither of the segments whose ends are /, and /a could consist entirely of interior points, contrary to Theorem 6.
FIG. 60
ITT, CONIC SECTIONS [CHAP.V
Tin- th.-oivnis al>ove are connected with the following algebraic considera- tions: Any involution can be written in the fm-in
(1) X'
—
If we regard a, b, c as a set of homogeneous coordinates in a projective plane. then for every involution (1) there is one and only one ]K»int (n, /-, <•); ami inversely for every point (a, ft, «•) there is a unique involution (1), provided that the point does not satisfy the condition
(2) «'- + be = 9. By § 18 the projectivities (1) for which
(3) a2 + be > 0 are opposite, and those for which
(4) a2 + be < 0
are direct.
The equation (2) represents a conic section of which the points satisfying (3) are the exterior and those satisfying (4) are the interior. This may be proved as follows :
The conic is given by the parametric representation (§ 82, Vol. I)
a : b : c = x : xz : — 1,
and any involution on the conic is given by the transformation (1) of the parameter x. The center of the involution is the point of intersection of the lines containing pairs of the involution. The point (0, 0, 1) of the conic is given by the value 0 of the parameter x and thus is transformed to the point given by the value x = — b/a, namely, the point (— oft, ft2, — o2). The point (0, 1, 0) of the conic is given by x = ao and thus is transformed to the point given by x = a/c, namely, the point (ac, a2, — c2). The point of inter- section of the lines joining (0, 0, 1) to ( — aft, ft2, — a2) and (0, 1, 0) to (or, a2, — c2) is manifestly ( — a, 6, c). Hence ( — a, ft, r) is the center of the involution (1), and therefore is interior to the conic if (4) is satisfied and the involution direct, and exterior to the conic if (3) is satisfied and the involu- tion opposite.
EXERCISES
1. Parabolic projectivities are direct.
2. Two of the three vertices of any self-polar triangle of a conic are exterior |K>ints.
3. The center of a hyperbola is an exterior point.
4. The center of a circle is an interior point.
5. In a Euclidean plane all points interior to a circle and all points on it (except the point of contact of the tangent in question) lie entirely on one side of any one of its tangents.
§§;.-.. ™j INTERIOR OF A CONIC 177
6. If a segment A1B1 is contained in a segment A9Bt, the circle the ends of whose diameter are Al and Bl is composed of joints interior to the circle the cuds of whose diameter are A0 and Bz.
7. In a Euclidean plane all i>oints interior to an ellipse lie entirely on one side of any line consisting entirely of exterior points.
8. Any two pairs of conjugate diameters of an ellipse separate each other. Two pairs of conjugate diameters of a hyperbola never separate each other.
9. If O is the center of a conic A'2, the polar reciprocal of a conic C* with respect to A"- will be an ellipse, parabola, or hyperbola according as 0 is interior to, on, or exterior to C'2.
10. Consider a conic C- in a planar net of rationality satisfying Assump- tion II. The points of the net exterior to the conic fall into two classes [E~\ and [F] such that two tangents to the conic can be drawn from any point E and no tangent can be drawn to the conic from any point F. On any line in which one E is conjugate to an F with regard to C'2, every E is conjugate to an F. On any line in which one E is conjugate to an E, every E is conjugate to an E and every Fto an F. The interior points fall into two classes [/] and [/] such that the pairs of conjugate lines on a point / either both meet C2 or both do not meet C2, whereas one member of any pair of conjugate lines ou a point J meets C- and the other member does not meet C2.
11. Let the equation of a conic bey(x0, xv x2) = 0 and let the determinant of the coefficients of f(x0, xv a:2) be
A =
*00 "01 "02
0,
A point (x'Q, x{, x'2) is interior or exterior according as A '/(XQ, x{, x'2) is greater or less than zero.
76. Double points of projectivities. The preceding theorems hold for any ordered* space. On specializing to a real space we have the additional theorem that a projectivity which alters sense has two double points (§ 4). In the case of involutions this result com- bined with the theorem that a hyperbolic involution is always opposite gives
THEOREM 8. The pairs of an elliptic involution always separate one another, and the pairs of a hyperbolic involution never separate one another.
The last half of this theorem, combined with Theorem 3, gives the condition for the intersection of a line with a conic, a condition which has already been given in a more special form in § 64.
178 CONK' SKCTIONS [CH.U-.V
TIIKORKM 0. On "ni/ /</"' tlu-uiKjh an interior point of a conic tJte ' conjiHjiite points is hyperbolic, and the line meets the conic in (In- double points of this involution.
By Cor. 2, Theorem 3, the polar of an exterior point is a line through an interior point. The lines joining the exterior jMunt tu the points of intersection of its polar with the conic are tangents. H'-nce
COROLLARY 1. Through any exterior point there pass two tangents to a conic.
COROLLARY 2. Two involutions, one at least of which is elliptic, have one and only one common pair.
Proof. The center of an elliptic involution represented on a conic is an interior point. The line joining this point to the center of any other involution meets the conic in two points which are pairs of both involutions. Since any pair of an involution is collinear with the center, the two points so constructed are the only pair common to the two involutions.
A special case of this corollary may he stated in the following form :
COROLLARY 3. In a given one-dimensional form there is one and only one pair of elements which are conjugate with respect to a given elliptic involution and harmonically separated by a given pair of elements.
Since a hyperbolic involution is determined by its double points, it is evident that any two hyperbolic involutions are equivalent under the group of all projectivities of a one-dimensional form. The corre- sponding theorem for elliptic involutions is best seen by representing the involutions on a conic. The two centers Ilt If are interior points, and the line joining them meets the conic in two points Ct, Ct which do not separate them (Theorem 5). Let O^ and O2 be the double points (Theorem 8) of the involution in which IJZ and CvCt are pairs. An involution with either of the points Ol or O2 as center \\ill evidently transform the one with II as center into the one with /8 as center. Hence
COROLLARY 4. Any two elliptic involutions in the same real one- dimensional form are conjugate under the projective group of that form.
§76] DOUBLE POINTS OF PKOJECTIVITIKS 179
EXERCISES
1. All involutions which are harmonic to (i.e. commutative with and distinct from) an elliptic involution are hyperbolic.
2. If two points A, B of a line separate each point P(P ^ A, P 5* B) of the line from its conjugate point in a given elliptic involution, .1 and B are conjugate in this involution.
3. A hyperbolic projectivity is opposite or direct according as a pair of homologous jKiints does or does not separate the double points.
4. Elliptic projectivities are direct.
5. The center of an ellipse is an interior point.
6. The involution determined on the, line at infinity of a Euclidean plane by an ellipse is elliptic, by a hyperbola, hyperbolic.
7. Any two ellipses are conjugate under the affine group.*
8. An involution in a flat pencil is either such that every pair of conju- gate lines is orthogonal or there is one and only one orthogonal pair of conjugate lines.
9. A conic having two pairs of perpendicular conjugate diameters is a circle.
10. If A! and A2 are the real limiting points of a pencil of circles, each circle of the pencil either contains At and is on the opposite side of the radi- cal axis from A2, or contains Aa and is on the opposite side of the radical axis from .1 r
11. Of two circles of a pencil, both containing the same* limiting point, one is entirely interior to the other.
12. For any angle, &ABC, there is one and only one pair I, I' of orthog- onal lines through B which separate the lines BA and BC harmonically. One line, I, of the pair contains points P interior to &ABC, and 4.ABP is con- gruent to £PBC. The line I is called the interior bisector, and the line I' the exterior bisector, of the angle &ABC.
13. The asymptotes of an equilateral hyperbola bisect any pair of conju- gate diameters. ^
14. The bisectors of the angles of a triangle ABC meet in four points, one in each of the four regions determined by ABC according to § 26. These four points are the centers of four circles inscribed in ABC and are the vertices of a complete quadrilateral of which ABC is the diagonal triangle. The mid- point of the pair BC is the mid-point of the points of contact of either pair of inscribed circles whose centers are collinear with A.
15. Let V and V be the vanishing points (§ 43) of a projectivity on a line, the notation being so assigned that the point at infinity is trans- formed to V. There exist two points A, B which are transformed to two points A', B' such that
AV= VB = A'V' = V'B'. *Cf. §87, Exs. 14 and 15.
IM> cnMC SKCT10NS [riiAi-. v
77. Ruler-and-compass constructions. The discussion in Chap. IX, Vol. I, reduces any quadratic problem to the problem of finding the points of intersection of an arbitrary line with a fixed conic. Accord- ing to Theorems 5 and 9 the necessary and sufficient condition that a line coplanar with a conic meet it in two points is that the line pass through an interior point of the conic. Hence this condition will serve to determine the solvability of any problem of the second degree in a real space. Thus the discussion of linear and quadratic constructions, under the protective meaning of these terms, may be regarded as complete.
When we adopt the Euclidean point of view, the fixed conic may be taken as a circle ; and therefore every problem of the second degree is reduced to the problem of determining the points of inter- section of an arbitrary line with a fixed circle (cf. § 86, Vol. I).
The constructions of elementary Euclidean geometry which are known as ruler-and-compass constructions involve the determination of the points of intersection (whenever existent) of two arbitrary lines, or of an arbitrary line with an arbitrary circle, or of two arbitrary circlea The last of these problems has been shown in § 65 to be reducible to the first and second. Hence any ruler-and-compass con- struction may be reduced to the problem of finding the intersection of an arbitrary line with a fixed circle.
On account of the special character of the line at infinity, there is not a perfect correspondence between the linear constructions of pro- jective geometry and the Euclidean constructions by means of a ruler. The operations involved in the linear constructions of protective geometry are
(a) to join two points by a (protective) line ;
(b) to take the point of intersection of any two lines.
These are evidently equivalent to the following Euclidean operations:
(1) to join two ordinary points by a line ;
(2) to take the point of intersection of two nonparallel lines ; (3') to draw a line through a given point parallel to a given line.
The first of these operations corresponds to the proposition that two points are on a unique line, the second to the proposition that two nonjwrallel lines determine a unique point. These operations
§77] RULER-AND-COMPASS CONSTRUCTIONS 181
may be thought of as carried out with a straightedge or ruler whose length is not limited.
The operation (3') can be effected by means of (1) and (2), together with the following operation :
(3) to find on any ray through a point A, a point C such that the point pair AC is congruent to a preassigned point pair AB*
For let A be the given point and let BC be the given line. Let 0 be a point on the line AB in the order {ABO} such that BA is congruent to BO. Let A be the point of the line OC in the order OCA such that CO is congruent to CA. Then AA is evidently parallel to BC.
Thus (1), (2), and (3) serve as a basis for all linear operations in the projective sense. They obviously yield also a certain class of quadratic constructions ; but they do not suffice for all quadratic constructions. The latter may be provided for, as explained above, by adjoining the operation of taking the point of intersection with a fixed circle of an arbitrary line through an arbitrary interior point.
For the proof that (3') is not a consequence of (1) and (2), and that (1), (2), (3) do' not provide for all quadratic constructions, the reader is referred to Hilbert, Grundlagen der Geometric, Chap. VII (4th edition, 1913).
EXERCISES
1. Given three collinear points A, B, C such that AB is congruent to BC, show how to construct a parallel to the line AB through an arbitrary point /' by means of the operations (1) and (2) alone.
2. (liven two parallel lines, show how to find the mid-point of any pair of points on either of the lines by means of (1) and (2) alone.
3. (Jiven a parallelogram and a point Pand a line / in its plane. Through P draw a line parallel to /, making use of the ruler only.
* It is important to notice that the pairs AB and AC have the point A in com- mon. Thus (3) provides merely for drawing a circle through a given point and with a given other point as center. The drawing instrument to which this corre- sponds is a pair of compasses which snaps together when lifted from the paper, so that it cannot be used to transfer a point pair AB to a point pair A' If unless A = A'. This will be understood by anyone reading the second proposition in Euclid's Elements, which shows how to lay off a point pair congruent to ;i -riven point pair on a given ray. The operation (8) may be replaced by the operation of finding on any ray .4 7? a point C such that the point pair A C is congruent to a fixed point pair OP. The instrument for this operation may be thought of as a measur- ing rod of fixed length (say unit length) without subdivisions. (Cf . the reference to Hilbert, below.)
]s-J CONIC SBTIOXS [CHAP. V
4. Civt-ii a jM.int j.air .1'' an<l its mid-point IS, using the ruler alone, construct tin- jKiint i -air .!/> such that
AC
5. (Jivni four collinear points A, A', B, B', construct the fixed j>oiut of tin- parabolic projretivity carrying A to A' and B to B'.
6. (liven a projectivity on a line, find a pair of corresponding points .1 ami I such that a given point M is the mid-point of the segment J.I'.
7. Inscrilte in a given triangle a rectangle of given area.
8. (liven four tangents of a parabola, construct a tangent parallel to a ijiven line.
9. (liven three jxrints of a hyperbola and a line parallel to each asymptote, find the jK)int of intersection of the hyperbola with a line parallel to one of tin- asymptotes.
10. Construct by ruler and compass any number of tangents to a conic given by five of its ]M>ints; also any number of points of a conic given by
ive of its tangents.
11. Construct any number of points of a parabola through four given points.
12. Construct any number of points of a parabola touching three given lines and passing through a given point.
13. Through a given point construct an orthogonal pair of lines conju- gate with regard to a conic. (If the point is exterior to the conic, these lines are the bisectors of the angles formed by the tangents to the conic from this i>oint.)
78. Conjugate imaginary elements. It has been shown in § 6 that a real projective space S can be regarded as immersed in a complex projective space S' in such a way that every line of S is a subset of a unique line of S'. Certain additional definitions and conventions have been introduced in § 70. But in both these places little use was made of the properties of imaginary elements beyond their existence and the fact that S' satisfies Assumptions A, E, P. We shall now prove some of the most elementary theorems about the relation be- tween elements of S and S'.
DEFINITION. Two imaginary points, lines, or planes are said to be conjugate relative to a real one-dimensional form of the first or second degree if and only if they are the double elements of an involution in the real form.
As an example consider a real conic C2 and a line I exterior to it. The conic and the line have in common the double points of an ellij)- tic involution on /. But these points are also the double jM.ints of
§78] CONJUGATE IM AGIN ARIES 183
the involution on C12 whose axis is /. Hence the points common to C2 and / are conjugate imagiuaries both with respect to C'2 and to I. Since any one-dimensional form of the first or second degree whose elements are points is a line or a point conic, and since the double points of any involution on a conic are the intersections of the axis of the involution with the conic, we have
THEOREM 10. Any two conjugate imaginary points are on a real line.
By duality we have that any two conjugate imaginary planes are on a real line.
Two conjugate imaginary lines are by definition on a real point, line conic, cone of lines, or regulus. If they are on a real line conic, the plane dual of the argument above shows that they are on a real point. By dualizing in space we obtain the same result for conjugate imaginary lines of a cone of lines. Hence we have
THEOREM 11. Any two conjugate imaginary coplanar lines are on a real point and any two conjugate imaginary concurrent lines are on a real plane.
Conjugate imaginary lines on a regulus will be considered in a later chapter.
THEOREM 12. The lines joining a real point to two conjugate imaginary points not collinear with it are conjugate imaginary lines.
Proof. The conjugate imaginary points are double points of an elliptic involution on a real line. From any point not on this line this involution is projected into an involution of lines whose double lines are the projections of the given points.
THEOREM 13. If A^AZ and B^BZ are two pairs of conjugate imagi- nary points on different lines, the lines A^B^ and AZB^ meet in a real point and are conjugate imaginary lines.
Proof. By hypothesis the lines A^A^ and BJt^ are real and hence they meet in a real point C. Let B be the conjugate of C in the elliptic involution with Al and A^ as double points. By Corollary 3, Theorem 9, there are two real points P and Q which are paired in this involution and separate B and C harmonically. Let A be the conjugate of C in the elliptic involution with Bl and B2 as double points, and let R and 8 be
1M CONIC SECTIONS [CHAP, v
the two real points which are paired iii this involution and separate A and C harmonically. Since any two harmonic sets are projective,
CBPQ — CARS and CBPQ = CASK.
A A
The centers of these two perspectivities are two real points Ct and Ct, and since each perspectivity transforms two pairs of the elliptic invo- lution on the line A^At into two pairs of the elliptic involution on the line B^Bt, it transforms Al and Ay to Bl and Bif Hence one of the points Cl and Ct is the intersection of the lines AlBl and A^B^ and the other that of the lines A^B% and A^B^ By Theorem 12 each of these pairs of lines is a pair of conjugate imagiuaries.
The complete quadrilateral whose pairs of opposite vertices are A^Aj -#,#,, and C^C, is analogous to the quadrilateral considered in § 71 whose vertices were /^ and the limiting points of two orthog- onal pencils of circles (cf. fig. 57). With regard to the existence of such quadrilaterals we have
THEOREM 14. Let AtAz, BJiat 0^ be the pairs of opposite vertices of a complete quadrilateral. If A^3 and B^ are pairs of conjugate imaginary points, then C\ and C*2 are real and the diagonal triangle of the complete quadrilateral is real. .If A and AZ are real and BI and Bf are conjugate imaginaries, then Cl and Cf are conjugate imaginaries and the diagonal triangle is real.
Proof. In the first case Cl and Cn are determined as in the proof of the last theorem and hence are real. The diagonal triangle has for its sides the three real lines A^,, #,#2, C/'...
§T«J CONJUGATE IMAGINARIES 185
In the second case let a IHJ the line through A^ which is harmon- ically conjugate to A^A^ with respect to the pair of lines AyJil and A .,/>'„. Since the latter two lines are conjugate imaginaries and AyAl is real, a is real. The harmonic homology with A1 as center and a as axis transforms Bl and BZ to Cl and Cf Hence Cl and C'a are conjugate imaginaries and the line C^C^ is real.
Relatively to a real frame of reference a real involution is repre- sented by a bilinear equation with real coefficients (§ 58, Vol. I), and its double points appear as the roots of a quadratic equation with real coefficients. Hence the coordinates of a pair of conjugate imaginary points are expressible in the form
where XQ, xlt x^, xa, y0, ylt y2, yB are real. Like remarks can be made with regard to the coordinates of a plane or a line, and Theorems 10-14 can easily be proved analytically on this basis. The following theorem appears to be easier to prove analytically than synthetically :
THEOREM 15. A complex line on a real plane contains at least one real point.
Proof. Let the equation of the line be
V;o+"i'ri+ V2=°- This may be expressed in the form
« + tie;') x0 + (u[ + t'<) x, + (u't + 1<) a?a = 0,
where u'Qt u", etc. are real This equation is equivalent, if x0, xv xt are required to be real,- to
u'0x0 + ttX + ufa = 0, u"x0 + u['xl + u"x3 = 0, two equations which are satisfied by at least one real point
EXERCISES
1. A conic section through three real and two conjugate imaginary points is real.
2. A pair of conjugate imaginary points cannot be harmonically conjugate with regard to another pair of conjugate imaginary point-.
3. An imaginary point is on one and only one real line and has one and only one conjugate imaginary point.
186 CONIC SECTIONS [CHAP.V
79. Projective, affine, and Euclidean classification of conies. Let us regard a ival plane TT as immersed in a complex plaue TT', and consider all conies in IT' with respect to which the polar of a real point is always a real line.*
Throughout the rest of this chapter the word "conic" shall be used in this sense. The involution of conjugate points with regard to such a conic is one in which real points are paired with real points. Hence, if a conic contains one real point, every real nontangent line through this point contains another point of the conic, and the conic is real. The conies under consideration therefore fall into two classes, the real conies t and those containing no real point.
By § 76, Vol. I, any two real conies are equivalent under the group of projective collineations. The same proposition holds also for any two conies of the other class, as we shall now prove. Let two such conies be denoted by C* and C%. On an arbitrary real line I they each deter- mine an elliptic involution of conjugate points. By Cor. 4, Theorem 9, there is a projectivity of the line I carrying the involution determined by C'* into that determined by Cf. Any projectivity of the real plane which effects this transformation on / will carry C22 into a conic C, which has the two conjugate imaginary points AV A^ on I in common with C\. A collineation leaving I invariant will now carry the pole of / with regard to C'\ to the pole of I with regard to C\ ; and therefore carries C, to a conic C* which has Av A.2 and the tangents at these points in common with C\. Let L be the pole of / with regard to C* and L^ be any real point of /. By Cor. 3, Theorem 9, there is a pair of pouits J!fJ/i which are conjugate with respect to C\ and harmonically separate L and Ll and also a pair M' M[ conjugate with respect to C* and harmonically separating L and Lr The homology with I as axis, L as center, and carrying M1 to M carries C'l to C\. Hence we have
THEOREM 16. Any two real conies or any two imaginary conies itnth real polar systems are conjugate under the group of real pro- jective collineations.
• In § 85 this condition is seen to be equivalent to the condition that the equa- tion of the conic relative to a frame of reference in TT shall be expressible with real coefficients. For the present discussion, however, we do not need the general theory of correlation which is used in § 85.
t According to some usage any complex locus which has a real equation is called real. Cf. Pascal'.- li.-jM-rtormiii ,icr HOheren Mathematik, Vol. II (1910), Chap. XI 1 1 (lirrznlari). According to this definition both of the above classes of conies would be called n-al.
§?.»] CLASSIFICATION OF CONICS 187
If the line/ be taken as the line at infinity of a Euclidean plane the argument above shows that any two imaginary conies are also conjugate under the affine group. Since these conies do not meet any real line in real points, they are analogous to ellipses no matter how the line at infinity is chosen. Hence we make the definition :
DEFINITION. An imaginary conic with a real polar system is called an imaginary ellipse.
The results just established, together with those stated in Ex. 7, § 76, and Exs. 14 and 15, § 37, may be summarized as follows:
THEOREM 17. Under the affine group the conies with real polar systems fall into four classes, parabolas, hyperbolas, real ellipses, imaginary ellipses. Any two conies of the same class are equivalent.
Under the Euclidean group conies must be characterized by their relations to the circular points /t, 7g. Since a real conic which does not meet /„ in real points meets it in conjugate imaginary points, any real conic through 1^ also contains /2 and is therefore a circle. For the same reason the imaginary conic determined by an elliptic polar system must contain /2 if it contains /r
DEFINITION. An imaginary ellipse with respect to which the pairs of conjugate points on lx are pairs of the absolute involution is called an imaginary circle.
THEOREM 18. Any two real circles or any two imaginary circles are similar.
Proof. Let the centers, necessarily real, of two circles Ca and A'2 be Ox and 02 respectively. The center Ol may be transformed to #2 by a translation Tr This carries C* to a circle C\. Any real line 'I through 02 meets C* in two points Cl and C2 and K2 in two points A'j and A'.,. Since each of these pairs is harmonically conjugate with respect to #2 and the point at infinity 0* of /, the homology T0 with 02 as center and I* as axis which carries Cl to Kl also carries Cz to A^. This homol- ogy evidently carries all real points to real points if Clf C , A^, A"3 are real. If CtC2 and KJ\Z are pairs of conjugate imaginary points, con- sider (§ 77) the real pair of points PP' harmonically conjugate with iv'j-ard to CjCj and OO« and the real pair QQ' harmonically conjugate with regard to KI K2 and OOX. The homology T2 must carry P and P' to Q and Q' and therefore carries all real points to real points in this case.
188 CONIC SECTIONS < ..AP.V
Now the conic C'f is fully determined by its points /„ /„ Clt ('.. ;ni.l its center Ot and K* is fully determined by /t, /2, Klt K^ and og. J lence T, carries C? to #a. The product T2T, carries C12 to K*.
TIIKOKEM 19. Any two parabolas are similar.
Proof. Let Ca and K* be two parabolas and let C. and Ka be their points of contact with L. Let Tj be any rotation carrying (7« to K* «nd let T1(Cra)=Cfj*. Let K* be the conjugate of K* in the absolute involution and let c be the ordinary line through Kx tangent to C* and C its point of contact ; also let k be the ordinary line through JSC. tangent to K*, and K its point of contact. The translation T8
FIG. 62
carrying (7 to K carries c to k and C^ to a conic C\ touching L at jfiT«. Any line £ through K, not containing 7C or Kx, meets C% in a point C' and A"2 in a point K'. The horaology T8 with K as center, L as axis, and carrying C1 to ^L' carries (722 to K2. The product TgT2T1 is a similarity transformation carrying (72 to 7f2.
No theorem analogous to the last two holds for ellipses and hyper- bolas. Suppose an ellipse or a hyperbola C2 meets /« in Cl and C3 and another ellipse or hyperbola K2 meets it in K^ and Kf In case a similarity transformation carries Cl and Ct into Kv and Jf2,
(5)
Conversely, if C2 and Jf2 satisfy the condition (5) there evidently exists a rotation carrying C, and C2 to A"t and A'2. This rotation carries C* to a conic Ca which passes through K^ and A"s. By an argument
§§7», so] FOCI 189
analogous to the proof of Theorem 18 it can be shown that if C\ and A"- are both real ellipses, or both imaginary ellipses, or both hyper- bolas, there is a similarity transformation carrying C2 to A'2. Hence THEOREM 20. Two real ellipses or two imaginary ellipses or two hyperbolas which meet lx in pairs of points C^C^ and A^A^ are similar if and only if R(V2, C&) =»(//*> KiKJ-
EXERCISE A hyperbola for which R (IJv KlKs) = — l *s rectangular (Ex. 3, § 73).
80. Foci of the ellipse and hyperbola. Let C2 be any hyperbola or real or imaginary ellipse, and let /t, Z2 be the tangents to C2 through II and 13, lf the tangents to C2 through If The circular points /t, /2 are one pair of opposite vertices of the complete quadrilateral l^l^- Let the other two pairs of opposite vertices be F^FZ and F[F£ respec- tively (fig. 63), let a be the line FJFZ, b the line F[F'^ and O the point of intersection of a and b. Also let An and Bn be the points at infinity of the lines a and b respectively. The triangle OA^B^ is self-polar with respect to C2. Hence 0 is the center of C2 and is therefore real.
Let X be any real point not on llt 12, ls, /4 or C2. By the dual of the Desargues theorem on conies (§ 46, Vol. I) the tangents to C2 through A' are paired in the same involution with XIlt XIZ and A^, XFZ and XF[, XF'Z. The double lines xlt xz of this involution are harmonically conjugate with regard to XI lt A72 and to the tangents to C2. Hence they are paired both in the involution of orthogonal lines at X and the involution of lines conjugate with respect to (72 at X. Hence by Cor. 2, Theorem 9, x^ and xz are real, and are the unique pair of orthogonal lines on X which are conjugate with regard to C3.
In particular, if X = 0 it follows that a and b are real and are the only pair of orthogonal and conjugate diameters of C2. Hence Ax and B* are also real. If X is not on a, b, or /«, the lines x^ and #2 meet a in a pair of real points Xlt Xz distinct from Ax and O. Since Fl and Ft are harmonically conjugate with respect to the real pairs A^A', and A* 0, they are either real or conjugate imaginaries. But since /( and 7, are conjugate imaginaries, by Theorem 14 if one of the pairs FJ\ and /','/•'.! is a pair of real points, the other is a pair of conjugate iniagi- naries, and conversely. Hence the notation may be so assigned that J'\ and F3 are real and F[ and F% are conjugate imaginaries.
190 TONIC SKCTIONS [CHAP.V
L.-t .1, and J_ IK? the points in which a meets C2 and Bv and 7?f the points in which b meets C*. Uy construction neither of the lines a and b can be tangent to C'1 so that each of the pairs A^A% and 1^/A. is either real or a pair of conjugate imaginaries.
In case C* is an imaginary ellipse, both A^A^ and BJit are neces- sarily pairs of conjugate imaginaries. In case 6'2 is a real ellipse, the line /. does not meet it in any real point, and hence 0, the pole of lm, is an interior point. Hence both a and b meet C2 in real points. Hence if C3 is an ellipse, Alt A^ BI} B2 are all real. Whether C2 is an ellipse or a hyperbola, the tangents to (72 from F^ are conjugate imaginary lines since they join the real point Fl to the conjugate
FIG. 63
imaginary points /t and /2. Hence Fl is interior to C2, as is also Ff by a like argument. Hence the line F^ meets <72 in real points. ' Hence if C2 is a hyperbola, A1 and A^ are real. But if C2 is a hyper- bola, O is an exterior point, and hence Ax, which is harmonically separated from O by A^ and A2, must be an interior point. Hence b, the pole of A*, does not meet C2 in real points, and consequently B^ and Bt are conjugate imaginaries.
Let the polars of Flt Fv F[, F£ relative to C3 be denoted by dv dv d[t d!i respectively. Then dl and ds being the polars of real points are real ; and since their point of intersection is polar to a, it is Bx, and hence they are parallel to b. In like manner d( and d'^ pass through A, and are conjugate imaginaries.
§80]
FOCI
191
DEFINITION. The lines a and b denned above are called the axe» of the conic C'2, a being called the major, or principal, axis and b the minor, or secondary, axis. Each of the points Fv F.2, F{, F^ is called a focus, and each of the points Alt A^, Blt B^ a vertex, of the conic (72. Each of the lines dv d.2> d[, d[ is called a directrix of (7a.
FIG. 64
FIG. 65
In the course of the discussion of the complete quadrilateral IJJ^ we have established the following propositions :
THEOREM 21. If Cz is a hyperbola or a real or imaginary ellipse which is not a circle, its axes are the unique pair of conjugate diam- eters which are mutually perpendicular. Two of the foci and two of the directrices are real. The real foci lie on the major axis and the real directrices are perpendicular to it. The other two foci are conjugate imaginaries and lie on the minor axis. If C2 is real, the real foci are interior points and the real directrices are exterior lines. If C2 is a real ellipse, all four of the vertices are real ; if C3 is a hyperbola, the two vertices on the major axis are real and those on the minor axis are conjugate imaginaries.
The two tangents to C2 through Fl pass also through 1^ and /2. Pairs of conjugate lines at Fl are separated harmonically by these two tangents and hence meet /«, in pairs of the involution whose double points are II and /2. If we limit attention to real elements, this may be expressed by saying that the pairs of conjugate lines with respect to C* which pass through a focus are orthogonal. Conversely, if the pairs of orthogonal lines at any point P are conjugate with respect to C2, the double lines of the involution of orthogonal lines at
192 CONIC SKIT IONS [CHAP.V
P would have to coincide with the double lines of the involution of ronjugate lines, and hence P would be a focus. Hence
'I' 1 1 KOI: KM 'I'l. The real foci of a hyperbola or a real or imaginary eHij>s? nre tin- unique pair of real points at which all jim'rx of con- jnijtitt /iiti-x itrc orthogonal.
The set of all conies tangent to the four minimal lines llf 12, /g, /4 form a range (§ 47, Vol. I). Hence the pairs of tangents to these conies through any point P not on the sides of the diagonal triangle OA^B^ form an involution among the pairs of which are the pairs of lines PIV PIt\ PFV PFj and PF[, PF!,. Now if P is on C\ there is only one tangent to Ca at P, and this tangent is therefore a double line of the involution. Tin's and the other double line have to be harmon- ically conjugate with respect to PIl and P/g; that is, if (7a and P are real, the two double lines have to be orthogonal. These double lines must be harmonically conjugate also with respect to PFV and PFf Thus we have a result which may be expressed as follows (cf.Ex.12, § 76):
THEOREM 23. The tangent and the normal to a real ellipse or hyperbola at any real point are the bisectors of the pair of lines joining this point to the real foci.
In the proof of this proposition we have excepted the vertices of the conic, but the validity of the proposition for these points is self- evident. Another proposition which follows directly from the discus- sion above is the following, in which we make use of the fact that the pair of real foci determines the pair of imaginary ones, and vice versa.
THEOREM 24 DEFINITION. The system of all conies having two real or two imaginary foci in common is a range of conies of Type I. The two conies of the set which pass through any real point have ortJtogonal tangents at this point. Such a range of conies is called a system of confocal conies or of confocals.
The construction for the foci which has been considered in this section, when applied to a circle, reduces to a very simple one. The tangents to the circle at II and 72 meet in the center of the circle. The center of the circle is therefore sometimes referred to as the focus and the line at infinity as the directrix.
The term "focus" is derived from the property stated in Theorem 251, in consequence of which, if the conic be regarded as a reflecting surface, all rays of light diverging from one focus will be reflected back to the other focus.
§§80,8i] FOCUS OF PARABOLA 193
In the rest of the chapter the foci, center, directrices, and axes of an ellipse or a hyperbola will be denoted by the same letters as in thia section. The notation has been assigned so that for an ellipse the points are in the order {D.A
and for the hyperbola in the order
where Z>1 and Z>2 denote the points of intersection of the principal axis with the directrices dl and d^ respectively.
81. Focus and axis of a parabola. Let C* be any parabola. Since it is tangent to L, there are two ordinary tangents to it through II and /a respectively ; let these be denoted by ^ and lf respectively. Let their point of intersection be denoted by F, their points of contact with (7a by LI and L^ respectively, and the line LJL% by d. Also let the point of contact of C2 with /«, be denoted by Ax, the line AXF by a, and the point, other than A*, in which a meets C3, by A.
DEFINITION. The point F is called the focus, the line d the direc- trix, the line a the axis, the point A the vertex, of the parabola C2.
FIG. 66
That the focus, directrix, etc. of a parabola are real may be proved as follows: The transformation from pole to polar with regard to C* transforms the absolute involution to an involution of the lines through A» and transforms II and /„ into AXL1 and A*L^ respectively. The involution in the lines at A* is perspective with an involution among the points of (7a which has Ll and L^ as double points. Hence Ll and Lt are conjugate imaginary points. .Hence by Theorem 10 the
CONK SKCTIONS [CHAP.V
line d is real. Hence its pole, F, is real. Hence the line a joining /•' to A» is real, and also the point A.
Since the two tangents to C* through F pass through 1^ and /2, any two conjugate h'nes through F are perpendicular. Conversely, if the pairs of conjugate lines at any point are orthogonal, the tangents tli rough this point must contain II and /2 respectively. Hence /' is the only such point. Since the tangents through F are imaginary, /•' is interior to 6'2, and hence all real points on d are exterior.
The tangent at A is parallel to d, and hence by the construction of d perpendicular to a. Since the tangent at any other ordinary point of C2 is not parallel to d, it follows that the line a is the only diameter of (7a which is perpendicular to its conjugate lines. These and other obvious consequences of the definition may be summarized as follows :
THEOREM 25. The axis of a parabola is real and is the only diameter perpendicular to all its conjugate lines. The focus of a parabola is real and lies on the axis. The focus is the unique point at which all pairs of conjugate lines are orthogonal. It is interior to the parabola. The directrix is real, is the polar of the focus, and is perpendicular to the axis. All real points of the directrix are exterior to the parabola. The vertex is real and is the mid-point of the focus and the point in which the directrix meets the axis.
The system of all conies tangent to /x and 12 and to L at Ax forms a range of Type II (§ 47, Vol. I) which consists of all parabolas having F as focus and a as axis. The pairs of tangents to these conies through any real point P of the plane are by the dual of Theorem 20, Chap. V, Vol. I, the pairs of an involution in which PIl is paired with P/2 and PF with PA*. The tangents to the two conies of the range which pass through P are the double lines of this involution and hence separate P/, and P/2 harmonically. Thus we have
TIIKOREM 26. The parabolas with a fixed focus and axis form a range of Type II. The two parabolas of the range which pass through a given point have orthogonal tangents at this point.
The tangent to either parabola through P is therefore normal to the other. Since these two lines separate PFand PA* harmonically, we have
TMK.URKM 27. The tangent and the normal to a parabola at any point are the bisectors of the pair of lines through this point of which one passes through the focus and the other is a diameter.
§81] FOCAL PROPERTIES 195
EXERCISES
1. If P is any point of an ellipse, the normal at P is the interior bisector of 4 I'\Pt'"-2- If P is any point of a hyperbola, the tangent at P is the interior I .i sector of 4-/''1/>-/'V
2. At any nonfocal point in the plane of a conic there is a unique pair of orthogonal lines which are conjugate with regard to the conic. In case of an ellipse or a hyperbola these lines harmonically separate the real foci. In case of a parabola they meet the axis in a pair of points of which the focus is the mid-point.
3. For any point P of an axis of a conic there is a unique point P/ on the same axis such that any line through P is orthogonal to its conjugate line through P'. The pairs of points P and ly are pairs of an involution (called a focal involution) whose double points are the foci of the conic, or, in case of a parabola, the focus and the point at infinity of the axis. If P and P" are on the minor axis, 4 PF-f is a right angle. If the conic is a parabola, F is the mid-point of the pair PP'.
4. Of two confocal central conies having a real point in common, one is an ellipse and the other a hyperbola.
5. The tangents at the points in which a conic is met by a line through a focus meet on the corresponding directrix.
6. If two conies have a focus in common, the poles with regard to the two conies of any line through this focus are collinear with the focus.
7. Let P be any point of a conic, and Q the point in which the tangent at P meets a directrix. If F is the corresponding focus, &PFQ is a right angle.
8. If a circle passes through the two real foci and a poinj P of a conic, it will have the two points in which the tangent and normal at P cut the other axis as extremities of a diameter.
9. If a variable tangent meets two fixed tangents in points P and Q respectively, and F is a focus, the measure of 4 PFQ is constant.
10. Let t1 and tz be two tangents of a central conic meeting in a point 7"; the pair of lines tv TFl is congruent to the pair TF2, tv
11. The line joining the focus to the point of intersection of two tangents to a parabola makes with either tangent the same angle that the other tangent makes with the axis.
12. Let p be a variable tangent of a parabola, and /' a point of p such that the line PF makes a constant angle with p. The locus of P is a tangent to the parabola.
13. The foci of all parabolas inscribed in a triangle lie on a circle.
14. A circle circumscribed to a triangle which is circumscribed to a parabola passes through the focus.
15. The circles circumscribing four triangles whose sides form a complete quadrilateral pass through a point which is the focus of the parabola having the sides of the quadrilateral as tangents.
19f>
CONIC SECTIONS
[CHAP. V
16. I. ft /' be any point coplanar with, but not on an axis of, a conic < "-. lines wliirh are at once |>erpeudicular to and conjugate with regard to C*
t«> tin- lines thniii^h /' arc the tangents of a parabola (the Steiner jmnilinln). The axes of < ** are tangents of this parabola.
17. If /' uiul />- are a pair of one focal involution of a central conic, and Q and (f a {>air of the other, P, P", Q, Q' are on an equilateral hyperbola, which may degenerate into a pair of orthogonal lines.
18. (ii\'-n the ] ..lints of a conic, construct by ruler and compass the center, the axes, the vertices, the foci, and the directrices. Construct the same elements when five tangents are given.
82. Eccentricity of a conic. Let F be a real focus, and d the cor- responding directrix, of a conic C"2 which is not a circle. Let a be the major axis of C'2, arid h the line parallel to d such that if a meets d in a point D, and h in a point If, D is the mid-point of the pair FH. Then d is the vanishing line (§ 43) of the harmonic homology F with F as center and h as axis.
FIG. 67
Since F is a focus, the tangents to C*2 through F pass also through the circular points. Hence the transformation F changes C* into a circle K* with F as center. Now if P is any point of the circle. P1 the point of G'2 to which P is transformed by F, and IX the point in which the line through P' parallel to FD meets d, it follows by Cor. 2, Theorem 21, Chap. Ill, that
Dist (P'F) Dist (PF) Dist (P'J)')~ Dist (FD)'
Since Dist (PF) and Dist (FD) are constants, it follows that
TiiK'MjKM 28. DEFINITION. The ratio of the distances of a point of a conic to a focus and to the corresponding directrix is a constant called the eccentricity.
ECCENTRICITY 197
The conic C*a is a parabola if and only if the circle K* is tangent to d, the vanishing line of F. In this case
Dist (FD) = Dist (PF ),
and hence the eccentricity is unity. The conic 6'a is a hyperbola if and only if K* meets d in real points. In this case
Dist (FD) < Dist (PF),
and hence the eccentricity is greater than one. Applying a like remark to the ellipse we have
THEOREM 29. A conic section is an ellipse, hyperbola, or parabola according as its eccentricity is less than, greater than, or equal to unity.
A circle is said to have eccentricity zero, because if P and F be held constant, and D be moved so as to increase FD without limit, the ratio Dist (PF)/ Dist (FD) approaches zero.
The eccentricity of a hyperbola or an ellipse is evidently the same relatively to either of its real foci, because the two foci and the corresponding directrices are interchangeable by an orthogonal line reflection whose axis is the minor axis of the conic.
As an immediate corollary of the definition of eccentricity we have
THEOREM 30. Two real conies are similar if and only if they have the same eccentricity.
On comparing this theorem with Theorem 20, it is evident that the eccentricity is a function of the cross ratio of the double points of the absolute involution and the points in which the conic meets /.. As an example of this relation we have (by comparison with § 72) the theorem that any two hyperbolas whose asymptotes make equal angles have the same eccentricity. The formula connecting the eccen- tricity of a hyperbola with the angular measure of its asymptotes is given in Ex. 7, below, and the formula for the eccentricity in terms of the cross ratio referred to in Theorem 20 is given in Ex. 9.
Since a real focus of any conic is an interior point, the line through a real focus (e.g. Fz, fig. 64) perpendicular to the principal axis meets the conic in two points, Q1Q2. The number Dist (#,#2) is evidently the same for both foci of an ellipse or hyperbola, and hence is a fixed number for any conic (7a.
198 cnMC SKCTIONS [CHAP.V
Tin- mimlier p= Dist (^//,) is called the parameter, or Itttitx rt-ctnm, of tin- conic C*.
In tin- following exercises e will denote the eccentricity and ;> tin- jwraraeter of any conic. For an ellipse or hyperbola a denotes (o.l ) and c denotes Dist (OF^. For an ellipse b denotes Dist (ORJ. For a hyperbola b denotes Vc2 — a2.
In all cases a radical sign indicates a positive square root.
EXERCISES
1. If P is any point of an ellipse, Dist (/•',/') + Dist (F,P) = 2 a.
2. If /' is any point of a hyi*-rbola. Dist (F,P) - Dist (F,P) = ± 2 a.
3. In :m ellipse Dist (/*,/•',) = « and a- = l>'2 + <•-.
4. l)i8t(.l1/-'I)-Dist.(/.'1,l2) = /A ( 2f>2
5. In an ellipse or hyperbola e — - and p = --
6. In a parabola Dist (AF) = p/\.
7. The measure $ (§ (57) of the pair of asymptotes of a hyperbola is
determined by the equation o
cos 6 — 1 -- - •
e2
8. For an equilateral hyperbola e = v2,
9. The cross ratio R (<-\C2, /t/2) = £2 referred to in Theorem 20 is con- nected with the eccentricity by the relation
k* -4)1-
in case of an ellipse, and by «2 =
1 ~"~ — A? ~f~ rC
in case of a hyperbola.
10. Let ,42 and B2 be the circles with O as center and passing through the vertices A^ and Br respectively, of an ellipse, and let a variable ray making an angle of measure 0 with the ray OA meet these circles in A' and Y respectively. Then the line through Y parallel to OA l meets the line through .Y parallel to OBl in a' point P of the conic. If x and y are the coordinates of P relative to the axes of the conic,
x = a cos 6, y = b sin.0. 6 is called the eccentric anomaly of the point P.
11. Relative to a nonhomogeneous coordinate system in which the prin- ci|>al axis of a conic is the ./--axis, and the tangent at a vertex the ?/-axis, the e< | nation of a parabola, ellipse, and hyperbola, respectively, can be put in the form .» _
§§«-', 83] FOCAL PROPERTIES 199
12. Relative to the asymptotes as axes, the equation of a hyperbola may be written . . ,2
xy =
4
13. Relative to any pair of conjugate diameters as axes, an ellipse has the equation 2 ..2
X- M2
and a hyperbola, — — — = 1.
If A' is a point in which the x-axis meets the conic, Dist (OA") = a'. In the case of an ellipse, if B' is one of the points in which the #-axis meets the conic, Dist (OB') = I/.
14. The measure of the ordered point triads OA'B' is a constant.
15. The numbers a and V satisfy the conditions a"2 + b"2 = a2 + b2 in case of an ellipse and a"2 — I/2 = u~ — IP in case of a hyperbola.
16. The equation of a system of confocal central conies relative to a sys- tem of nonhomogeneous point coordinates in which the axes of the conies are x = 0 and y = 0 is 2 2
x 4. y - \ a»_A + a»-x~
where X is a parameter. In the homogeneous line coordinates such that MjX + u.2y + u0 = 0 gives the condition that the point (x, y) be on the line [MO, uv i/2], the equation of a system of confocals is M^ = (a2 — X) u2 + (1>~ — X)M|.
17. Relative to point coordinates in which the origin is the focus, y = 0 the axis of the parabolas, and x = 0 perpendicular to the axis, the equation of a system of confocal parabolas is
In the corresponding homogeneous line coordinates this is (cf. Ex. 16) ^u|-2«1M0-X(u-2-f «22) = 0.
83. Synoptic remarks on conic sections. An inspection of the literature will convince one that it would not be practical to include a complete list of the known metric theorems on conic sections in a book like this one. The theorems which we have derived, however, are sufficient to indicate how the rest may be obtained either directly as special cases of projective theorems or as consequences of the focal and affine theorems given in this chapter and Chap. III.
The theorems on conic sections have been classified according to the geometries to which they belong. The most general and elemen- tary which we have considered are those which belong to the proper projective geometry (§ 17), the geometry corresponding to the projec- tive group in any space satisfying Assumptions A, E, P. Theorems
200 CONIC SKCTIONS [CHAI-. V
of this class are given in Vol. I, particularly in Chaps. V, V II I, X. A second large class contains those theorems which l>elong to the atline geometry in any proper protective space. These are treated somewhat fully in Chap. III.
The theorems of the class considered in §§ 74, 75 of this chapter belong to the projective geometry of an ordered space. The theorems of § 76 belong to the projective geometry of a real space. Finally, in §§ 80~82 we have been considering theorems of the Euclidean geometry of a real space.
It is quite feasible to make a much finer classification of theorems on conies. This would mean, for example, distinguishing those proper- ties of foci which hold in a parabolic metric geometry in a general space, then those which hold in an ordered space, and then those which are peculiar to the real space.
The theorems which have been under discussion in the remarks above refer in general to figures composed of one conic section and a finite number of points and lines. Theorems regarding more than one conic at a time have not been considered in any considerable number, and the theory of families of conies has not been carried beyond pencils and ranges. For an outline of this subject the reader is referred to the Encyclopadie der Math. Wiss., Ill Cl, §§ 56-90.
EXERCISES
1. The diagonals of the rectangle formed by the tangents at the vertices of an ellipse are conjugate diameters for which a' — //. The angle between this pair of conjugate diameters is less than that between any other pair of conjugate diameters. For this pair of conjugate diameters a'+ b' is a maxi- mum. It is a minimum for a' = a, V = l>.
2. If two orthogonal diameters of a conic meet it in P and Q,
. ... ,
for an ellipse, and
for a hyperr>ola.
3. The locus of a point from which the two tangents to a conic C2 are orthogonal is a real circle in case C2 is an ellipse or a hyperbola for which a> b; is a pair of conjugate imaginary lines through the center and the cir- cular points in case C2 is a hyperbola for which a = 6; is an imaginary rin If in case C* is a hyperbola for which a < It ; is the directrix in case C2 is a parab- ola. The circle thus denned is called the director circle of C2. Construct it by ml«T and compass.
§§*•., *0 FOCAL PROPERTIES 201
4. A variable tangent to a central conic is met by the lines through a focus which make a fixed angle with it in the joints of a circle. In particular, the locus of the foot of a perpendicular from a focus to a tangent is a circle.
5. If I is a variable tangent of a central conic, Dist (/'V) * Dist (Ftf) = lr. If t' is the other tangent parallel to /, Dist (F,f) • Dist (F/) = £/2.
6. If /•' is a focus of a conic and Pv P2 the points of intersection of an arbitrary line through F with the conic,
J_ J_ P^F FI\ is a constant.
7. If the tangent to a conic af. a variable point P meets the axes in two points Tl and 7*2, and the normal at P meets them in N^ and Nv then
Dist (P7\) • Dist (Pr2) = Dist (PN^ - Dist (PNJ = Dist (PFJ • Dist (PF2).
8. There is a unique circle which osculates* a given conic at a given point P. This is called the circle of curvature at P. Its center is called the center of curvature for P and lies on the normal at P.
9. Construct by ruler and compass the center of the circle of curvature at an arbitrary point of a given conic.
10. The circle of curvature of a conic C2 at a point P meets C2 in one and only one other point, Q. The line PQ is the axis and the point P the center of an elation which transforms K2 into C2. The center of curvature is transformed by this elation into the center of the involution on C2 in which the pairs of orthogonal lines at P meet C2.
11. The tangent and normal at any point P of a conic C2 are both tangent to the Steiner parabola (Ex. 16, § 81) determined by this point. The point of contact of the normal with the parabola is the center of the circle of cur- vature of C2 at /', and the point of contact of the tangent with the parabola is the pole of the normal with respect to C2. (For further properties of the circle of curvature, cf. Encyclopadie der Math. Wiss., Ill Cl, § 36.)
12. The polar reciprocal of a circle with respect to a circle having a point 0 as center is a conic having O as a focus. (A set of theorems related to this one will be found in Chap. VIII of the book by J. W. Russell referred to in § 73.)
84. Focal properties of collineations. The focal properties of conic sections are closely related to a set of theorems on collineations some of which are given in the exercises below. A good treat- ment of the subject is to be found in the Collected Papers of H. J. S. Smith, Vol. I, p. 545, and further references in the Encyclopadie der Math. Wiss., Ill AB 5, § 9.
• Cf. § 47, Vol. I.
CONIC SECTIONS [CHAF.V
h-t FI be any real projective collineation which does not leave /. invariant, and let p and q be its vanishing lines; so that II (p) = /. and !!(/.) = q. If /, and /„ are the circular points, let n~1(/1) = JJ, II-' (/,) = /;, 0 (/,) = Qlt n (/„) = Qv By the theorems of § 78 the lines /^/, and 7£/a meet in a real point Alt and JJ/2 and ^/, meet in a real point Af If II (A^ = BV and II (AJ = BZ, it is clear that the complete quadrilateral whose pairs of opposite vertices are /,/,, P^t AtAt is transformed into one whose pairs of opposite vertices are QtQt, /,/2, ^A- The following propositions are now easily verifiable, and are stated as exercises.
EXERCISES
1. A i is such that any ordered pair of lines meeting at Av is transformed by II into a congruent pair of lines. A2 is such that any two lines meeting in At are transformed by II into a symmetric pair of lines. No other points have either of these properties.
2. Every conic having a focus at A^ or A2 goes to a conic with a focus at BI or .B2 respectively.
3. The range of conies having Al and A2 as foci is transformed by II into the range of conies with Bl and Bz as foci ; and this is the only system of confocals which goes into a system of confocals.
4. The pencil of circles with Av A2 as limiting points is transformed by II into that having Bv Bz as limiting points ; and these are the only two pencils of circles homologous under II. The radical axes of the two pencils are the two vanishing lines.
5. If P is any point and II (P) = P', then the ordered point triad A^PA^ is similar (but not directly similar) to the ordered point triad B1P'BV
6. At a point of a Euclidean plane there is in general one and only one pair of perpendicular lines which is transformed into a pair of perpendicular lines by a given affine collineation.
7. In any two projective pencils of lines there is a pair of correspond- ing orthogonal pairs of lines. The line pairs which are homologous with congruent line pairs form an involution.
8. Any projective collineation which does not leave /OP invariant is express- ible as a product of a displacement and a homology.
85. Homogeneous quadratic equations in three variables. Revers- ing the process which is common in analytic geometry, it is possible to derive certain classes of algebraic theorems from the theory of conic sections. We shall illustrate this process in a few important cases and leave the development of further algebraic applications to the reader.
§*-,]
ALGEBRAIC THEORY
203
The general homogeneous equation of the second degree can be written in the form
a x3 4- a xx -4-
(6) + a^a
where ay = aj(. Let us first suppose that
(7) A;
In § 98, Vol. I, it has been shown, from the point of view of general protective geometry, that every protective polarity is .represented by a bilinear equation of the form
(8)
-f- a^X^X^ -f- tt^XpK^ -f- d^X^X^ :=: «»
where afj = a^ and where A
0.
It was also shown that every bilinear equation of this form, subject to the condition A =£ 0, represents a polarity ; that the equation in point coordinates of the fundamental conic of the polarity is (6), which is obtained from (8) by setting x[ = xi ; and that the equation of this conic in line coordinates is
(9)
AiJuiuJ=Q,
where Atj is the cof actor of a;j in A.
The coefficients a{J are elements of the geometric number system. Therefore in the case of the real plane they are real numbers, and we have
THEOREM 31. Every equation of the form (6) with real coefficients such that aij= ajf and A =£ 0 represents a conic whose polar system transforms real points into real lines. Conversely, every conic with regard to which real points have real polars has an equation of the form (6) with real coefficients such that au = aj. and A 3= 0.
HH CONIC SKCTIONS [CHAP.V
In § 79 we have seen that any conic having a real polar system is in one of two classes, and that any two conies of the same class are projectively equivalent. Now it is obvious that
(10) *«+
is the equation of an imaginary conic, and that
(11) x*+x*-x*=Q
is the equation of a real conic. Hence we have
THEOREM 32. Any quadratic equation in three homogeneous vari- ables whose discriminant A does not vanish is reducible by real linear homogeneous transformation of the variables to the form (10) or to the form (11).
Algebraic criteria to determine whether a given conic C2 whose equation is in the form (6) belongs to one or the other of these classes may easily be determined by the aid of simple geometric considera- tions. In case C2 contains no real points, the line XQ= 0 has no real point in common with it, and the point wt= 0 (which is on the line x0= 0) is on no real tangent to it. On the other hand, if the line x0 = 0 contained no real point of C2, and C2 were real, this line would consist entirely of exterior points, and hence there would be a tangent to C'2 through the point 7^=0. Hence a pair of necessary and sufficient conditions that C2 contain no real points are (1) xo= 0 is on no point of C,2 and (2) Wj= 0 is on no tangent of C2.
Substituting #0= 0 and x'0 = 0 in (8), we have the equation of an involution
(12) Wi + W*
+ «,,«,«! + aMaV*J = 0,
which, by § 4, is elliptic if and only if Aw > 0. By a dual argument applied to (9), the necessary and sufficient condition that there be no real tangents to C2 through the point wt= 0 is
(13)
^00 ^02
By a well-known theorem on determinants .(or a simple computation)
this reduces to
«.. • A > 0.
§85] ALGEBRAIC THEORY 205
Hence we have
THEOREM 33. The imaginary conies are tJwse for which Aw > 0 and an-A> 0,
and the real ones are thvse for which not both of these conditions are satisfied and for which A 3= 0.
In these conditions it is obvious that Am and an may be replaced by A(i and a-, where i,j= 0, 1, 2, provided that i =t=j.
Let us now investigate the cases where A = 0, and first the case in which not all the cofactors AM, An, AN are zero. To fix the notation, suppose that Aw=£ 0. Then the bilinear equation (8) is satisfied
ty* ico=-^oo' x\ = Ai' x* = ^02' 110 matter what values are taken by x'0t x[> xr Hence in this case (8) determines a transformation, F, of all the points (x'0, x{, ,v!2) distinct from (Am, Aol, A^) into lines through (Aw, AQl, J02). A collineation which transforms ( A^, AOI, AM) to (1, 0, 0) must reduce (8) to
*n J12
It is to be noted that
because if this determinant vanished, F would transform all points (x'0, x{, x'z) into a single line, and hence AW would vanish. Hence F transforms any point (x'0, x[, x!2) into the line paired in a certain invo- lution with the line joining (x^, x{, x^) to (A^, Aol, Am). The double lines of the involution must satisfy the quadratic equation (6).
Comparing with the definitions in § 45, Vol. I, we have that when A=0 and not all the cofactors A^, Au, A^ are zero, (6) represents a degenerate conic consisting of two distinct lines and that (8) represents the polar system of the conic. Since the lines represented by (6) are the double lines of a real involution, they are either real or a pair of conjugate imaginaries. In the first case (6) can evidently be trans- formed by a collineation to
(15) *«-** = (), and in the second case to
(16) x* + x* = 0.
CONIC SECTIONS
[('HAP.V
The criteria to distinguish the two cases may be found by considering tlie intersection with (6) of a line #,= 0. This yields imaginary points (just as in the nondegenerate case) if and only if Ati> 0, and real iH.ints if and only if Aa S 0. Hence the case where (6) represents a pair of real lines occurs if and only if ^,,- = 0, for i= 0, 1, 2.
Finally, suppose that Aw= An = An =A = 0. In view of the identity,
(17) AaAjf-A*, = att • A, (i *j*k* i)
this implies that all the cofactors A^ are zero, and hence that (8) represents the same line, no matter what values are substituted for x'0, x(, x'r Hence (6) represents a single real line (i.e. two coincident real lines), and the polar system (8) transforms all points not on this line into this line. If this line be- transformed to 2^= 0, (6) obviously becomes
(18) xl = 0.
A degenerate point conic is two distinct or coincident lines. These may always be represented by a quadratic equation which is a product of two linear ones. For such a quadratic A = Q, because if A =£ 0, the equation has been seen to represent a nondegeuerate conic. Hence the theory of degenerate point conies is equivalent to that of homo- geneous quadratic equations for which ^4=0.
The complete projective classification of conies, degenerate or not, may now be stated as an algebraic theorem in the form :
THEOREM 34. Any homogeneous quadratic equation in three vari- ables may be reduced by a real linear homogeneous transformation,
(19)
to one of the normal forms (10), (11), (16), (15), (18). The criteria which determine to which one of these forms an equation (6) is reducible may be summarized in the following table :
|
lM MilNAICV ' '>MC |
HEAL, CONU |
lMA<iINARY I.IN i: PAIR |
REAL LINE PAIR |
COINCIDENT REAL LINE PAIR |
|
"i i . 1 > 0 |
aiiA =s 0 |
^oo >0 |
4oo <0 |
-4oo = 0 |
|
or AH > 0 |
-or An <0 |
An = 0 |
||
|
/loo > 0 |
or AM s 0 |
or yl22 > 0 |
or An < 0 |
^22 = 0 |
§85] ALGEBRAIC THEORY 207
Since the algebraic expressions in the above criteria determine conditions on the conic which are independent of the choice of coordinates and thus are invariant under the projective group, it is natural to inquire whether they are algebraic invariants in the sense of § 90, Vol. I. A direct substitution will readily verify that .1 is a relative invariant of (0).
Suppose we regard the coefficients of (6) as homogeneous coordinates (aoo» aii> a22» aoi' aio> aia) °f a point in a five-dimensional space. Then .4=0 determines a certain cubic locus in this space the points on which represent degenerate conies. Now if there were any other invariant of (6) under the projective group, say <f> (fly), the equation <£ (a^-) = 0 would represent a locus in this five-dimensional space. But since each nondegenerate conic is projec- tively equivalent to every other nondegenerate conic, this locus would have to be contained in the locus of A = 0. From this it can be proved, by the general theory of loci represented by algebraic equations, that the locus of <f> (o^-) = 0 coincides with that of A = 0, and that hence <fr (a^-) is rationally expressible in terms of A. Thus A is essentially the only invariant of (6) under the projective group.
The question, however, arises whether there are not other rational func- tions of the coefficients of (6) which are invariant whenever .4 = 0. If there were such a function, say <£ («„•), the conies for which <f> (a^-) = 0 would be a subclass of the degenerate conies which is transformed into itself by all complex projective collineations. The only class of this sort consists of the coincident line pairs which are given by two conditions, AM = 0, An = 0. In view of the theorem that a locus represented by two inde- pendent algebraic equations cannot be the complete locus of a single algebraic equation, this shows that there is no other invariant of (6) even for the cases in which ^4 = 0.
This reasoning could be expressed still more briefly by saying that, while the set of all conies is a five-parameter family, and the set of degenerate conies a four-parameter family given by one condition, the only invariant subset of the degenerate conies is the two-parameter set of coincident line pairs which have to be given by two conditions and so cannot correspond to a single invariant in addition to A.
EXERCISES
1. In case A = 0, the lines represented by (6) intersect in the point ~^AIV "^^22)' unless the three cof actors A ,-,- vanish, in which case (6) represents the coincident line pair
2. In case (6) represents a pair of distinct lines, (9) represents their point of intersection counted twice. In case (6) represents a |>air of coincident lines, Av = 0 (f,/ = 0, 1, 2).
m Mr SECTIONS
[CHAP.V
86. Nonhomogeneous quadratic equations in two variables. The affine theory of point conies corresponds to the theory of
(20)
where the ay's satisfy the same conditions as in the last section. The theorem that any nondegeuerate conic is an imaginary ellipse, real ellipse, hyperbola, or parabola, and that any two conies of the same class are equivalent under the affine group, translates into the following : Any quadratic equation in two variables, for which A =£ 0? is transformable by a transformation of the form
(21)
x' = a^x + bty +
y =
a. b.
into one of the following four forms : (22) (23) (24) (25)
To know this it is merely necessary to observe that these equations represent conies of the four types respectively.
The criteria to determine in which class a given conic C2 belongs may be inferred from the discussion in the last section if we set * = x1/x0 and y = xJxQ. It is then evident that Aw > 0 for an ellij>s.-, AW = 0 for a parabola, and AM < 0 for a hyperbola. Hence the attine classification of cases where A =£ 0 may be summarized in the following table:
A jtO
|
IMACINAKV Ki.i.ir>i. |
REAL ELLIPSE |
HYPKKHOLA |
PAKAHOI.A |
|
^oo>0 "n-l>0 |
AQO >0 filial =0 |
^oo <0 |
^4oo = 0 |
The cases where ^4 = 0 correspond, as in the last section, to degener- ate conies. Geometrically the types of figures are obvious, and to obtain the algebraic criteria we need only combine with considera- tions already adduced, the observation that when Aw = 0 and either = 0, then a = a = 0.
au = 0 or
580]
ALGEBRAIC THEORY
209
|
CONJUGATE IMAGINARY LINES |
DISTINCT REAL LINK- |
COINCIDENT REAL. LINES |
||||
|
Concurrent Ht ordinary point |
Parallel pair |
Concurrent at ordinary point |
Parallel pair |
One at infinity |
Ordinary |
At infinity |
|
4oo>0 |
;loo = 0, |
AM<Q |
-4 00 |
= 0, |
•^00 = -^11 |
= ^4,2 = 0; |
|
Au >0 |
An<Q |
|||||
|
or |
or |
|||||
|
^22 >0 |
^«<0; |
|||||
|
anj±0 |
On 7*0 |
|||||
|
or |
011 = 082 = 0 |
or |
011 = 022 — 0 |
|||
|
flfsa * 0 |
«22 *0 |
As normal forms for the first six cases we may take
(26) *2 + /=0,
(27) a*+l =0,
(28) **-y2=0,
(29) x2-! =0,
(30) x=Q,
(31) ^2=0.
The case of coincident real lines at infinity does not correspond to any equation in nonhomogeneous coordinates.
Summarizing these results we have the following algebraic theorem :
THEOREM 35. Any quadratic equation in two variables may be reduced to one and only one of the normal forms (22)-(31) by a transformation of the form (21). The normal form to which it is reducible is determined by the criteria in the two tables above.
The question of invariants of (20) under the affine group may be investigated in the manner indicated for the corresponding protective problem in the fine print at the end of the last section. The results of such an investigation are given in the exercises below.
There are no absolute invariants of conies under the protective and affine groups, because two conies would fail to be equivalent under the one group or the other if they determined different values of an absolute invariant, and this would contradict the fact that there are only a finite number of conies distinct under the affine group.
210 COMC SKCTIONS LCHAP.V
EXERCISES
1. A and AM are invariants of (20) under the affine group.
2. In case .-I = -4^, = 0, .4u/«22 and AM/an are invariants of (20) under the affine group.
3. The homogeneous coordinates of the center of (20) are (Aw, Aol, Aoz).
A A
4. If ^oo ^ 0, the translation x = x — — ^ » y = y — ^-^ transforms (20) into
a.,x2 + 2 a,~rli + a,,w2 + -— = 0.
5. If A 5* 0 and Aw 5* 0, the asymptotes of (20) are given by the equation
anz2 + 2 al2xy + a22#2 = 0.
6. Any diameter of a parabola is parallel to anx + a12y = 0 and to
87. Euclidean classification of point conies. With respect to a non- homogeneous coordinate system in which the pair of lines x = 0 and y = 0 is orthogonal and bisected by the lines x = y and x = — y, the transformations of the Euclidean group take the form (21) subject to the conditions
(32) a* + a* = ll + bl, «A + «-A=0,
and the displacements are subject to the additional condition
(33)
-i
Since any ellipse or hyperbola is congruent to one whose principal axes are x = 0 and y = 0, and since any parabola is congruent to a parabola with the origin as vertex and y = 0 as its principal axis, it follows that any conic is congruent to a conic having one of the following equations :
T2 W2
(34) £. + |- + 1 = 0,
(35) f! + |!_i=o,
(36) ^_^_i=0,
/OT\ 2 f\
(oi) y — px = 0.
The normal forms to which degenerate point conies can be reduced by displacements are evident when one recalls that two pairs of non- parallel lines are congruent when they have the same cross ratio with
§87] ALGEBRAIC THEOKY 211
the circular points and that two pairs of parallel lines are congruent if the lines of each pair are the same distance apart.* By comparison with the second table (A = 0) in § 86 we find
(38) ^+y2=0'
(39) #2+c2=0,
(40) $-y*=Q>
(41) a?-c2=Q,
(42) £=0,
(43) y? = 0.
The group of displacements is extended to the group of similarity transformations by adjoining transformations of the form
(44) k * 0.
y' = %,
Transformations of this sort will reduce the equations (34) — (43) to normal forms in which b, c, and p are all unity.
The criteria for determining to which of these normal forms a conic is reducible under the group of displacements or that of simi- larity transformations are the same as those already found for the affine group. Two conies whose equations can be reduced to the same normal form are evidently equivalent under the group of displace- ments if and only if they determine the same values for a and b or c or p, and under the Euclidean group if they determine the same value for a. The numbers a, b, c, p are evidently absolute invariants of the corresponding conies under the group of displacements, and a in (38) and (40) also under the Euclidean group.
The problem of determining a, b, c, p in terms of the coefficients of (20) presents no special difficulty, and will be left to the reader to be considered in connection with the exercises below and those at the end of the next section.
When b, c, p are all unity, a is a function of the eccentricity given by the equations in Exs. 7 and 9, § 82. The same reference gives the con- nection between the eccentricity and the invariant v^T ~A^/(an + a.^).
•The distance apart is the distance of an arbitrary point on one of the parallel lines from the other line. The formula for distance is applied to the case of a pair of conjugate imaginary lines as explained in § 70.
CONIC SKCTIoNS [CHAI-. \
EXERCISES
•
1. If A * 0 and Aw ^ 0, the angular measure of the asymptotes is 0, where
Moreover, 6 = - ^ log B ( C\ Cv IJJ,
where f| and C, are the points in which the conic meets /», and 7t and 72 are the circular points. If A = 0 and ^400 ^ 0, these formulas give the angular measure of the lines represented by (20). Derive from this the formula for o in (M) and (40) in terms of the coefficients of (20).
2. .loo and au -f «22 are absolute invariants of (20) under the group of displacements, and V— .I00/(nu + «22) under the Euclidean group. If A * 0 and nu + an = 0, (20) represents an equilateral hyperbola; if A = 0 and aii + an ~ 0» it represents a pair of orthogonal lines or /. and an ordinary line.
3. If A * 0 and AM * 0, the axes of (20) are
°i2 (** + F) + ("22 - a\\) *y = °»
where x and y are defined as in Ex. 4, § 86. < - — - . - —
4. For an ellipse the constants a and b are -%/- - — and -%/- — , where Xt
* •" 00^1 * •"• 00^2
and for a hyperbola a and ib are -\— -- and "\rr-r-' The discriminant of (45) is (au - a22)* + 4 a\,.
5. If A ^ 0 and AM= 0, the parabola (20) touches /« at (0, o12, — a^^ which is the same as (0, a22, — a12). The axis is
(46) a
88. Classification of line conies. The protective classification of line conies is entirely dual to that of point conies and so need not be con- sidered separately. The affine classification, however, corresponds to a new algebraic problem. If the line coordinates are chosen so that
1^+1^+1^=0
is the condition that the point (#0, xlt #2) be on the line [UQ, ult u^\, the point coordinates being the same as already used, we have the problem of reducing equations of the form (9) to normal forms by means of transformations of the form
(47) u[=
U= —
a. b
i i
*«. b.
ALGEBRAIC THEORY
213
These are the transformations which leave the line [1, 0, 0] invariant. If
and c?» =
c. a.
(47) is the same collineation as (21).
The affine classification of nondegeiierate line conies is of course the same as that of nondegenerate point conies. To express the criteria in terms of the equation (9) regarded as given primarily,* let us write
(48)
cc =
where the A^'s are the coefficients of (9), and let aif denote the
cofactor of AfJ in a. The point conic associated with (9) must have the equation
(49) 2X^.= 0.
By the criteria already worked out, this is an ellipse, hyperbola, or parabola according as the value of
is greater than, less than, or equal to zero; and, in the case of an ellipse, real or imaginary according as an > 0 or «u s 0. Thus we have
a 5* 0
|
IMAGINARY ELLIPSE |
REAL ELLIPSE |
HYPERBOLA |
PARABOLA |
|
a • .loo > 0 au>0 |
a-AM> 0 an ^0 |
a.^oo<0 |
^loo = 0 |
The normal forms for these four classes are respectively
(50) < + < + «• = <),
(51) un--ul-u'i = Q,
(52) < - ul + ul = 0,
(53) < - u2 = 0.
The projective classification of degenerate line conies is dual to that of degenerate point conies, and therefore yields the following three cases: (1) two distinct real points, a = 0, a,,. S 0, one at least
•Instead of in terms of the coefficients of (6).
214
CONIC SECTIONS
[CRAP, v
of (i^, an, «n being different from zero; (2) coincident real points, a = an = «„= «„ = 0 ; (3) conjugate imaginary points, a = 0, aw > 0 1« >r at least one value of t.
For the affiue classification let us observe that since [1, 0, 0] is the line at infinity, the condition that at least one factor of (9) represent a point at infinity is Aw= 0. The following criteria are . now evident.
|
CONJUGATE iMAiiiNAuv POINTS |
DISTINCT REAL POINTS |
COHtCrDCXT KKAI, POINTS |
||||
|
Ordinary |
At infinity |
Both ordinary |
One ordinary |
Both at iiiliniiy |
Ordinary |
At infinity |
|
an>0 |
aoo>0 |
1 |
<Too<0 |
«00 = au = 0-22 = 0 |
||
|
or |
or |
an = 0 |
||||
|
an >0 |
an = aii=0 |
a22 |
<0 |
«22 = 0 |
AM ^0 |
^oo = 0 |
The normal forms for these cases are respectively
(54) ttt + M« = of
(55) ut + < = 0,
(56) < - uf = 0,
(57) V! = 0,
(58) u^ = 0,
(59) < = 0,
(60) uf = 0.
EXERCISES 1. The two pairs of foci of (0) are the degenerate conies of the range
+ -I 2,,"-",, + -'--.".j"! + (-'-.a - P) "I = 0S
which are given by the values of p satisfying
(62> AooP* ~ («ii + 022) P + o = 0.
The discriminant of this quadratic is (a,, — a^)2 +
§§»*,«»] POLAR SYSTEMS iM-'>
2. In case o = 0 and A^ ^ 0, the distance between the points represented
3. The normal forms for line conies under the group of displacements are
(63) u2 + a2u* + A»u| = 0,
(64) «02 - a*u« - **«! = 0,
(65) u2 - aX + 6*«J = 0,
(66) 4 uottj + />«2 = 0,
(67) «2 + *»i/f = 0,
(68) «2 + c*«| = 0,
(69) ti« - *X = 0,
(70) «0«i = 0,
(71) «f - c2u| = 0,
(72) «08 = 0,
(73) u,2 = 0.
Here a, b, p have the same significance as in (34) -(37); 2 ki is the distance between the two points represented by (67) ; 2 k is the distance between the two points represented by (69) ; c is expressible in terms of the cross ratio of the circular points and the two points represented by (68) or (71).
* 89. Polar systems. The theorems on the classification of conies (§ 79) may be regarded as completing the discussion of protective polar systems in a real plane. There is, however, a certain amount of inter- est in making the discussion of polar systems without the intervention of complex elements, and basing it entirely on the most elementary theorems about order relations. This treatment will hold good for a projective space satisfying Assumptions A, E, S, P.
THEOREM 36. In any projective polar system in an ordered plane the involutions of conjugate points on the sides of a self-polar triangle are all direct, or else one involution is direct and the other two opposite.
Proof. Let ABC be the self-polar triangle (fig. 68), and let PP' be a pair of points on the side BC and QQ' a pair on the side CA. Let R be the point of intersection of the lines PQ and AB, O that of AP' and BQ', and R' that of CO and AB. Then AP' is the polar of P, BQ> of Q, PQ of 0, and CO of R. Hence R and R' are paired in the involution of conjugate points on AB. Let R" be the point in which P'Q' meets AB ; R" is the harmonic conjugate of R' with respect to A and B.
216 CONIC SECTIONS [CHA, v
If the involutions on BC and CA are direct, P and P' separate />' and C, and (J and </ separate C and A It follows by Theorem 1!», Chap. II, that K and R" do not separate B and A Hence by Theorems 7 and 8, Chap. II, R' is separated from R by ,4 and B, and hence the involution on the line AB is direct.
On the other hand, if the involutions on BC and CA are not direct, P and P1 do not separate B and 6', and Q and </ do not separate C and J. Hence R and .K" do not, and therefore R and R' do, separate .1 and B. Hence again the third involution is direct.
We have thus shown that at least one of the three involutions is direct ; and that if two are direct, so is also the third. From this the statement in the theorem follows.
The reasoning above is valid in any ordered projective space. Specializing to the real space, we have
COROLLARY 1. The involutions on the sides of a self-polar triangle of a projective polar system in a real plane are all three elliptic, or else two are hyperbolic and the third is elliptic.
THEOREM 37. If the involutions of conjugate points on the sides of one self -polar triangle of a projective polar system in an ordered plane are direct, the involution of conjugate points on any line is direct.
Proof. Let the given self-polar triangle on the sides of which the involutions of conjugate points are direct be ABC. The theorem will
§«>] POLAR SYSTEMS 217
follow if we can prove that the involution of conjugate points on any line through a vertex of such a triangle is direct. For any line / meets BC in a point M which has a conjugate point N on EC. By the prop- osition \vlrich we are supposing proved, the involutions on the sides of the self-polar triangle, A My, are direct; and by a second application of the same proposition, the involution of conjugate points on / is direct. Thus the proof of the theorem reduces to the proof that the involution of conjugate points on any line through A is direct.
Let such a line meet BC in a point P', and let P be the conjugate of P' in the involution on BC. Let Q and Q' be a conjugate pair distinct from A and C on the line A C, and let 0, R, R', R" have the same meaning as in the proof of the last theorem (tig. 68). Also let CX be the conjugate of 0 on the line AP', i.e. let Cf be the intersec- tion of AP' with PQ. Applying Theorem 19, Chap. II, to the triangle ABP' and the lines O'R and OR1, it follows that, since C and P do not separate B and P', and R and R' do separate A and B, 0 and O1 are separated by A and P'. Hence the involution of conjugate points on the line AP' is direct.
COROLLARY 1. If the involutions on two sides of a self -polar triangle of a polar system in an ordered plane are opposite, then two of the involutions on the sides of any self-polar triangle are opposite and the third is direct.
Proof. If there were any self-polar triangle not satisfying the con- clusion of the theorem, this would, by Theorem 36, be one for which all three involutions were direct. By Theorem 37 it would follow that the involutions on all lines were direct, contrary to hypothesis.
The propositions stated in the last two theorems and in the last corollary may evidently be condensed into the following :
COROLLARY 2. Any protective polar system in an ordered plane is either such that the involution of conjugate points on any line is direct, or such that on the sides of any self-polar triangle two of the involu- tions are opposite and the third direct.
Applying this result in a real plane, we have that every projective polar system is either such that all involutions of conjugate points are elliptic, or such that on the sides of any self-polar triangle two involutions are hyperbolic and the third elliptic. In the latter case let ABC be a self-polar triangle, AB and AC being the sides upon
218 CONIC SECTIONS [CHAP.V
which the involutions are hyperbolic. Let the double points of the involution on AH be Cl and Ct, and those of the involution on A C be /.' and It . The polar of Cl is then the line C^C. The conic section A'* through (',, C\2, /*,, #a and tangent to the line C^C at Cl has a polar svst.-iii in which A/>C is a self-polar triangle, and in which the given involutions are involutions of conjugate points. By § 93, Vol. I, these conditions are sufficient to determine a polarity. Hence the given polarity is the polar system of K2. Thus we have
THEOREM 38. DEFINITION. A protective polar system in a real plane if eithtr the polar system of a real conic, or such that the involution of conjugate points on any line is elliptic. A polar system of the latter type is said to be elliptic.
The existence of elliptic polar systems is easily seen as follows: Let ABC be any triangle, O any point not on a side of this triangle, P1 the point of intersection of OA with BC, Q' the point of intersec- tion of OB with CA, and P and Q any two points separated from P' and Q' by the pairs BC and CA respectively. By the theorems in § 93, Vol. I, there exists a polar system in which the triangle ABC is self-polar and the point 0 is the pole of the line PQ, and by the theorems in the present section this polar system is elliptic.
CHAPTER VI
INVERSION GEOMETRY AND RELATED TOPICS*
90. Vectors and complex numbers. The properties of the addition of vectors have been derived in § 42 from those of the group of translations. If the operation of multiplication is to satisfy the dis- tributive law,
a (b -\- c) = ao + <w,
multiplication by a vector, «, must effect a transformation on the vector field such that b + c is carried into the vector which is the sum of those to which b and c are carried. Since the group of trans- lations is a self-conjugate subgroup of the Euclidean group, any similarity transformation of the vector field satisfies this condition.
Let us then consider the transformations effected on a vector field by the Euclidean group. Any similarity transformation is a product of a translation by a similarity transformation leaving an arbitrary point 0 invariant. But a translation carries every vector into itself. Hence any similarity transformation has the same effect on the field of vectors as a similarity transformation leaving 0 invariant. Hence the totality of transformations effected on the vector field by the Euclidean group is identical with the totality of transformations effected on it by the similarity transformations leaving 0 invariant. Since no such transformation changes every vector into itself, any two of them effect different transformations of the field of vectors. Hence we have
THEOREM 1. The group of transformations effected by the Euclidean group in a plane upon the Jicld of vectors is isomorphic with the group of similarity transformations leaving an arbitrary point invariant.
To obtain a definition of multiplication we restrict attention to the group of direct similarity transformations and make use of the fact that if OA and OB are any two nonzero vectors, there is one and but
» The main part of Chap. VII is independent of this chapter. The two chapters may therefore be taken up in reverse order if the reader so desires.
219
INN KRSION GEOMETRY [CHAP. VI
one transformation of this group carrying the points 0 and A to 0 and B respectively.
liKKiMTK'N. Kfliitive to an arbitrary vector OA, which is called the unit vector, the product of two vectors OX (where X ^ 0) and OY is the vector OZ to which OF is carried by the direct similarity trans- formation carrying OA to OX, and is denoted by OX • OY. In case X=(), OX'OY denotes the zero vector.
As obvious corollaries of this definition we have the following two theorems : .
THEOREM 2. The triad of points OA Y is directly similar to the triad OXZ if and only if
OZ=OX-OY.
THEOREM 3. The equation
OZ=OX-OY FIG. 69
is satisfied if and only if £AOX+AAOY=4.AOZ and Dist (OZ} = Dist (OX) -Dist (OY), the unit of distance being OA.
Since the direct similarity transformations leaving a point 0 invari- ant form a group, the operation of multiplication must be associative, i.e.
OX • (OY • OZ) = (OX- OY) • OZ,
and also such that there is a unique inverse for every vector OB for which 0 3= B, i.e. there must be a vector 0 Y such that
OB-OY=OA.
The group of direct similarity transformations leaving O invariant is commutative because it consists of the rotations about O (which form a commutative group by § 58) combined with dilations with 0 as center. Hence the operation of multiplication is commutative, i.e.
OX-OY=OY-OX.
The fact that the group of translations is self-conjugate under the group of displacements translates into the distributive law,
OX-(OY-\- OZ) = OX- OY+ OX- OZ.
Recalling the definition of a number system given in Chap. VI, VoL I, we may summarize these results by saying,
§90] VKCTOIIS AND COMPLEX M'Mr,i:i;s 221
THEOREM 4. With respect to the operation of addition described in § 42 and of multiplication defined in this section, a planar vector field is a commutative number system.
In proving this theorem we have made use of no properties of the Euclidean yroup except such as hold for any parabolic metric geometry for which the absolute involution is elliptic. In case the absolute involution were hyper- bolic, exceptions would have to be made corresponding to properties of the minimal lines.
The definition of multiplication of vectors as given here does not conflict with the notion of the ratio of colliuear vectors as developed in Chap. III. For the quotient of two collinear vectors is a vector collinear with the unit vector OA, and the system of vectors collin- ear with OA constitutes a number system isomorphic with the real number system. Thus, if we denote the unit vector by 1, any vector OA' collinear with it may be denoted by
x\,
where, according to the definition of § 43, a? is a real number and where, according to our present definition, x denotes OX itself.
Let us denote a vector OB such that the line OB is perpendicular to the line OA and such that Dist (OB) = Dist (OA), by /. Then by the definition of multiplication,
;« = - 1.
Any vector collinear with I is expressible in the form xi, where a; is a vector parallel to 1, and by Theorem 8, Chap. Ill, any vector whatever is expressible uniquely in the form
«1 + bi
The product of two vectors may be reduced by the associative, distributive, and commutative laws as follows :
(al + bi) (cl + di) = (rtl + li) cl + (al + bi) di = (uc — Id) 1 + (be + ad) L
By comparison with §§3 and 14 this shows that
THEOREM 5. A planar Jield of vectors is a member system isomor- phic with the complex number system, i.e. the geometric number system of a complex line.
23J INVKKSloN (JKO.MKTKV [CHAP. VI
The isomorphism in question is that by which the complex number o + bi corresponds to the vector al-f bi Supposing that the funda- mental ]H)ints of tlie scale on the complex line are 7J, /;, /;, this means that there is a correspondence between the complex line ami the Km-lidean plane in which JFJ corresponds to 0, Pl to A, and every point whose coordinate relative to the scale P0, Plt P*, is
a + bi
corresponds to the point Q of the Euclidean plane such that
OQ = al + bi
One obvious property of this correspondence which we shall have to use later is that the points of the complex line which have real coordinates relative to the scale PQ, Plf P^ correspond to the points of the line OA, or, in other words, that the points of the chain* C (P^P*), other than P*, correspond to the points on the real line OA.
Theorem 5 may be made the basis of a method for the investigation of theorems of Euclidean geometry, particularly those relating to n-lines and circles. The complex numbers may be regarded as the coordinates of the points of the Euclidean plane and many interesting theorems obtained by interpreting simple algebraic equations. Compare the articles by F. Morley, Transactions of the American Mathematical Society, Vol. I, p. 97; Vol. IV, p. 1 ; Vol. V, p. 467 ; Vol. VIII, p. 14.
The whole subject is closely related to certain elementary parts of the theory of functions of a complex variable. Cf. an article by F. N. Cole, Annals of Mathematics, 1st Series, Vol. V (1890), p. 121.
91. Correspondence between the complex line and the real Euclidean plane. The operation of addition of vectors has been so defined that
OX' = OX+ OP,
where 0 and P are fixed and X and X' variable points, may be taken as representing a translation carrying X to X'. The operation of multiplication has been defined so that
OX' = OP • OX
may be taken to represent a direct similarity transformation carrying O into itself and X to X'. Thus the general direct similarity trans- formation may be written
OX' = OP-OX+OQ.
• Cf . §11. The reader who has omitted the starred sections in Chap. I may take a chain C (P0P,P.) an by definition consisting of those points of a complex line which have real coordinates relative to the scale P0, P,, P«.
§91]
COMPLEX LINK AND EUCLIDEAN PLANE
223
The last theorem may therefore be stated in the following fonu :
THEOREM 6. Let (>0, Qiy Q* be three arbitrary points of a complex protective line I, and let P, and Pl be two arbitrary points of a Euclid- ean plane TT in whose line at infinity lx an elliptic absolute involution is given. There exists a one-to-one and reciprocal correspondence F in which P0 corresponds to Q0, Pl to Qlt lx to Qx, and every ordinary point of TT to a point of I distinct from Qx. This correspondence is such that to every protective transformation of I leaving Q*> invariant, i.e. to every transformation of the form
(1) x' = ax + b, a =£ 0,
there corresponds a direct similarity transformation of TT, and conversely.
The question immediately arises, What group of transformations of TT corresponds to the general protective group on I, i.e. to the set of transformations
b
ax
ex -
d
The transformation of TT corresponding to (3) x' = l/x
must change any point P to a point P' such that
PP' • PP — PP Jo* *** — fkfv.
Hence, by Theorem 3, 4-PP^ is congruent to AP^P'. Therefore the orthogonal line reflection with P^ as axis must carry P to a point P" of the line P0P'. If P be re- garded as a variable point of a line through P0, it follows that the correspond- ence between P' and P" is projective. In this correspondence P0 corresponds to the point at infinity of the line P0P', and each of the points in which this line meets the circle through P^ with P0 as center corre- sponds to itself. Hence the correspond- ence between P' and P" on a given line through P0 is an involution, and P' and P" are conjugate points with respect to the circle. Hence (§ 71), if P be a variable point of the plane, the correspondence between P1 and P" is an inversion. Hence
Ki... 70
1NVKKSION (iKo.METRY [CHA,-. vi
the transformation of TT corresponding to a/ = \/x is the product of the orthogonal liin- n-llrrtion with l'J\ as axis and the inversion H-ith respect to the circle through /,' with J* as center.
NII\V any transformation (2) is evidently (cf. § 54, VoL I) a product of transformations of the forms (1) and (3). But the transformation (1) has been seen to correspond to a direct similarity transformation, Le. to a product of a dilation and a displacement. A displacement has been proved in Chap. IV to be a product of two orthogonal line reflections ; and a dilation will now be shown to be a product of two or four inversions and orthogonal line reflections.
For consider a dilation A with a point 0 as center and carrying a point A to a point B. If 0 is not between A and B, there exists (Theorem 8, Chap. V) a pair of points C^C^ which separate A and B harmonically and have 0 as mid-point. Let ll be the inversion with respect to the circle with 0 as center and passing through C^. The transformation IjA leaves invariant all points of the circle through A with 0 as center, and effects a projectivity on each line through 0 which interchanges 0 and the point at infinity. The projectivity on each line through 0 is therefore the involution carrying each point to a conjugate point with regard to the circle through A with O as center. Hence ItA is an inversion, I2, with respect to this circle. From IjA = I2 follows A = l^0. If 0 is between A and B, let A be the point reflection with O as center. The product AA is a dilation such that 0 is not between A and AA (A). Hence AA is a product of two inversions Ix, I2 and A = AI2Ir Since A is a product of two orthogonal line reflections, A is a product of four inversions and orthogonal line reflections.
Hence any protective transformation of a complex line I corresponds under T to a transformation of a real Euclidean plane TT •//•// ir/t is a product of an even number of inversions and orthogonal line reflections.
The converse of this proposition is also valid. In order to prove it we need only verify (a) that the product of two orthogonal line reflec- tions in TT corresponds to a projectivity of /, (#) that the product of an orthogonal line reflection A and an inversion P of TT corresponds to a projectivity of I, and (7) that the product of two inversions P^ of TT corresponds to a projectivity of /. The first -of these statements is a corollary of Theorem 6.
J§yi,i«] COMPLEX LINE AND EUCLIDEAN PLANE 225
To prove (£) let us first consider the case where the axis of A passes through the center O of P. Let O1 be one of the points in which the axis of A meets the invariant circle of P, X be any point of TT, and A"' = AP (A'). The considerations given above in connection with the transformation (3) show that
OX
and hence that AP corresponds to a transformation of / of the same type as (3), i.e. to an involution. Moreover, AP is obviously the same as PA. In case the axis of A does not pass through the center of P, let A' be an orthogonal line reflection whose axis passes through the center of P. Then
AP = AA' - A'P and PA = PA' • A' A.
The products AA' and A' A correspond to projectivities by Theorem 6, and PA' = A'P corresponds to an involution by what has just been proved. Hence AP and PA correspond to projectivities.
To prove (7) let A be an orthogonal line reflection whose axis contains the centers of Pl and P2. Then
The products PXA and AP2 correspond to projectivities by (y9). Hence PjP2 corresponds to a projectivity. Thus we have the important result :
THEOREM 7. A protective transformation on a complex line corre- sponds under F to a transformation of the real Euclidean plane which is a product of an even number of inversions and orthogonal line reflections, and, conversely, any transformation of the real Euclidean plane of this type corresponds to a projectivity of the complex line.
92. The inversion group in the real Euclidean plane.
DEFINITION. The transformations of a Euclidean plane and its line at infinity which are products of orthogonal line reflections and inversions are called circular transformations, and any circular transformation which is a product of an even number of inversions and orthogonal line reflections is said to be direct.
THEOREM 8. DEFINITION. The set of all circular transformations of a Euclidean plane and its line at infinity in which an absolute
226 1NVKKS10N OEOMETBY [CHAP.VI
involution is given constitute a group which is called the inversion group. The set of direct circular transformations form a sulnji-mijt of the inversion group, which, if the Euclidean plane is real, is isomorphic with the protective group of a complex line.
The first part of this theorem is an obvious consequence of the definition, and the second is equivalent to Theorem 7. That not all circular transformations are direct is shown by the special case of an inversion. An inversion is not a direct circular transformation, because it leaves invariant all points of a circle and hence cannot correspond under F to a projectivity. Combining Theorems 8 and 6 we have
COROLLARY. In a real Euclidean plane the group of circular transformations leaving I*, invariant is the Euclidean group, and the direct circular transformations leaving /„ invariant are the direct similarity transformations.
The isomorphism between the group of direct circular transforma- tions and the projective group on the line may be used as a source of theorems about the former. Thus the fundamental theorem of projective geometry (Assumption P) translates into the following theorem. about the real Euclidean plane:
THEOREM 9. A direct circular transformation which leaves three- ordinary points, or two ordinary points and L, invariant is the identity. There exists a direct circular transformation carrying any three distinct ordinary points A, B, C respectively into three distinct points A', B', C' respectively, or into A', B', and /„ respectively.
Now consider a circular transformation II which is not direct and which leaves three distinct points A, B, C invariant. By definition
n= A2n + i- A2B---A2- Alf
where A, (i = 1, 2, • • •, 2 n + 1) is an inversion or an orthogonal line reflection. Let A be an orthogonal line reflection whose axis contains I . //, C, if these points are collinear, or an inversion with respect to the circle containing them in case they are not collinear. Then All is a direct circular transformation leaving A, B, C invariant. Hence
All = 1. Since A is of period two, this implies
n = A.
§'.'->] HEAL CIRCULAR TRANSFORMATIONS 227
The same argument applies in case one of the points A, B, C is replaced by /.. Hence we have
THEOREM 10. A circular transformation which is not direct and leaves invariant three distinct ordinary points A, If, C, or two ordinary points A, B, and /„, is an orthogonal line reflection or an inversion according as the invariant points are collinear or not.
THEOREM 11. If II is a circular transformation and A an inver- sion or orthogonal line re/lection, UAH"1 is an inversion or orthogonal line refection.
Proof. Let A, B, C be three of the invariant points of A ; then HAH-1 leaves II (A), II (B), II (C) invariant. If
where Al? • • •, Am are orthogonal line reflections or inversions, then
and is thus a product of an odd number of orthogonal line reflections or inversions. Hence by the last theorem it is an orthogonal line reflection or an inversion.
The invariant elements of II AH"1 are those to which the invariant elements of A are carried by II. Since II All"1 is an inversion or an orthogonal line reflection, we have
COROLLARY 1. Any circular transformation carries any circle into a circle or into the set of points on an ordinary line and on I*,. It carries the set of points on /„ and an ordinary line into a set of this sort or into a circle.
COROLLARY 2. If C'2 and K* are any two circles and I any line, there exists a direct circular transformation carrying C* to K* and one carrying C2 to the set of all points on I and fc.
Proof. Let A, B, C be any three points of C2, let A', B', C? be any three points of A'2, and let A', B' be any two points of I. By Theorem 9, there exist direct circular transformations II and II' such that
U(ABC) = A'B'C' and U' (ABC) = A'B'L.
Since A', B', C' are not collinear, the set of points into which II carries C'2 must be a circle ; and since there is only one circle containing A', B', C', this circle is K*. Since there is no circle containing A', B', and lx, the set of points into which II' carries <7a must be the set of
228 INYKKSION CKOMKTKV LCMAF.VI
points on /. and an ordinary line. Since the ordinary line contains A' and fft it must be I
An inversion (§71) transforms all lines through its center into themselves and interchanges the center with /.. Hence, by the last two corollaries, we have at once
COROLLARY 3. An inversion carries a circle through its center into the set of points on /, and a line not passing through the center.
COROLLARY 4. A pair of circles which touch each other is carried by an inversion into a pair of circles which touch each other, or into a circle and a tangent line together with /„, or into two parallel lines and I*.
Proof. Let C* and K* be two circles which touch each other. Since an inversion is a one-to-one reciprocal correspondence except for the origin and /„, if neither (72 nor JT2 passes through the origin, they must be carried into two circles having only one point in common and which therefore touch each other. If C2 passes through the origin and X'1 does not, (72 is carried into /. and an ordinary line I, while K* is carried into a circle K* which has one and only one point in common with the line pair 1*1. Since I* cannot meet K* in a real point, I meets it in a single point and therefore is tangent. If C*2 and K* both pass through the center of inversion, they are transformed into /. and a pair of ordinary lines /, m. Since (72 and K* have only the center of inversion in common and this is transformed into L, the lines / and m can have no ordinary point in common. Hence I and m are parallel.
It was remarked in § 90 (just before the fine print at the end) that the correspondence F between the complex line and the real Euclidean plane is such that the points of a certain chain C (P^P^), with the exception of J£, correspond to the points of a certain Euclid- ean line/. Since Px corresponds to /„, the chain C (P0P^PX} corresponds to the line pair II*. Under the protective group on a line any two chains are equivalent ; and under the group of direct circular trans- formations any circle is equivalent to any circle or any line pair II* (Cor. 2). Hence we have
THEOREM 12. Tlie correspondence T is such that chains in the com- plex line correspond to real circles or to line pairs U9t where I is ordinary and I* the line at infinity of the Euclidean plane.
§9-.-] CIRCLES AND CHAINS 229
The theory of chains on a complex line is therefore equivalent to the theory of the real circles and lines of a Euclidean plane. In view of this equivalence we shall freely transform the terminology of the complex line to the Euclidean plane, and vice versa. Thus we shall speak of the cross ratio of four points in the Euclidean plane and of pencils of chains in the complex line. The exercises below contain a number of important theorems some of which can be obtained directly from the definitions in § 71 and some of which can be proved most simply by translating projective theorems on the complex line into the terminology of the Euclidean plane.
DEFINITION. An imaginary circle is an imaginary conic through the circular points such that its polar system transforms real points into real lines.
The definition of an inversion given in § 71 applies without change to the case of imaginary circles.
On the geometry of circles in general the reader is referred to the papers by Mobius in Vol. II of his collected works; to those by Steiner in Vol. I (especially pp. 16—83, 461-527) of his collected works; to Vol. II, Chaps. II, III, of the textbook by Doehlemann referred to in Ex. 4 ; and to the forthcoming book by J. L. Coolidge, A Treatise on the Circle and the Sphere, Oxford, 1916.
EXERCISES
1. An inversion with respect to an imaginary circle is a product of an inversion with respect to a real circle and a point reflection having the same center as the circle.
2. The inverse points on any line through the center 0 of a circle C* are the pairs of an involution having O as center. If Al and A2 are any two inverse points, OAl • OA2 is a constant, which in case of a real circle is equal to (OC)3, C being a point of C2.
3. Two pairs of points A A' and DB' are inverse with respect to- a circle with 0 as center if and only if (1) O is collinear with the pairs A A' and /'>/>', and (2) the ordered triads OAB and OB' A' are similar, but not directly similar.
4. A linkage which consists of a set of six bars OA, OC, AB, BC, CD, DA, jointed movably at the points 0, A, B, C, D, and such that Dist (OA) = Dist (OC) and A BCD is a rhombus, is called a "Peaucellier inversor." If 0 is held fixed and B varies, the locus of D is inverse to that of B with resect to a circle with O as center. If B be constrained, say by an additional link, to move on a circle through O, D describes a line. On the general
2*4 QfVKKS I ON GEOMETRY [CHAI-.VI
subject of linkages, of. K. Poehlemann, (ieoinetrisclie Tr:insfonn:it.i<mrii. Vol. II. p. '.MI. l..-i[./i.u. r.'H>, and A. Kinrli. I'n.jertive (Jeometry, ^ (5'J-(57,
V..rk. 1 !•«»:•. 5. If .1, /.'. ^', 1> are four joints of a Euclidean plane,
Dist (/IQ I)ist(#n
where * = rr . ' •*- . »./ an<1 0 = a -
Dist (A If)
when- a and /J an- the measures of &CAD aud &CBD respectively. The number I- is invariant under the inversion group, and 6 under the group of direct circular transformations. The four points are on a circle or collinear
if e = o.
6. Construct a }»oint having with three given points a given cross ratio.
7. If Ilis any circular transformation, the points O = n~1(/«) and 0'= II (Ax.) are called its vanishing jKrints. The lines through O are transformed by II into the lines through O'. If A" is any point of the plane, and X' = II (A'), then IH^t (OX) • Dist (O'A"') is a constant, called the power of the transformation (of. § 43>
8. Let A and K be two points not collinear with O and let II (A) = A', II (7?) = &. The ordered point triads OAR and O'B'A' are directly similar if II is direct, and similar, but not directly so, if II is not direct.
9. The equations of an inversion relative to rectangular nonhomogeneous coordinates, having the center of inversion as origin, are
kx kii
X = - y V = - - - •
x2 + y2 x2 + y*
The circle of inversion is real or imaginary according as k > 0 or k < 0.
10. The coordinate system for the real Euclidean plane obtained by means of the isomorphism of the Euclidean group with the projective group leaving a jK)int invariant on a complex line is such that the coordinate z of any point is z 4- iy, where x and y are the coordinates in a system of rectangular non- homogeneous coordinates and z2 = — 1. The points 2 of a circle satisfy the condition
2 =
at + b ct + (I '
a b c d
where t is real and variable and a, b, c, d are complex and fixed. If c = 0, this circle reduces to a line.
11. The circles orthogonal to z = are
ct + d
_ (a + bft) it + b+ aa
~ (c + i{ft)it + d+ ca' where a and ft are real.
12. The circles through two points zv z2 are given by
§§«.«, j»] GENERALIZATION BY INVERSION 231
13. A circle with zt as center is given by
z-Sl = k&t where 0 s $ < 2 rr and £ is a real constant.
14. The centers of the circles circumscribing the four triangles formed by the sides of a complete quadrilateral are on a circle. This circle is called the cmhr circle of the complete quadrilateral. The centers of the center circles of the five complete quadrilaterals formed by the sides of a complete five-line are on a circle called the center circle of the five-line. Generalize this result.
93. Generalization by inversion. By the corollary of Theorem 8 the set of direct circular transformations leaving /„ invariant is the group of direct similarity transformations, and the set of all circular transformations leaving /. invariant is the Euclidean group. This is the basis of a method of generalization by inversion entirely analo- gous to the generalization by projection employed in § 73.
In case a figure Fl which is under investigation can be trans- formed by one or more inversions into a known figure F2, then such of the relations among the elements of F2 as are invariant under circular transformations must hold good among the corresponding elements of F^
In order to apply this method it is necessary to know relations which are left invariant by the circular transformations. The most elementary of these are given in the last section, but perhaps the most important property of an inversion for this purpose is that of isogonality, or " preservation cf angles."
DEFINITION. If Cf and C\ are two circles having a point Q in com- mon, and ml and ra2 are the tangents to Cf and C* respectively at Q, the measure (according to § 72) of the ordered line pair rn^rn^ is called the angular measure of the ordered pair of circles at Q, or simply the angle between the two circles at Q. If Cf is any circle, m2 a line meeting it in a point Q, and rat the tangent to Cf at Qt the measure of the ordered line pair mjii^ is called the angle between m.2 and Cf, and the measure of m^m.^ is called the angle between Cf and ra2. The measure of a line pair w1m2 is called the angle* between wij and w2.
THEOREM 13. An angle a between two circles or a circle and a line or between two lines is changed into IT — a by an inversion or
* In accordance with common usage, we are here using the term " angle " to denote a number, in spite of the fact that we use it in § 28 to denote a geometrical figure.
282
i n \i-. VI
an ort/toi/onul line reflection and is left unaltered Inj a iuj circuit r t /•<> it .</'« wnation.
Proof. The statement with regard to direct circular transforma- tions is an obvious consequence of the one with regard to inversions and orthogonal line reflections. What we have to prove is, therefore, the following:
Let II be an inversion or an orthogonal line reflection, and let ^ and /8 be two lines meeting in a point P such that II (P) = Q is an ordinary point. If /: is carried by II into a line, let this line be deii< .let 1 by m ; and if l^ (together with L) is carried to a circle C'j2, let ml denote the tangent to C'[ at Q; likewise, if 12 is carried by II into a line, let this line be denoted by m2 ; and if /2 (together with L) is carried to a circle C$, let m.^ denote the tangent to C* at Q. The two ordered pairs of lines l^ and m^^ are symmetric.
In case II is ail orthogonal line reflection, mt = II (^) and w2= II (/,)f and the proposition is a direct consequence of the definition of the term "symmetric" (§ 57). Suppose, then, that II is an inversion hav- ing a point 0 as center.
One of the lines llt l^ say llt can be transformed into itself if and only if lv is on 0. By hypothesis O^P; hence if II (Zj) = llf the line 1.2 goes into the set of points dif- ferent from O on a circle C$ through 0 and Q. Then m2 is the tangent to C$ at Q. Any line through 0 which meets 1.2 in an ordinary point X meets C% in the point which corresponds to X under the inversion. Hence the line n.t through 0 and tangent to C\ cannot meet l.t in an ordinary point, and is therefore parallel to /2. Hence the line pair /^ is congruent to the pair Iji^. The line in., is the tangent to C\ at Q. Since Ijn^ is carried to ^m.^ by the orthogonal line reflection whose axis is the perpendicular bisector of OQ, the pair /,na is symmetric with /tm2. Hence /,/2 is symmetric with l^n^.
FIG. 71
§w] NINE-POINT CIRCLE 233
If neither of the lines ^, /2 is transformed into itself, neither passes through O. Let / denote the line OP. Then by the last paragraph II l is symmetric with lmlt and //2 with lma. But by Theorem 13, Chap. IV, the symmetry which carries II l to lml must be identical with that which carries //a to lmz. Hence l^ is symmetric with ra^.
As an exercise in generalization by inversion let us prove the following : THEOKKM 14. If three circles Cf, C2, C£ meet In a point O in such a way (tint
each pair of them makes an angle — , and also meet oy pairs in three other points
oi
/', (}, 11, the circle (or line) through 7J, Q and 11 makes with each of the other circles an angle — •
I 'roof. The pair of circles which meet at O obviously make the angle — at
o
each of the points P, Q, R. An inversion II with respect to a circle having O as center must therefore change them into the sides of an equilateral triangle.
The circle circumscribing this triangle makes the angle - with each of the
o
sides. But since this circle is the transform of the circle PQR by II, the conclusion of the theorem follows.
As a second application of the theory of inversion, in combination with project! ve methods, we may consider the theorem of Feuerbach on the nine point circle (cf. Ex. 2, § 73).
THKOHKM 15. The nine-point circle of a triangle touches the four inscribed circles.
Proof. Let the given triangle be ABC, and let the mid-points of the pairs ]'•<'. CA, AB be Av Bv Cl respectively. The nine-point circle is the circle containing AVBV Cr
Let A'j2 and A'| be the two inscribed circles whose centers are on one of the bisectors of 4 CAB. In case K2 and A'.| touch the line BC at the same point, this is the mid-point Al of the pair BC, the triangle ABC is isosceles, and the nine-point circle obviously touches A",2 and K\ at A r In every other case there
INYKKSIMN (IKOMKTKY [CHAP. Vi
is one line, /, U-si.les .1 /;. !',< ', ( '.I. which touches both A'f and A'.;. Let A'B'C' be the joints in which/ meets tin- sides ]',(', ('A, AH resj>ectively. Then .1.1', /.'/.'', ' ' " are tin- pairs uf opposite vertices of a complete quadrilateral circum- scribing both A"f ami A'.?, and the diagonal triangle of this quadrilateral is a srlf-jmlar triangle Itoth for A'f and A'.r (§ 44, Vol. I). Since the side .1.1' of this triangle is the line of ••••liters of A'f and A'|, the other two sides, Btt' and ;ire parallel to each other and perpendicular to .1.1'. Let their point * of intersection with .1.1' !>«• /*'„ and < '„ respectively. These two points are con- jugate with respect to both circles, and hence must be the limiting points of the jM-ncil of circles containing A'f and A'.j. The radical axis of the pencil of circles is the perpendicular bisector of the pair B0C0, and hence (§ 40) passes through the mid-points of all the pairs BC, B'C', BC', B'C, B0C0. In partic- ular the radical axis of A'f and A'| passes through Av the mid-point of /.'''. Hence there is a circle G2 with Al as center and passing through B0 and ' ',,.
Let F be the inversion with respect to G~. Since this circle passes through B0 and C0, it is orthogonal both to A'j2 and A'| (Theorem 34, § 71), and hence F transforms each of these circles into itself. We shall now prove that F transforms / into the nine-point circle.
Let B2 be the point in which AlBl meets /. Since A1B1 is parallel to AB, it is not parallel to /, and hence B.2 is an ordinary point. Since AlBl contains the mid-point Al of the pair CD and is parallel to BC', it contains the mid- point C0 of the pair CC'. The involution which F effects on the line AlBl must have C'0 as one of its double points and Al as its center ; hence the other double point must be the point Ba in which AlBl meets BB', because Al is the mid-jxiint of the pair C0B3. Thus G2 passes through Bs as well as through CQ. But since n/
50J'C0.4=£8J52C0BP
Bv and B2 are harmonically conjugate with respect to C0 and B3. Hence F transforms B2 to Rv
In like manner it can be shown that if C2 is the point in which AlCl meets /, F transforms C, to Cr Since any line whatever is transformed by F to a circle through „!,, it follows that / is transformed to the circle through A v Ilv and C,, i.e. to the nine-point circle. By Theorem 11, Cor. 4, since / is tangent
to A',2 and A'*, the nine-point circle touches A'j2 and A"|. Since it has not \ n
specified which of the bisectors of 4 CAB contains the centers of A'j2 and A'rj, this argument shows that the nine-point circle touches all four inscribed circles.
EXERCISES
1. Any three points can be carried by an inversion into three collinear points.
2. Two non intersecting circles can be carried by an inversion into concentric circles.
3. Any direct circular transformation is a product of an inversion and an orthogonal line reflection.
M«»,94] ro.MI'LKX INVK1ISIONS
4. A product of two iii\er>i.,ns i> an involution if and only if the cir<-le> are orthogonal.
5. Of four circles mutually perpendicular by pairs, three can be real.
6.. The nine-}K>iut circle meets the circle through ' ',, having . lt as center in points of the line A' 11'.
7. Tin- nine-point circle of a triangle touches the sixteen circles inscribed to the triangle or to any of the triangles formed by pairs of its vertices with the orthocenter. I 2. 3
8. Let three circles ( ',-', / ';. ( ; meet in a j>oint (>, and let Pv I\, J\ l»e the other points of intersection of the pairs (.'£('£, f£C'f, C'fL'% respectively. If (^ be any }x>int of {"*, (J., the point of ('% collinear with and distinct from Qt and /'.. and Qs the point of C% collinear with and distinct from Q2 and /',, then (j... J'.,, and Ql are collinear.
9. The prohlt'in of Apolloniux. Construct the circles touching three given circles. Cf. Pascal, Repertorium der Hohereu Mathematik, II 1, Chap. II, on this and the following exercise.
10. The problem of Malfatti. (liven a triangle, determine three circles each of which is tangent to the other two and also to two sides of the triangle.
94. Inversions in the complex Euclidean plane. Thus far we have dealt only with a .real Euclidean plane. The definition of an inver- sion given in § 71, however, applies without change in the complex Euclidean plane; i.e. two points A^ A2 are inverse with respect to a circle C'2, provided they are conjugate with respect to C'1 and collinear with its center. The transformation thus defined is obviously one to one and reciprocal for all points of the complex protective plane except those on the sides of the triangle OIJ^, where 0 is the center of C2, and /t and /2 are the circular points at infinity. Any point of I* is carried to O by the inversion, and 0 is carried to every point of L. The circular point It is transformed to every point of the line O/1? and every point of the line 07, is transformed to 1^ In like manner /„ is transformed to every point of the line 0/2, and every point of this line is carried to /,.
DEFINITION. The sides of the triangle OIJ^ are called the singular lines of the inversion with respect to 6'2, and the points on these lines are called its singular points.
The principal properties of an inversion may be inferred from the following construction: If AI is any jnmit not on a side of the triangle OIJ», let J^ and I!a be the points distinct from II and /2 (fig. 73) in which the lines AJ^ and ./,/., respectively meet Ca. Let Af be the point of intersection of /j/?a and /..//,. The points Al and At
INVERSION (iKo.MKTRY
[CHAI-. VI
are mutually inverse because, by familiar theorems on conies, they
'iijugate with regard to C2 and collinear with 0. From this construction it is evident in the first place that all points, except 7j of the line AJ.^ are transformed into points of the line At/t, and vice versa. Hence an inversion transforms the minimal lint's through II into the minimal lines through 72, and vice versa. More- over, the correspondence between the two pencils of minimal lines is such that if B is a variable point of f2, the line IJ$ always corresponds to 72& In other words, the correspondence effected by an inversion between the two pencils of mini- mal lines is a projectivity generating the invariant circle C"2.
The definitions of circular and of direct circular transformations, given in § 92, apply without change in the complex Euclidean plane. The result just obtained therefore implies that any direct circular transformation transforms each pencil of minimal lines projectively into itself, and any nondirect circular transformation transforms each pencil of minimal lines projectively into the other.
Now suppose that Al is a variable point on any line I not contain- ing 7, or 72.
FIG. 73
Since Bl and /?2 are always on the conic Ca,
<5> 7iKlA
and
<6> A MA
Hence
<7> W] A
But corresponding lines of these two pencils intersect in the vari- able point At, which is therefore always on a conic through II and It
COMPLEX INVERSIONS 237
or on a line. In the projectivity (5) the line 1^0 corresponds to lm ; in (4) /. corresponds to itself; and in (6) /. corresponds to IV0. Hence in (7) the line I^O corresponds to 1^0, and so the circle or line generated by (7) passes through 0.
This result may be stated in a form which takes account of the singular elements, as follows: Any degenerate conic consisting of i* a nd a nonminimal line is carried by an inverswn with respect to C* into a conic (degenerate or not) which passes through I , /2, and 0.
Next suppose A1 to be a variable point on any nondegenerate conic throuh / and /. In this case
(8) JI^A
and hence by the projectivities (5) and (6) we have
Hence A2 is again on a conic through /t and /2, which can degenerate only if Zo,, corresponds to itself under (9). The latter case implies, by (5) and (6), that 1^0 and 72O correspond under (8) or, in other words, that the locus of Al passes through 0. Hence any nondegenerate conic K* through 1^ and /2 corresponds by the inversion with respect to C* to a conic through 1^ and /2, which degenerates into a pair of lines, one of which is lx, only in case K* passes through 0.
This result, together with the other statement italicized above, amount to an extension of Cors. 1 and 3 of Theorem 1 1 to the com- plex Euclidean plane. From our present point of view we can also establish the following theorem, which did not come out of the reasoning in § 92.
THEOREM 16. The correspondence between two circles which are homologous under an inversion is projective.
Proof. If Al is a variable point of one circle and A^ of the other, then, in the notation above, IJil = I^An, and hence by (5)
which is a necessary and sufficient condition that the correspondence between the two circles be projective (cf. the corollary and definitions following Theorem 10, Chap. VIII, Vol. I).
INVERSION (JKO.MKTKV [CHAP. VI
Tin- same reasoning also applies in case one <>r l><>th of the which are the loci of J( and J2 degenerate. We thus have
COROLLARY. A protective correspondence is established by an in- vergion bet iron "//// two homologous lines or between a line and its homologous set of points on a circle.
The proof of Theorem 13 on the preservation of angles under a cir- cular transformation applies without change in the complex Euclidean plane. This theorem can also be proved by the use of considerations with regard to the circular points. We shall give the argument for the case of orthogonal circles, leaving it as an exercise for the reader to derive the proof along these lines for the general case.
It has been proved in § 71 that the circles through two points Alt A are orthogonal to the circles through two points Blt 7?2 if and only if the pairs A^A^ T^TJ^, and Illn are pairs of opposite vertices of a complete quadrilateral (cf. fig. 73). The sides 7^, 7^, 7.,.^, lnA^ of such a quadrilateral are transformed by an inversion relative to any circle into four lines through II and 72. Hence the points A^ A2, i:^ B.2 are transformed into four points A[t A!2, B[, B!2 such that 7^, A(A!2, and B[B!2 are pairs of opposite vertices of a complete quadri- lateral. Hence the pencils of circles through Av A^ and BV /.'., respectively are transformed into two pencils such that the circles of one pencil are orthogonal to those of the other.
With this result it is easy to prove that Theorems 8-11, 13, and their corollaries hold in the complex Euclidean plane, proper excep- tions being made so as to exclude minimal lines and pairs of points on minimal lines. This is left as an exercise.
95. Correspondence between the real Euclidean plane and a complex pencil of lines. The correspondence between a complex one-dimen- sional form and the points of a real Euclidean plane, together with /., can be established in a particularly interesting way if the onr- dimensional form be taken as the pencil of lines on one of the circular points of the line at infinity of the Euclidean plane.
Let /. l>e the line at infinity, and II be one of the circular points. By Theorem 15, Chap. V, each line through II contains at least one real point. No line through 7t, except /„, can contain more than one real point ; for otherwise it would be a real line, and hence would meet L in a real point contrary to the fact that 7X is imaginary. Then cadi
§«:.] INVOLUTIONS IX A EUCLIDEAN PLANE 239
line through 1^ except /., contains one and only one real point of the Euclidean plane. Let us denote by F' the correspondence by which /. corresponds to itself and the other lines through 1^ correspond each to the real point which it contains.
By § 94 a direct circular transformation transforms the pencil of lines on /t protectively into itself. Hence every direct circular trans- formation corresponds under F' to a projectivity of the lines on / .
By Theorem 9 there is one and only one direct circular trans- formation carrying an ordered triad of distinct points to an ordered triad of distinct points; and by Assumption P there is one and only one projectivity carrying an ordered triad of lines of a pencil to any ordered triad of the pencil. Hence a given projectivity of the pencil of lines on II can correspond under F' to only one direct circular trans- formation. In other words, T' sets up a simple isomorphism between the projective group of a complex one-dimensional form and the group of direct circular transformations.
The correspondence between the points of a real line and the lines joining them to II is evidently projective. Since the cross ratio of four points of a real line is real, so is the cross ratio of the lines join- ing them to /r Hence any real line together with /„ corresponds under F' to a chain. Since any two chains of a one-dimensional form are projectively equivalent, and any circle of the Euclidean plane is equivalent under the inversion group to an ordinary line and /„, it follows that under F' any chain corresponds to a circle and any circle to a chain.
The correspondence F' may be used to transfer the theory of invo- lution from the complex pencil of lines to the Euclidean plane. Let A A', BB1 ', CC' be pairs of opposite vertices of a complete quadrilateral of the Euclidean plane. The pairs of lines joining these point pairs to /t are pairs of an involution. Hence
THKOREM 17. TJi e pairs of opposite vertices of a complete quadri- lateral are pairs of an involution, i.e. they are pairs of homologous points in a direct circular transformation of period two.
In other words, the pairs of opposite vertices of a complete quad- rilateral constitute the image under F' (and hence under F) of a quadrangular set. "While the converse of this proposition is not true, the proposition can be generalized by inversion so as to give
240 1NVKKSION GEOMETRY [CHAI-. vi
a construction for the most general quadrangular set in which no four of the six points are on the same circle or line (cf. K.\. 1. In-low). We shall state the construction in terms of chains.*
TIIKOKKM 18. Given two pairs of points AA' and KB' and a point C such that no four of the Jive points are on the same chain. The chat us C (AB'C) and C (A'BC) either meet in a point D other than C or touch each other at C. In the latter case let D denote C. The chains C(DAB) and C(DA'B') meet in a point C' such that A A', /:/:', CC' are pairs of an involution.
Proof. Consider the figure in the Euclidean plane (together with /.) corresponding under F" to the figure described in the theorem. If r'(/>)^=/., F'(Z>) can be transformed to fc by an inversion I. Under IF the four chains C (AB'C), C (A'BC), C (DAB), and C (DA'Ji') correspond to Euclidean lines (with £»), and hence AA', BB', CC' corre- spond to the vertices of a complete quadrilateral ; so that the theorem reduces to Theorem 17. If T'(D) = lx, the theorem reduces directly to Theorem 17.
COROLLARY. Three pairs of points on a complex line AA', BB'} CC', such that the chains C (A'B'C'), C (A'BC), C (AB'C), C (ABC1) are distinct, are pairs of an involution if and only if the four chains have a point in common.
EXERCISES
1. Three pairs of points of the same chain A A', llli', Cf" are in involution if for any point D not in the chain the chains C (DA A'). C (DIM!'), C (DCC) are in the same pencil.
2. Derive Ex. 15, § 81, from the theory of involutions in a plane.
3. If A A', BB', C(" are pairs of opposite vertices of a complete quadri- lateral, the three circles having A A', BB', CC' respectively as ends of their diameters belong to the same pencil, and the radical axis of this pencil passes through the center of the circle circumscribing the diagonal triangle of the quadrilateral.
4. Construct the double points of an involution in a Euclidean plane with ruler and compass.
• This puts in evidence the fact that while the geometry of real one-dimensional forms depends essentially on constructions implying the existence of two-dimru- sinnal forms, the geometry of the complex projective line could be developed without supposing the existence of points outside the line.
§§'-'.-,,96] REAL INVERSION PLANK 241
96. The real inversion plane. In a real Euclidean plane an inver- sion has been seen to be a one-to-one and reciprocal transformation except in that it transforms /„ to the center of inversion, and the center to /„. An inversion, therefore, is strictly one to one if we regard it as a transformation of the set of objects composed of the points of the real Kuclidean plane together with /. regarded as a single object.
DEFINITION. The set of points in a real Enclidean plane, together with the line at infinity regarded as a single object, is called a real i/ir-rsion plane; /„ is called the point at infinity of the inversion plane. The set of points on a real circle, or on a real line / together with /«, is called a circle of the inversion plane. An inversion is either an inversion in the sense of § 71 with respect to a real or imaginary circle or an orthogonal line reflection. Circular transformations, etc. are denned as in § 92. The set of theorems about the inversion plane, which remain valid when the figures to which they refer are sub- jected to every transformation of the inversion group, is called the real inversion geometry.
Although the point at infinity receives special mention in this definition, from the point of view of the inversion geometry it is not to be distinguished from any other point of the inversion plane. For any point of the inversion plane can be carried to any other point of it by an inversion. In a set of assumptions for the inversion geometry as a separate science, there would be no mention of a point at infinity; just as there is no mention of a line or a plane at infinity in our assumptions for projective geometry.
The inversion geometry has a relation to the Euclidean geometry which is entirely analogous to the relation of the projective geometry to the Euclidean ; namely, the set of transformations of the inversion group which leaves one point of the inversion plane invariant is a parabolic metric group in the Euclidean plane obtained by omitting this point from the inversion plane.
A large class of theorems about circles can be stated with the utmost simplicity in terms of the geometry of inversion. For exam- ple, the propositions that three noncollinear ordinary points determine a circle and that two ordinary points determine a line combine into the single proposition :
THEOREM 19. In the inversion plane any three distinct points are on one and but one circle.
INVERSION GEOMETRY [CHA.-.YI
The thcoivm that there is one and only one circle touching a given circle C'1 at a given point A, and passing through a given point B not on <'*, may be put in the following form, which also includes the proposition that through a given point not on a given line I there is one and but one line parallel to I.
THF.OKKM 'JO. There is one and but one circle through a point A «n a iircle C* and a point B not on C2, and having no point except A in a> in //i o/t urith C'1.
The theory of pencils of circles makes no special mention of the radical axis (§ 71), for the radical axis (with /„) is merely one circle of the pencil and is indistinguishable from the other circles. In like manner the center of a circle is not to be distinguished from any other point; for the center is merely the inverse of L, with respect to the circle, and the inversion group does not leave /. invariant.
Thus the theory of pencils of circles in the inversion geometry involves no reference to the radical axis or to the line of centers. A pencil of circles may be defined as follows:
DEFINITION. A pencil of circles is either (a) the set of all circles through two distinct points, or (b) the set of all circles orthogonal to the circles of a pencil of Type (a), or (c) the set of all circles through a point of a given circle C2 and meeting C2 in no other point. A pencil of circles is said to be hyperbolic, elliptic, or parabolic, according as it is of Types (a), (b), or (c). Any point common to all circles of a pencil is called a base point of the pencil.
By comparison with the theorems in the preceding sections it is evident that the pencils of circles of these three types include all the pencils referred to in § 71 and also certain pencils of circles which are regarded as degenerate, from the Euclidean point of view. Thus, consider a pencil of lines through an ordinary point of a Euclidean plane. Each of these lines, with L, constitutes a degenerate circle, and the set of degenerate circles is a pencil according to the definition above. Again, a pencil of parallel lines in the Euclidean plane deter- mines a set of circles [A"2] in the inversion plane which have in common only the one point Z«. By Theorem 11, Cor. 3, any inver- sion F with a center 0 transforms [A"2] into a set of circles [A7] through 0 which have in common no other real points than 0.
§«6] KEAL INVERSION PLANE 243
Since there is one and only one circle of the set [A"*] through every point of the Euclidean plane, [Kf] must be a pencil of circles of Type (c).
The fundamental theorems about circular transformations may be stated as follows :
THEOREM 21. A circular transformation is a one-to-one transfor- mation of the inversion plane which carries circles into circles. There is a unique direct circular transformation carrying three distinct / K il/its A, B, C to three distinct points A', li', C' respectively. A circular transformation leaving three points invariant is either an inversion relative to the circle through these three points or the identity.
The theorems on orthogonal circles in §71, together with the corresponding propositions on circles, lines, and orthogonal line reflections, become:
THEOREM 22. Two circles are orthogonal if and only if one of them passes through two points which are inverse with respect to the other.
COROLLARY 1. Two circles are orthogonal if and only if they belong respectively to two pencils of circles such that the limiting points of one pencil are the common points of the circles of the other pencil.
COROLLARY 2. If A l and A2 are inverse with respect to a circle C*, all circles through Al and orthogonal to C2 pass through A^.
The correspondence F, which was established in §§ 90, 91, between the Euclidean plane and the complex protective line, is one to one and reciprocal between the inversion plane and the complex line. Since circles and chains correspond under F, the inversion geometry is identical with the geometry of chains on a complex line. The direct circular transformations of the inversion plane correspond to the projectivities of the complex line.
It follows from § 90 that the inversion with respect to the chain C (Q^Q^Q^) transforms every point z = x + iy into the conjugate imagi- nary point z = x — iy. Hence an inversion with regard to any chain is a transformation projectively equivalent to that by which each point goes to its conjugate imaginary point (cf. § 78). For this reason we make the definition :
DEFINITION. Two points are said to be conjugate with respect to a chain if they are inverse with respect to it.
1>44 IN YKKSloN ( i KoMETRY [CHAV. V.
It is easily seen that any nondirect circular transformation is ;i product of a particular inversion and a direct circular transformation. Hence any nondirect transformation may be written in the form
d We shall return to this subject in § 99.
EXERCISES
1. Construct a set of assumptions for the inversion geometry as a separate science.*
2. Work out the theorems analogous to those of §§ 71, 90-96 for the parabolic metric group in a modular space. Thus obtain a modular inversion geometry. The number of points in a finite inversion plane is p* + 1 if the number of points on a circle is /> + 1.
3. The double points of an involution leaving a chain invariant are inverse with respect to the chain.
97. Order relations in the real inversion plane. The more elemen- tary theorems on order relations in the inversion plane follow readily from the corresponding theorems for the Euclidean and protective planes. Suppose we start with a projective plane TT'. By leaving out a line /» of IT', a Euclidean plane TT is determined; and by regarding /, as a point, an inversion plane TT is determined. Any line / of TT' which is distinct from I* determines a circle of the inversion plane TT ; and we now define the order relations on this circle as identical with the projective order relations of /, the point lx taking the place of the point in which / meets £„. The order relations on any circle which does not contain /„ are determined by § 20.
Since the correspondence effected between any two circles by an inversion is projective (Theorem 16), it follows that the order relations among the points on any circle are unaltered by inversion. Hence order relations on circles are unaltered by circular transformations.
On a complex line the order relations in a chain are identical with the order relations on a real line as developed in §§18, 19, 21-24. The correspondence F (§§ 90, 91) is such that the order relations of corresponding sets of points on a chain C (Q^Q^Q*,) and the circle PJ\l» are identical. Since order relations on circles are unaltered by
•This question lias been treated for the three-dimensional case by M. Fieri, Giornale di Matematidie, Vol. XLIX (1911), p. 49, and Vol L, p. 100.
S'->7] ORDER RELATIONS 245
circular transformations, and order relations on chains are unaltered by projectivities, it follows that F is such that the order relations of corresponding sets of points on any chain and the corresponding circle are identical. Therefore the theory of order in the inversion plane applies also to the complex line.
Returning to the Euclidean plane TT', we know by § 28 that the points not on an ordinary line / fall into two classes such that any two points of the same class are joined by a segment not meeting /, whereas a line joining two points of different classes always meets I. By § 64 any circle containing two points of different classes meets I in two points. We thus have
THEOREM 23. DEFINITION. The points of an inversion plane not on a circle C* fall into two classes, called the two sides of C2, such that two points on the same side of C'2 are joined by a segment of a circle ivhich does not contain any point of C*,and such that any circle containing two points on different sides of C2 contains two points of C*.
Since order relations on circles are not altered by inversion, there follows :
COROLLARY 1. If two points are on opposite sides of a circle C*f the points to which they are transformed by an inversion H are on opposite sides of II (C).
On a complex line the points on one side of the chain C ((^(^(M are evidently those whose coordinates relative to the scale QQ> Qlt Q* are x + iy, where x is real and y real and positive, and those on the other side are those whose coordinates are x — iy. Hence, in general,
COROLLARY 2. The points D and D' are on opposite sides of a circle through A, B, C if and only if y and y' are of opposite sign in the following two equations :
R (AB, CD} = x + iy, R (AB, CD') = xf + iy', where x, y, x', y' are all real.
DEFINITION. A throw T (AB, CD) is said to be neutral if R (AB, CD) is real. Two throws T (AB, CD) and T (A'B1, C'D1) are similarly or oppositely sensed according as y and y' are of the same or of opposite signs in the equations
R (AB, CD) = x + iy and R (A'B', C'D') = x1 -f iy', x> H> x\ y1 being real.
•jj»; INYKKSION (JKOMKT1IY L«'HAI-. VI
From this dftinition it is obvious that a direct circular transforma- tion transforms any non-neutral throw into a similarly sensed throw. It is also obvious that an inversion which reduces in the Euclidean I -lane tr to an orthogonal line reflection changes non-neutral throws into oppositely sensed throws. Hence we have
THI.OKKM 24. A direct circular transformation carries non-neutral I h rows into similarly sensed throws, and a nondirect circular trans- formation carries them into oppositely sensed throws.
EXERCISES
1. Two circles C2, A'2 intersecting in two distinct points separate the inver- sion plane into four classes of points such that two points of the same class are joined by a segment of a circle containing no points of C2 and A'2, whereas any circle containing points of different classes contains points of f '2 and A'2.
2. Two points which are inverse with respect to a circle are on opposite sides of it.
3. What is the relation between the sense of throws as denned above and the sense of noncollinear point triads in a Euclidean plane as denned in § 30^
4. In a Euclidean plane if a triangle ABC is carried to a triangle A'R'C' by an inversion, the sense S (ABC) is the same as or different from S (A'B'C") according as the center of the inversion is or is not interior to the circle A BC.
5. In the notation of Ex. 7, § 92, if 0 is interior to a circle C2, then O' is interior to II (C2), and every point interior to C2 is transformed by II to a point exterior to O'.
98. Types of circular transformations. By § 5 every projectivity on a complex line has one or two double points. On account of the correspondence F the same result holds for the direct circular transformations of the real inversion plane.
Let us consider first a transformation II having but one double point. In the theory of projectivities such a transformation has been called parabolic; and it has been proved that there is one and but one parabolic projectivity leaving a point M invariant and carrying a point AQ to a point A^ We have also seen that if A_^ is the point which goes to AQ, R (MA0, A1A_l) = — 1. Hence A_lt AQ, Al are on the same chain through M. Since A_v A0, M are transformed into A0, Alt M respectively, this chain is left invariant by II.
In like manner any other point BQ not on the chain C (A^.l^lf) determines a chain which is left invariant by H. These two chains cannot have another point than M in common, because this point
§98] TYPES OF CIRCULAR TRANSFORMATIONS 247
would have to be left invariant by II. Thus II leaves invariant a set of chains through M no two of which have a point in common, and such that there is one and only one chain of the set through any point except M.
If II be regarded as a transformation of the inversion plane, this means that II leaves invariant each circle of a pencil of circles of the parabolic type. In the Euclidean plane e, obtained by leaving M out of the inversion plane, this pencil of circles is a system of parallel lines and II is a direct similarity transformation. Now let us regard € from the projective point of view. The transformation II leaves all points of the line at infinity of e invariant, because it leaves each of the circular points invariant as well as the point at infinity of the system of parallel lines. Hence II is a translation in the Euclidean plane e.
This result may be expressed in terms of the inversion plane as follows :
THEOREM 25. Any direct circular transformation with only one invariant point transforms into itself every pencil of circles of the parabolic type having this point as base point. One and only one of these pencils is such that each circle of the pencil is invariant.
Returning to the Euclidean plane we have
THEOREM 26. Any direct similarity transformation which is not a translation or the identity leaves invariant one and only one ordinary point.
Proof. Regard the Euclidean plane as obtained by omitting one point from an inversion plane. A .direct similarity transformation effects a transformation of the direct inversion group and leaves this point invariant. In case it leaves only this point invariant, it has just been seen to be a translation in the Euclidean plane. If not, by the first paragraph of this section it has one and only one other invariant point unless it reduces to the identity.
A similarity transformation leaving an ordinary point O invariant must transform into itself the pencil of lines through this point and the pencil of circles having this point as center.
Two important special cases arise, namely, a rotation about 0 and a dilation with 0 as center. Moreover, since there is one and only one direct similarity transformation leaving O invariant and carrying
INVERSION GEOMETRY [CHAP.VI
a point P, distinct from 0, to a point P', distinct from 0, any non- parabolic direct similarity transformation is expressible as a product of a rotation and a dilation.
A rotation which is not a point reflection leaves all circles with 0 as center invariant, and changes every line through 0 into another line through 0. A dilation which is not a point reflection leaves every line through O invariant, and changes every circle with O as center into another such circle. Hence a product of a dilation and a rotation, neither of which is of period two, leaves invariant no line through 0 and no circle with O as center. Since either a rotation or a dilation of period two is a point reflection, any direct circular transformation falls under one of the three cases just mentioned or else is a point reflection. Stated in terms of the inversion plane these results become (cf. fig. 56, p. 158):
THEOREM 27. A direct circular transformation having two fixed points transforms into itself the pencil of circles through the fixed points and also the pencils of circles about these points. The trans- formation either leaves invariant every circle of one pencil and no circle of the other pencil, or it leaves invariant no circle of either pencil, or it leaves invariant every circle of both pencils and is of period two.
DEFINITION. A direct circular transformation is said to be parabolic if it leaves invariant only one point ; to be hyperbolic if it leaves in- variant two points and all circles through these points ; to be elliptic if it leaves invariant two points and all circles about these points; to be loxodromic if it leaves invariant two points and no circle through the invariant points or about them.
The theorems above are all valid for the complex line if circles be replaced by chains and direct circular transformations by projec- tivities. The definition is to be understood to apply in the same fashion. Since every nonidentical projectivity on the complex line has one or two double points, the discussion above gives the theorem :
THEOREM 28. A direct circular transformation (or a projectivity on a complex line) is either parabolic, hyperbolic, elliptic, or loxodromic.
COROLLARY. An involution on a complex line is both hyperbolic and elliptic ; and any projectivity which is both hyperbolic and elliptic is an involution.
$i*J TYPES OF CIRCULAR TRANSFORMATIONS 249
EXERCISES
1. A projectivity whose double joints Zj and xt are distinct from each other and from tlie point P. of a scale P0, I\, P., and whose characteristic cross ratio (§ 7:), Vol. I) is k, may l>e written
(10)
x ' —
If one of the double points is P. and the other is xv the projectivity may be written
(11) x'-xl = k(x-xl).
The projectivity is hyperbolic if k is real, elliptic if k = eif, where 6 is real, and loxodromic if neither of these conditions is satisfied.
2. The parabolic projectivities with Zj as double point may be written in the form
or, in case the double point is P«=, in the form
x' = x + at.
In either case a svibgroup is obtained by requiring / to be real. The locus of the points to which an arbitrary point is transformed by the transformation of this subgroup is a chain, and the set of such chains constitutes a parabolic pencil of chains.
3. The projectivities (10) and (11) for which
k = a',
where a is constant and t a real variable, form a group (a continuous group of one real parameter, in fact). The locus of the points to which a given point is carried by the transformations of this group or the group considered in Kx. 2 is called a path curve. In the nonparabolic cases, if a is real the path curves are chains through the double points. If a is complex and | a | = 1, tln-v are chains about the double points. If a satisfies neither of these con- ditions, and the double points are P0 and /•*», the path curves are the loci of x = re'9 satisfying the condition
(13) r = o«*,
where a and ft are real constants ; if the double points are not specialized, the j>ath curves are projectively equivalent to the system (13). Diagrams illus- trating the three types of path curves will be found in Klein and Fricke's Klliptische Modulfunktionen, Vol. I, Abschnitt II.
4. From the Euclidean point of view the r and 0 in Ex. 3 are polar coordi- nates, and the loci (13) are logarithmic gpiral* nifcting the lines through the origin at the angle tan-1 (1//8). (A generalization of the notion of angle analogous to that in § 03 is here taken for granted.) The path curves of a
•j:.n INVERSION GEOMETRY [CHAP. VI
one-parameter group of Euclidean transformations may be a pencil of par- allel lines or a pencil of concentric circles or a set of logarithmic spirals congruent to (Itf).
5. A projectivity having a finite jx'riod must lie elliptic. A direct similarity transformation having a finite jx-riod must he a rotation.
6. A loxodromic projectivity is a product of an elliptic and a hyperbolic projectivity.
7. A projectivity leaving a chain invariant is either hyperbolic or elliptic.
99. Chains and antiprojectivities. The theory of chains on a com- plex line has been developed in the sections above by combining the general theory of one-dimensional projectivities with the Euclidean theory of circles. It is of course possible, and from some points of view desirable, to develop the theory of chains entirely independently of the Euclidean geometry. The reader is referred for the outlines of such a theory to an article by J. W. Young in the Annals of Mathe- matics, 2d Series, Vol. XI (1909), p. 33. Many of the properties of chains may be generalized to n dimensions, an n-dimensional chain or an n-chain being defined as a real ^-dimensional space contained in an ?wlimensional complex space in such a way that any three points on a line of the real space are on a line of the complex space. (This is the relation between S and S' in §§6 and 70.) A discussion of the theory of these generalized chains will be found in the articles by C. Segre and C. Juel referred to below, and also in those by J. W. Young, Transactions of the American Mathematical Society, Vol. XI (1910), p. 280, and H. H. MacGregor, Annals of Mathematics, 2d Series, Vol. XIV (1912), p. 1.
The transformations,
(14)
az
cz + d
a
c d
= 0,
of the complex line which were mentioned at the end of § 96 are analogous to the following class of transformations of the complex projective plane :
x* = Vo (15) x{ = a10x0
where x{ denotes the complex number conjugate to x(. These trans- formations are collineations, because they transform collinear points
§99] ANTIPKOJECTIVITIES 251
to collinear points,* but they are not projective collineations. If x'ot x(, x!2 be replaced by u'0> it[, u',, (15) gives the equation of a non- projective correlation. The analogous formulas in four homogeneous variables will define nonprojective collineations and correlations in space.
DEFINITION. A nonprojective collineation or correlation or a one-dimensional transformation of the type (14) is called an anti- projectivity.
The theory of antiprojectivities has been studied by C. Juel, Acta Mathematica, Vol. XIV (1890), p. 1, and more fully by C. Segre, Torino Atti, Vol. XXV (1890), pp. 276, 430 and VoL XXVI, pp. 35, 592. Their rOle in projective geometry may be regarded as defined by the following theorem due to G. Darboux, Mathematische Annalen, Vol. XVII (1880), p. 55. In this paper Darboux also points out the connection of the geometrical result with the functional equation,
THEOREM 29. Any one-to-one reciprocal transformation of a real projective line which carries harmonic sets into harmonic sets is projective.^
Proof. Let II be any transformation satisfying the hypotheses of the theorem, A, B, C any three points of the line, II (ABC) = A' B' C' t and H' the projectivity such that IT (A'B'C'} = ABC. Then ITII (ABC} = ABC. If we can prove that Il'II is the identity, it will follow that II = II'"1, and hence that II is a projectivity.
If Il'II were not the identity, it would transform a point P to a point Q distinct from P, while it left invariant all points of the net of rationality R (ABC). Let L^ L2, Lg be points of this net in the order
By Theorem 8, Chap. V, there would exist two real points S, T which harmonically separate the pairs PL^ and -£a£g. The transformation Il'II must carry S and T into two points harmonically separating the pairs QLl and L^La. But since the latter two pairs separate each
*Cf. §28, Vol. I.
t Von Staudt, Geometrie der Lage (Ntirnberg, 1847), § 9, defined a projectivity of • a real line as a transformation hsiviim tliis property. We are using Cremona's definition of a projectivity as a resultant of perspectivities (cf. Vol. I, § 22).
25S INYKKSION GEOMETRY [CHAI-. vi
otlu-r, l>y Theorem 8, Chap. V, there is no pair separating them both harmonically. Hence the assumption that IITI is not the identity It-ads to a contradiction.
COROLLARY 1. Any collineation or correlation in a real projective space is project it f.
Proof. Since a collineation transforms collinear points into col- linear points, it transforms nets of rationality into nets of rationality in such a way that the correspondence between any two homologous nets is projective (cf. §§ 33~35, Vol. I). Hence, according to the theo- rem above, the correspondence effected by the collineation between any two lines is projective. Hence the collineation is projective.
A like argument proves that a correlation is projective. The reason- ing holds without change in a real projective space of n dimensions.
COROLLARY 2. Any one-to-one reciprocal transformation of the real inversion plane which carries points into points and circles into circles is a transformation of the inversion group.
Proof. Kegard the inversion plane TT, minus a point P*,, as a Euclid- ean plane TT' ; let II be any transformation satisfying the hypotheses of the corollary, let II (K) = P', and let II' be an inversion carrying P' to P*. Then II 'II is a transformation satisfying the hypotheses of the corollary and leaving P^ invariant.
Since H'll carries circles through 7i into circles, it effects a collin- eation in TT. By the first corollary this collineation is projective. Since it carries circles into circles, it is a similarity transformation. Hence H'n is a transformation, say II", of the inversion group in IT'. Since II = II'"1!!", II is also in the inversion group.
Translated into the geometry of the complex projective line the last corollary states :
COROLLARY 3. Any transformation which carries chains into chains is either a projectivity or an antiprojectivity.
In the light of Corollary 2 it is clear that the whole theory of the inversion group can be developed from the definition of a circular transformation as one which carries points into points and circles into circles. This is the point of view adopted by Mobius in his Theorie der Kreisverwandtschaft, where, h'owever, he used also the unnecessary assumption that the transformation is continuous.
§§y.M<»] TETRACYCLIC COORDINATES 253
EXERCISES
1. Derive the formulas for antiprojectivities in a modular geometry. Cf. O. Veblen, Transactions of the American Mathematical Society, Vol. VIII (1907), p. 366.
2. Which if any of the following propositions are true? Any one-to-one and reciprocal transformation of a complex projective line which carries harmonic sets of points into harmonic sets of points is either projective or antiprojective. Any one-to-one and reciprocal transformation of a complex projective line which carries quadrangular sets of points into quadrangular sets is either projective or antiprojective. Any collineation or correlation of a complex projective space is either projective or antiprojective.
3. An antiprojectivity carries four collinear points having an imaginary cross ratio into four points whose cross ratio is the conjugate imaginary.
100. Tetracyclic coordinates. The general equation of a circle in a Euclidean plane TT with respect to the coordinate system employed in Chap. IV is
(16) ao(x2 + y*)+2aix+2azy + aa=Q.
DEFINITION. A degenerate circle is either a pair of lines joining an ordinary point to the circular points at infinity or a pair of lines //., where L is the line at infinity.
Thus (16) represents a nondegenerate circle, provided that the following condition is not satisfied :
(17) 0 =
-*?-*«*)•
The condition a0= 0 clearly means that (16) represents a degenerate circle consisting of ln and an ordinary line, unless al = «2 = 0 also, in which case (16) reduces to «3= 0. The condition
(18) «0a3 -<-< = ()
means in case ao ^= 0 that (16) represents a pair of ordinary lines through the circular points. In case «0, alt «2, ag are real, these two lines must be conjugate imaginaries. In the rest of this section the a's are supposed real.
Let us now interpret the ordered set of numbers («0, alt at, ag) as homogeneous coordinates of a point in a projective space of three dimensions, S8. For every point of Sg, except those satisfying (18), there is a unique circle or line pair llmt where / is ordinary, and vice versa. Hence there is a one-to-one and reciprocal correspondence
•_'.-, 1 INVERSION GEOMETRY [CHAP. VI
U't \veeii the points of S, not on the locus (18) and the circles of the inversion plane TT obtained by adjoining L (regarded as a point) to TT.
The points of S, which are on the locus (18) and not on «0= 0 represent pairs of conjugate imaginary lines joining ordinary points of TT to / and /, respectively. There is one such pair of conjugate imaginary lines of TT through each ordinary point of TT. The points of S, on the locus (18) and not on aQ= 0 may therefore be regarded us corresponding to the points of TT, with the exception of L. The only point of S8 common to «„= 0 and (18) is (0, 0, 0, 1), and this point may be taken to correspond to L. Thus the points of S3 not on (18) represent circles of the inversion plane TT, and the points of S3 on (18) represent the points of TT.
Stated without the intervention of S,, this means that the ordered
•
set of numbers («0, ax, «2, «3) taken homogeneously and subject to the relation (18) may be regarded as coordinates of the points of TT. When not subject to the relation (18) they may be regarded as coordinates of the circles and points in TT.
DEFINITION. The ordered sets of four numbers (aQ, a^ «2, ag) subject to (18) are called tetr acyclic coordinates of the points in TT. The same term is applied to any set of coordinates (£0, /31} /?2, /83) such that
A = 2 «««,, | «,v =* °- (* = °> !» 2> 3)
j=<>
The circles (real or imaginary or degenerate) represented by (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) are called the base or fundamental circles of the coordinate system.
A second particular choice of tetracyclic coordinates is given below.
The points of S3 on (18) evidently constitute the set of all real points on the lines of intersection of corresponding planes of the two protective pencils
(19) ao=o-(ai + V-la2) and al - V- 1 «2 = <nr3,
where the planes determined by the same value of a- are homologous. For (18) is obtained by eliminating <r between these two equations. The lines of intersection of homologous planes are all imaginary, but each contains one real point. This system of lines is, by § 103, Vol. I, a regulus, and the set of points on the lines, by § 104, Vol. I, a quad- ric surface. The locus (18) is therefore a real quadric surface all of whose rulers are imaginary (cf. also § 105, Vol. I).
§100] TETRACYCLIC COORDINATES
The correspondence between the points of S8 and the circles and points of the inversion plane TT is such that a range of points corre- sponds to a pencil of circles. For the points of the line joining (aQ, alt «a, «3) and (y8o, filt #2, y9a) correspond to the circles given by the equation
(X«0 + p00) (x2 + ya) + (\ai + /i/fy x + (\at + ;*#,) y + (Xag + /*£,) = 0,
which represents a pencil of circles, together with its limiting points in case the latter are real.
Any collineation F of S3 which carries the quadric (18) into itself must correspond to a transformation F of TT which carries points into points, circles into circles, and pencils of circles into pencils of circles. F therefore has the property that if a point P of TT is on a circle C'a of TT, then T(P) is on r(C"). By Theorem 29, Cor. 2, F is a circular transformation. Conversely, any circular transformation of TT carries points to points, circles to circles, and pencils of circles to pencils of circles, and therefore corresponds to a colliiieation of S3 which carries the quadric into itself. By Theorem 29, Cor. 1, this collineation is protective. In other words,
THEOREM 30. The real inversion geometry is equivalent to the protective geometry of the quadric (18).
COROLLARY-. The projective geometry of the real quadric (18) is equivalent to the complex projective geometry of a one-dimensional form.
A one-to-one correspondence between a complex line and the real quadric (18) may also be set up as follows: Let I be any complex line in the regulus conjugate to that composed of the lines (19). Each of these lines contains one real point, P, of the quadric (18) and one point, Q, of /. The correspondence required is that in which Q corresponds to 7*.
By properly choosing the constants which enter in the equation of a circle, we may set up the correspondence between the circles of the inversion plane and the points of an S8 in such a way that the equa- tion of the quadric surface corresponding to the points of the inversion plane has a particularly simple form. The equation of a circle in TT may be written
(20) £„(*» + /+ 1) + £(*»+ f- 1) + 2 £*+ 2 f.y = 0.
INVERSION CKO.MKTKY [CHAP. VI
The points (£0, £lf £s, £8) which correspond to points of the inversion plane now satisfy the equation
(21) # = £,' + # + &
and the circles corresponding to the four points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) are mutually orthogonal, one of them being imaginary. The coordinates (£0, ^, £2, £8) are connected with («0, «lf a«» rti) ky the equations
ao=£o+£1> ai = £2> «•--*•» a3=£o-£i>
which represent a collineation carrying the quadric (18) into the quadric (21).
If £i/£o» £j/£o» £s/£o are regarded as nonhomogeneous coordinates with respect to a properly chosen frame of reference in a Euclidean space of three dimensions (cf. Chap. VII), (21) is the equation of a sphere. Hence the real inversion geometry is equivalent to the projective geometry of a sphere.
The latter equivalence may be established very neatly, with the aid of theorems of Euclidean three-dimensional geometry, by the method of stereo- graphic projection. This discussion would naturally come as an exercise in the next chapter. It is to be found in books on function theory. On the whole subject of inversion geometry from this point of view, compare Bocher, Reihenentwickelungen der Potentialtheorie (Leipzig, 1894), Chap. II.
DEFINITION. A circle (782 is linearly dependent on two circles (72 and C.* if and only if it is in the pencil determined by (72 and C2. A circle C12 is linearly dependent on n circles Cf, • • •, C*B2 if and only if it is a member of some finite set of circles C*+l, • • •, C*+t such that C*+i is linearly dependent on two of (72, • • •, C*+i_l(i= 1, 2, • • •, /•). A set of n circles is linearly independent if no one of them is linearly dependent on the rest. The set of all circles linearly dependent on three linearly independent circles is called a bundle.
EXERCISES
1. The tetracyclic coordinates of a point are proportional to the powers of the point with respect to four fixed circles. If the four circles are mutually orthogonal, the identity which they satisfy reduces to (21).
2. A homogeneous equation of the first degree in tetracyclic coordinates n-jin-sents a circle.
3. What kind of coordinates are obtained by taking as the base (a) two orthogonal circles and the two points in which they meet? (b) four points?
4. Two points of S3 correspond to orthogonal circles if and only if they are conjugate with regard to the quadric (21).
§§100,101] INVOLUTORIC COLLINEATIONS 257
5. What set of circles corresponds to the conies in which the quadric (21) is met by the planes of a self-polar tetrahedron ?
6. The direct circular transformations of v correspond to collineations of S3 which leave each imaginary regulus of (21) invariant, while the others correspond to collineations interchanging the two reguli. The direct circular transformations of TF correspond to direct collineations of Ss in the sense of §31, Chap. II.
7. The circles of a bundle correspond to the points of a plane of S8.
8. The circles common to two bundles constitute a pencil and hence corre- spond to a line of S8. Determine the projectively distinct types of pencils of circles on this basis.
9. All circles are linearly dependent on four linearly independent circles.
10. For any bundle of circles there is a point O which has the same power, c2, with respect to every circle of the bundle. The radical axes of all pairs of circles in the bundle pass through 0. In- case there is more than one point O, the radical axes of all pairs of circles of the bundle coincide.
11. A bundle of circles may consist of all circles through a point (the set of all lines in a Euclidean plane is a special case of this). In every other case there is a nondegenerate circle orthogonal to all circles of the bundle. This circle has the point O (Ex. 10) as center and consists of the points C such that Dist (OCT) = c. It is real if and only if c is real. In case c is imaginary let C"a be the real circle consisting of points C' such that Dist (0C") = c; any circle of the bundle meets C'1 in the ends of a diameter.
101 . Involutoric collineations. In view of the isomorphism between the real inversion group and the protective group of the real quadric (21), a further consideration of the group of a general quadric will be found apropos. In this connection we need to define certain particular types of involutoric collineations in any projective space. The theorems are all based on Assumptions A, E, P, HQ.
It is proved in § 29, Vol. I, that if a> is any plane and 0 any point not on w, there exists a homology carrying any point P to a point P', provided that 0, P, P' are distinct and collinear and P and P' are not on w. It follows by the constructions given in that place that if one point P is transformed into its harmonic conjugate with regard to O and the point in which the line OP meets o>, every point is transformed in this way. It is also obvious that a homology is of period two if and only if it is of this type. Hence we make the following definition :
DEFINITION. A homology of a three-space is said to be harmonic if and only if it is of period two. A harmonic homology is also called a point-plane reflection and is denoted by {Oca} or {&>#}, where 0 is the center and to the plane of fixed points.
258 INVERSION GEOMETRY [CHAP.VI
DEFINITION. If I and /' are two nonintersecting lines of a projective space S,, the transformation of S3 leaving each point of I and /' inva- riant, and carrying any other point P to the point P' such that the line PP1 meets / and /' in two points harmonically conjugate with repaid to P and /*', is called a skew involution or a line reflection in I ami I'. It is denoted by {//'}, and / and /' are called its axes or directrices.
THEOREM 31. A line re/lection {II1} is a product of two point-plane reflections {0o>} • {Pir}, where O and P are any two distinct points of I, & is the plane on P and I', and TT is the plane on 0 and I'.
Proof. Consider any plane through I, and let L be the point in which it meets /'. In this plane {Oca} and {Pir} effect harmonic homologies whose centers are 0 and P respectively and whose axes are PL and OL respectively. The product is therefore the harmonic homology whose center is L and axis I. Hence the product {Oca} • {Pir} satisfies the definition of a line reflection whose axes are I and /'.
COROLLARY. A line reflection is a projective collineation of period two, and any projective collineation of period two leaving invariant the points of two skew lines is a line reflection.
EXERCISES
1. A projective collineation of period two in a plane is a harmonic homology.
2. A projective collineation of period two in a three-space is a point-plane reflection or a line reflection.
3. Let .4, B, C, D be the vertices of a tetrahedron and a, /?, y, 8 the respec- tively opposite faces. The transformations obtainable as products of the three harmonic homologies Ma}, {B{$}, {Cy} constitute a commutative group of order 8 consisting of four point-plane reflections, three line reflections, and the identity. If the transformations other than the identity be denoted by 0, 1, 2, 3, 4, 5, 6, the multiplication table may be indicated by the modular plane given by the table (1) on p. 3, Vol. I, the rule being that the product of any two. transformations corresponding to points i, j of the modular plane is the one which corresponds to the third point on the line joining i andy.
4. (ieneralize the last exercise to n dimensions. The group of involutoric transformations carrying n +1 independent points into themselves is commu- tative, and such that its multiplication table may be represented by means of a finite projective space of n — 1 dimensions in which there are three points on each line.
5. A projectivity F of a complex line such that for one point P which is not invariant, rn(/>) = 7) is such that F" is the identity. If n is the least positive integer for which F* = 1, F is said to be cyclic of degree n ; the
§§101,102] GROUP OF A QUADRIC 259
characteristic cross ratio of F is an »th root of unity ; in case n = 3, this cross ratio is said to he equianharmonic, and a set of four points having this cross ratio is said to be eyviankarmonic. As a transformation of the inversion group, F is equivalent to a rotation of period n.
6. A plauar projective collineatioii of period n(n>2) is of Type I and the set of transforms of any point is on a conic, or else the collineation is a limnology. In the first case, it is projectively equivalent to a rotation ; in the second case, to a dilation (in general, imaginary). Consider the analogous problem in three dimensions. (For references on this and the last exercise cf. Encyclo^die des Sc. Math. Ill 8, § 14. The statements in the Kncyclopgdie on the planar case are not strictly correct, since they do not sufficiently take the existence of homologies of finite period into account.)
102. The projective group of a quadric. According to the definition in § 104, Vol. I, a quadric may be regarded as the set of points of intersection of the lines of two conjugate reguli. These two reguli may be improper in the Sense of Chap. IX, Vol. I, and in the following theorems improper elements are supposed adjoined when needed for the constructions employed.
DEFINITION. If there are proper lines on a quadric, the quadric is said to be ruled, otherwise it is said to be unruled.
THEOREM 32. A harmonic homology whose center is the pole of its plane of fixed points with regard to a quadric surface </ transforms Q* into itself in such a way that the two lines of Q* through any fixed point are interchanged.
Proof. Let 0 be a point not on Q*, and a> its polar plane. Any line / of Q2 meets o> in a unique point K. The plane 01 contains one other line /' of Q*, and (cf. § 104, Vol. I) I' passes through K. Any line join- ing 0 to a point L of I other than K must meet V in a point L' such that L and L' are harmonically conjugate (§ 104, Vol. I) with regard to 0 and the point in which OL meets <u. Hence {Oca} interchanges I and /'. From this result the theorem follows at once.
Comparing Theorems 31 and 32, we have
COROLLARY. A line reflection {ab} such that a and b are polar with respect to a quadric Q2 transforms Q* into itself in such a way that each regulus on Q* is transformed into itself.
THEOREM 33. A projective collineation of a quadric which leaves three points of the quadric invariant, no two of the three points being on the same ruler, is either the identity or a liarmonic homology whose, center and plane of fixed points are polar with respect to the quadric.
260 INVERSION GEOMETRY [CHAI-.VI
Proof. Denote the three points by A, B, C, the plane containing them by o>, and the pole of to by O. Since no two of A, B, C are on a line of </, to contains no line of Q? and hence is not on 0. Since three points of the conic in which <o meets the quadric are invariant, all such points are invariant, as is also 0. Hence the given collineation is either the identity or a homology. In the latter case it must be a harmonic homology, since any two points of the quadric collinear with 0 are harmonically conjugate with respect to 0 and the point in which the line joining them meets o>.
THEOREM 34. There exists one and only one projective collineation transforming each line of a regulus into itself and effecting a given projectivity on one of these lines. Such a collineation is a product of two line reflections whose axes are lines of the conjugate regulus.
Proof. Let R* be a regulus and R* the conjugate regulus. A projec- tivity on a line, I, of R* is by § 78, Vol. I, a product of two involutions, say I and I'. Let {w^wij be a line reflection such that ml and raa are lines of R% through the double points of I, and let {m(m^} be a line reflection such that m( and ra^ are lines of R% through the double points of I'. The product of {m[m(} and {m1m2} effects the given projectivity on I and transforms each line of Rf into itself.
Conversely, any projectivity F leaving all lines of JK* invariant effects a projectivity on I which is a product of two involutions I and I'. The line reflections {m^m^ and {m(m!2} being defined as before,
leaves all points of I invariant and hence leaves all lines of R* as well as a
ftnd
as all lines of R* invariant. Hence
{w^m J = T.
COROLLARY. The group of permutations of the lines of a regulus effected by the projective collineations transforming the regulus into itself is simply isomorphic with the projective group of a line.
DEFINITION. A collineation of a quadric which carries each regulus on the quadric into itself is said to be direct.
THEOREM 35. There is one and but one direct collineation of a quadric surface Q2 carrying an ordered triad of points of Q3, no two of which are on a line of QP, to an ordered triad of points of Q3 no two of which are on a line of Q*.
§ 102] GROUP OF A QUADRIC 261
Proof. Let ABC and PQR be the given ordered triads of points, let a, b, c,p, q, r be the lines of one regulus through the points A, B, C, P, Q, R respectively, and let a', b', c', p't q', r' respectively be the lines of the conjugate regulus through the same points. By the last theorem there is a projective collineation F carrying a, b, c to p, q, r respectively while leaving all lines of the conjugate regulus invariant, and also a projective colliueation F' carrying a'b'c' to p'q'r' respec- tively while leaving all of the lines a, b, c, p, q, r invariant. The product of F and F' carries A, B, C to P, Q, It respectively. That there is only one direct colliueation having this effect is a corollary of Theorem 33.
Let .Kf be the regulus containing the lines a, b, c, and JK2 the regulus containing a', b', c'. The two collineations F and F' which have been used in the proof above are commutative as transformations of Jtf because F' leaves all lines of R* invariant, and are commutative as transformations of JR2 because F leaves all lines of J?2 invariant. Hence
rr'=r'r.
By Theorem 34, FF' = {lm} • {rs} - {I'm1} - {rV},
where I, m, r, s are lines of R?, and I', m', r', s' are lines of R%. The collineations {rs} and {I'm'} are commutative for the same reason that F and F' are commutative. Hence
FF'= {lm} • {I'm'} • {rs} • {r's1}.
The pairs lm and I'm' are two pairs of opposite edges of a tetra- hedron the other two edges of which may be denoted by a and b. The product {lm} • {I'm'} leaves each point of a and b invariant and is involutoric on each of the lines /, I', m, m'. Hence
The lines a and b are polar with respect to R? because one of them is the line joining the point IV to the point mm', and the other the line of intersection of the plane IV with the plane mm' (cf. § 104, Vol. I). In like manner {pq} . {p,ql} = {cd } ?
where c and d are polar with respect to J?22. Hence we have
THEOREM 36. Any direct projective collineation T of a quadric surface is expressible in the form
T={ab}-{cd}, where the line a is polar to the line b, and the line c is polar to the line d.
INYKKS10N C1EOMETKY [CHAP. VI
Since any line reflection whose axes are polar with respect to a quadric is a product of two harmonic homologies whose centers are polar to their planes of fixed points (cf. Theorem 31), the last theorem implies
COROLLARY 1. Any direct protective collineation of a quadric is a product of four harmonic hotnologies whose centers are polar to their respective planes of fixed points.
COROLLARY 2. Any nondirect protective collineation of a quadric is a product of an odd number of harmonic homologies whose centers are polar to their respective planes of fixed points.
Proof. If a protective collineation F interchanges the two reguli, and A is a harmonic homology of the sort described in the statement of the corollary, then FA= A is a projective collineation leaving each regulus invariant. By Cor. 1, A is a product of an even number of harmonic homologies of the required sort, and hence F = AA is a product of an odd number.
103. Real quadrics. The isomorphism between the real inversion group and the projective collineation group of the real quadric (or sphere) (21) may now be studied more in detail. Since a circular transformation leaving three given points of the inversion plane TT invariant is the identity or an inversion (Theorem 21), and since a collineation of S3 leaving three points of the quadric (21) invariant is the identity or a harmonic homology whose center is polar to its plane of fixed points, it follows that inversions in TT correspond to homologies of S8- Hence the direct circular transformations of TT cor- respond to the direct colliueations of S3 transforming (21) into itself.
An involution in TT is a product of two inversions whose invariant circles intersect and are perpendicular. To say that the invariant circles intersect and are perpendicular is to say that they intersect in such a way that one of the circles is transformed into itself by the inversion with respect to the other. Now suppose that {Oa>} and {Pir} are the harmonic homologies corresponding to the two inver- sions. If the points of the quadric on the plane o> are to be trans- formed among themselves by {Pir}, o> must pass through P. In like manner TT must pass through 0. Hence
KKAL QUADRICS
where / is the line OP, V the line wr, and the lines I and I' are polar with respect to the quadric. Hence tin- involutions in the group of direct circular transformations correspond to the line reflections whose axes are polar with respect to (21).
Thus the theorem that any direct circular transformation of TT is a product of two involutions is equivalent to Theorem 36 applied to the quadric (21). Since an involution in TT always has two double points, we have the additional information, not contained in § 102, that every line reflection transforming the quadric (21) into itself has two and only two fixed points on the quadric. The line joining these two points is obviously one of the axes of the line reflection. Hence the line reflection has two real axes one of which meets the quadric (21) and the other of which does not.
These remarks are enough to show how the real inversion geometry can be made effective in obtaining the theory of the real quadric (21). We shall now show that any real nonruled quadric is protectively equivalent to the quadric (21), from which it follows that the real inversion geometry is equivalent to the projective geometry of any real nouruled quadric.
A nonruled quadric is obviously nondegenerate. In the complex space any two nondegenerate quadrics are protectively equivalent, because any two reguli are projectively equivalent. Since (18) repre- sents a quadric, it therefore follows that every noudegenerate quadric may be represented by an equation of the second degree.
Now let Q2 be any quadric whose polar system transforms real points into real planes, and let the frame of reference be chosen so that (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) are vertices of a real self-polar tetrahedron. The plane section by the plane #0=0 must be a conic whose equation is of the form
a!xi + atx9 + asx* = °»
and similar remarks can be made about the sections by the planes 2^= 0, #2= 0, and x^ = 0. From this it follows that (f has the equation
(22) arf + atf + atf + a^ = 0,
where «0, aY, «2, ag are real. The projective collineation
transforms (f into a quadric having one of the following equations
'
264 INVERSION GEOMETRY [CHA. vi
Any one of the eight quadrics thus represented is obviously equiva- lent projectively to one of the following three:
(24) x% + x* + xl + xl = 0,
(25) - xl + x* + xl + x* = 0,
(26) -x;-x* + xf + a:f = Q.
It is also obvious that (24) is imaginary, that (26) has real rulers, and that (25) is equivalent to (21).
EXERCISES
1. Determine the types of collineations transforming into itself (1) a real unruled quadric, (2) a real ruled quadric, (3) an imaginary quadric having a real polar system.
2. Discuss the projective groups of the three types of quadrics enumerated in the last exercise.
104. The complex inversion plane. A projective plane may be obtained from a Euclidean plane (cf. Introduction, VoL I) by adjoining ideal points and an ideal line in such a way as to make it possible to regard every collineation as a one-to-one reciprocal transformation of all points in the plane. In like manner the real inversion plane has been obtained from the real Euclidean plane by adjoining a single ideal point which serves as the correspondent of the center of each inversion. Similar considerations will now be adduced showing that an inversion in the complex plane may be rendered one to one and reciprocal by introducing two intersecting ideal lines.
In the complex projective plane an inversion has been seen (§ 94) to be a one-to-one reciprocal transformation of all points not on the sides of the singular triangle 0/,/2, and to effect a projective transfor- mation interchanging the pencil of lines on /t with the pencil of lines on If In this projectivity the line I^IZ is homologous both with 01^ and with 0/2.
In the Euclidean plane obtained by omitting the line /^ from the projective plane, it follows that the inversion is one to one and recip- rocal except for points on the two minimal lines, pQ and mQ> through O. Moreover, it effects a projective correspondence between the set of minimal lines [p] parallel with and distinct from p0 and the set of minimal lines [m] parallel with and distinct from mQ.
§104] COMPLEX INVERSION PLANE
The correspondence between any line p and the homologous line m is incomplete because there is no point on p corresponding to the intersection of m with pQ and no point on m corresponding to the intersection of p with mQ. This correspondence, however, may be made completely one to one and reciprocal by introducing an ideal point M, on m as the correspondent of the point pmQ and an ideal point JFi on p as the correspondent of the point mpQ. In order to treat all the minimal lines symmetrically, ideal points /£,' and Mi must be introduced on p0 and ra0, respectively, as mutually correspond- ing points. Also one other ideal point 0. is introduced as the correspondent of 0.
According to these conventions the line p0 together with its ideal point PJ[ is transformed into a set of points consisting of #„, ML, and all the points Mx. This set of points is therefore called an ideal line m*,. In like manner the line mo together with its ideal point Mx is transformed into a set of points consisting of #«,, P^, and all the points Px ; and this set of points is called an ideal line p*. The Euclidean
plane with the lines p*, and mx adjoined is called an inversion plane. Or to state the definition formally and without reference to a partic- ular inversion :
DEFINITION. Given a complex Euclidean plane TT and in it two pencils of minimal lines [p] and [m]. By a complex inversion plane TT is meant the set of all points of TT (referred to as ordinary points) together with a set of elements called ideal points of which there is one, denoted by J^, for each/), and one, denoted by M*, for each m, distinct JP'S and 7/i's determining distinct ideal points, and also one other ideal point which shall be denoted by #„. By a minimal line of TT is meant (1) the set of points on a p together with the corres- ponding Px, or (2) the set of points on an m together with the cor- responding 3f«, or (3) the set of all /i's together with #„, or (4) the set of all MJ$ together with O*. The minimal lines of Types (1) and (2) are called ordinary, and the lines (3) and (4) are called ideal.
'2M INVERSION GEOMETRY [CHAP.VI
A minimal line of Type (1) or (4) will lie denoted by p, of Type (2) or (3) by m ; the minimal lines of Types (3) and (4) are denoted by m,. and pv respectively.
Tbis definition is evidently such that each point of 7? is on a unique p and on a unique m.
IMIFIMTION. By an inversion I of TT is meant a transformation drliiH'd as follows by an inversion I of TT : If p0 and mQ are tin- singu- lar lint's of I, I interchanges p^ with mx, m0 with p*, and each p con- tain ing a p with the m containing the m to which p is transformed by I. A point of TT which is the intersection of a p and an m is transformed to the point which is the intersection of I (p) and I (m). The set of points of TT left invariant by an inversion is called a nondegenerate circle of TT. A pair of minimal lines, one a p and the other an m, is called a degenerate circle of TT.
By reference to § 94 it is evident that every circle of TT is a subset of the points on a circle of TT.
The complex inversion plane is perhaps best understood by setting it in correspondence with a quadric surface, the lines of one regulus on the quadric being homologous with [p] and those of the other with [wi]. This correspondence may be studied by means of tetracyclic coordinates as in § 100, but it can also be set up by means of a geometric construction as follows :
Regard the complex Euclidean plane TT with which we started as immersed in a complex Euclidean space. Let Q2 be a quadric surface such that OIl is a line of one ruling and 0/2 of the other (fig. 74). Through Jt and /2 there are two other lines of the two rulings which intersect in a point 0«. Any point P of the Euclidean plane is joined to 0. by a line which meets the quadric Q* in a unique point Q other than O* and, conversely, any point of <?2 which is not on either of the lines O^/j or O*,!^ is joined to 0* by a line which meets the Euclidean plane in a point P. Thus there is a correspondence T between the Euclidean plane and the points of Q2 not on Oxll or Ox/a. This correspondence is such that every minimal line in TT of the pencil on II corresponds to a line of the quadric which is in the same ruling with 01 lt and every line of TT of the pencil on /2 corre- sponds to a line of the quadric which is in the same ruling with O/2. From this it is evident that if ideal elements are adjoined to TT as explained above, the ideal points can be regarded as corresponding to
§104] COMPLEX INVERSION PLAM1 267
the points of the lines O./j and O./a so that there is a one-to-one reciprocal correspondence between TT and </.
Now any noudegenerate circle of TT is a conic through /t and If This is projected from 0« by a cone of lines having in common with </ the two lines O*,Il and 0^1^ It follows that the cone and Q? have also a (tonic section in common. For let Qlf <?2, Qt be three of the common points which are not on the lines O.J^ and 0«/2; the plane ^1^-2^8 meets the cone in a conic A'* and Q3 in a conic A'*. Tliese two conies have also in common the points in which they meet the lines Ooo/j and 0,0/2 (if these points coincide, Xf and K* have a common tan- gent at this point), and hence K* = A"2. The conic A"2 is nondegenerate, because a uondegenerate cone through Ox can have no other line than 0,/j and 0»/a in common with Q2. Hence every nondegenerate circle of TT corresponds under T to a section of $2 by a nontangent plane.
Conversely, if A'2 is any nondegenerate conic section which is a plane section of O2, it is projected from Ox by a cone two of whose lines are Oa>Il and 0«/2. Hence A"2 corresponds under T to a non- degenerate circle of IT.
An inversion in TT with respect to a circle Ca transforms every minimal line of the pencil [p] into that one of [ra] which meets it on (72. Let AT2 be the conic section on Q2 corresponding under T to (72. The inversion corresponds under T to a transformation of Q2 by which every line of one regulus is transformed into the line of the other regulus which meets it in a point of A"2. This is the transformation (Theorem 32) effected by a harmonic homology whose plane of fixed points contains A"2 and whose center is the polar to this plane with respect to 0A Hence every inversion in TT corresponds under T to a collineation of O2 effected by a harmonic homology whose center and plane of fixed points are polar with regard to Q2. Conversely, every such collineation of Q* evidently corresponds under T to an inversion in TT. Hence (Theorem 36, Cors. 1 and 2) the inversion group in TT is isomorphic under T with the group of projective collineations of O2, and the direct circular transformations of TT correspond to the projec- tive collineations of Q2 which carry each regulus into itself.
EXERCISE
Develop the theory of the modular inversion plane, using improper elements j'n the sense of Chap. IX, Vol. I.
268 1NYKKSION (JKOMKTKY [CHAP. VI
105. Function plane, inversion plane, and projective plane. In the theory of functions of two complex variables
F(xy}
the two variables x and y are thought of as completely independent of each other. The domain of each is the set of all complex numbers, including oo. This domain is therefore equivalent to the complex line or to the real inversion plane. Thus the domain of x may be taken to be a real unruled quadric (in particular, a sphere) and the domain of y another real unruled quadric. Or the pair of values (x, y) may be regarded as an ordered pair of points on the same real unruled quadric.
Now consider a regulus in the complex projective space and, adopt- ing the notation of the last section (fig. 74), let a scale be established on the lines p0 and m0 so that 0 is the zero in each scale. Let x be the coordinate of any point on p0 and y of any point on m0. Then a pair of values (x, y) determines a unique point on the quadric, i.e. the point of intersection of the line m through the point with x as its coordinate, and the line p through the point with y as its coordinate. Conversely, the same construction determines a pair of numbers (x, y) for each point of the quadric.
DEFINITION. The set of all ordered pairs (x, y) where x and y are complex numbers, including oo, is called a complex function plane, or the plane of the theory of functions of complex variables, or the com- plex plane of analysis. The ordered pairs (x, y) are called points. Any point for which x = oo or y = oo is said to be ideal or at infinity, and all other points are called ordinary.
The points at infinity of the function plane can be represented conveniently by replacing x by a pair of homogeneous coordinates x0, xv such that xjx9= x, and y by a pair (y0, yt) such that y{/y0 = y. Thus the points of the function plane are represented by
K'^; y0>yJ>
and the ideal points are those satisfying the condition
^o=°-
The set of ordinary points of the function plane obviously forms a Euclidean plane in which a line is the locus of an equation of the form
ax + ly + c = 0.
IH»] FUNCTION PLANE 269
This is equivalent in homogeneous coordinates to
an equation which is linear both in the pair of variables x , x and in the pair y0, y^ The most general equation which is linear in both paii-s is
(28) ax0yo + fa + 7 + 8x = 0.
This reduces to (27) if the condition be imposed that the locus shall contain the point (oo, oo) which in homogeneous coordinates is (0, 1; 0, 1).
DEFINITION. The set of points of the function plane satisfying (28) is called a circle (or a bilinear curve), and any circle of the form (27) is called a line.
The group of transformations which is indicated as most important by problems of elementary function theory has the equations
(29)
y' =
or, in homogeneous coordinates,
/OA\ 1 f\\ 21 n' "1
'- , l=r + s '
This group of transformations clearly transforms circles into circles. The subgroup obtained by imposing the conditions,
r = 0, r0 = 0,
1 2
transforms lines into lines because it leaves (oo, oo) invariant.
Returning to the interpretation of the coordinates x and y on a quadric, it is clear (cf. § 102) that every transformation (29) represents a direct collineation of the quadric, the formula in x determining the transformation of one regulus and the formula in y the transforma- tion of the conjugate regulus. Hence the fundamental group of the function plane is isomorphic with the group of direct projective collineations of a quadric surface.
270 INVERSION GEOMETRY [CHAI-.VI
The parameters x and y which determine the points of a refill us may be connected with the three-dimensional coordinates (£0, £t, £„ fa) by means of the following equations :
where ta = — 1. For the set of all points (£0, £1? £2, £3) given by these equations are the points on the quadric,
(21) *.*=*,' + # + #.
Any plane section of this quadric is given by a linear equation in £o' £i> £z' £8> which by (31) reduces to a relation of the form (28) among the parameters #0, a^; y0, y^ Hence the circles of the func- tion plane correspond to the plane sections of the quadric (21). In view of the relation already established between the groups it follows that the geometry of a quadric in a complex projective space is identical with that of a complex function plane. In view of § 104 both these geometries are identical with the complex inversion geometry.*
The complex projective plane may be contrasted with the complex inversion plane or function plane in an interesting manner as follows : The homogeneous coordinates (a0, a{, «2) may be regarded as the coefficients of a quadratic equation
(32) «X + Wi + V'=0-
Every such equation determines two and only two values of «,/«„, which may coincide or become infinite (if «2= 0); and, moreover, two distinct points of the projective plane determine distinct quadratic equations and hence distinct pairs of values of zjz$
•If one were to confine attention to real values, the definition of the plane of analysis given above would determine a set of elements abstractly equivalent to a real ruled quadric. This is distinct from the real inversion plane, because the latter is equivalent to a real nonruled quadric. For the purposes of the theory of func- tions of a real variable, however, it is usually desirable to distinguish between + oo and —QD. If this be done, the function plane is easily seen to be a figure analogous to a rectangle in a Euclidean plane. The group of transformations of such a func- tion plane does not seem to be of great interest from the projective point of view.
§§105,106] ONE-DIMENSIONAL PKOJECTIVITIES 271
The numbers (z0, z^) may be taken as homogeneous coordinates on a projective line. Thus there is a one-to-one and reciprocal corre- spondence between the points of a complex projective plane and the pairs of points on a complex projective line. It is important to notice that the pairs of points on the line are not ordered pairs, because a pair of values of zjz^ taken in either order would be the pair of roots of the same quadratic.
Now representing the points of a complex line on a real unruled quadric (e.g. a sphere), we have that the projective plane is in one- to-one reciprocal correspondence with the unordered pairs of points of the quadric. On the other hand, we have already seen that the com- plex projective plane is in one-to-one reciprocal correspondence with the ordered pairs of points of the quadric. In either case the points of a pair may coincide.
For further discussion of the subject of this section see " The Infinite Regions of Various Geometries" by M. Bocher, Bulletin of the American Mathematical Society, VoL XX (1914), p. 185.
106. Projectivities of one-dimensional forms in general. The theorems of the last four sections have established and made use of the fact that the permutations effected among the lines of a regulus by protective collineations form a group isomorphic with the projective group of a line. Now a regulus is a one-dimensional form of the second degree,* and the notion of one-dimensional projective trans- formation has been extended to all the other one-dimensional forms (Chap. VIII, Vol. I, particularly § 76). It is therefore to be expected that an analogous extension can be made to the regulus. This we shall now make, but instead of dealing with the regulus in particular, we shall restate the old definition in a form which includes the cases where the regulus is in question.
DEFINITION. A correspondence between any two one-dimensional forms whose elements are of different kinds and not such that all elements of one form are on every element of the other form is said to be perspective if it is one-to-one and reciprocal and such that each element of either form is on the corresponding element of the other form.
•The one-dimensional forms of the first and second degrees in three-space are the pencil of points, the flat pencil of lines, the pencil of planes, the point conic, the line conic, the cone of lines, the cone of planes, and the repulus.
i'7-J DIVERSION GEOMETRY [CHAP.VI
This covers the notion of perspectivity as defined in Vol. I between a pencil of points and a pencil of lines or between a pencil of lines and a point conic, etc. It also defines perspectivities between (1) the lines of a regulus and the points on a line of the conjugate regulus, (2) the lint's of a regulus and the planes on a line of the conjugate regulus, (3) the lines of a regulus and the points of a conic which is a plane section of the regulus, (4) the lines of a regulus and the planes of a cone tangent to the regulus.
DEFINITION. A correspondence between two one-dimensional forms or among the elements of a single one-dimensional form is protective if and only if it is the resultant of a sequence of perspectivities.
This definition comprehends that made in § 22, Vol. I, for forms of the first degree, and extended in § 76, Vol. I, so as to include those of the second degree equivalent under duality to a point conic. In order to justify the new definition, it is necessary to prove that it does not lead to any modification of the relation of perspectivity between one- dimensional forms of the first degree. In other words, we must prove that any correspondence between two one-dimensional forms of the first degree is protective according to the new definition only if it is protective according to the definition of § £2, Vol. I.
To prove this theorem it is sufficient to show that a sequence of perspectivities beginning and ending with forms of the first degree and involving forms of the second degree can be replaced by one involving only forms of the first degree. This follows directly from the fact that each one-dimensional form of the second degree is generated by projective one-dimensional forms of the first degree. For example, if a pencil of points [P] is perspective with a regulus [/] and the regulus with a point conic and the point conic with some- thing else, it follows by the theorems of § 103, Vol. I, that [P] is perspective with the pencil of planes [ml], where m is a line of the conjugate regulus and [ml] is perspective with the point conic. Thus the regulus [/] in this sequence of perspectivities is replaced by the pencil of planes [ml]. In similar fashion it can be shown by a con- sideration of the finite number of possible cases that however a form of the second degree may intervene in a sequence of perspectivities, it can be replaced by a form or forms of the first degree. The enumeration of the possible cases is left to the reader, the argument required in each case being obvious.
§§ KM, 107] ONE-DIMENSIONAL PROJECTIVITIES 273
From this theorem it follows that the group of protective corre- spondences of any one-dimensional form with itself is isomorphic vith the projective group of a line. For let F be any projectivity of a OIM-- dimensional form F' of the second degree (e.g. a regulus), and let II represent a perspectivity between f* and a one-dimensional form /• : of the first degree (e.g. a line of the conjugate regulus). Then FiFD""1 is a projectivity of F1. In like manner, if F' is a projectivity of /'*, n^'FTI is a projectivity of F'\ Hence II establishes an isomorphism between the two groups.
*107. Projectivities of a quadric. An involution on a regulus is the transformation of the lines of the regulus effected by a line reflection whose axes are the double lines of the involution. Since any projectivity of a regulus is a product of two involutions, it may be regarded as effected by a three-dimensional projective collineatiou which transforms the regulus into itself. Conversely, any direct pro- jective collineatiou transforming a quadric into itself is a product of two line reflections (Theorem 36) each of which effects an involution on each of the reguli on the quadric.
This relation between the theory of one-dimensional projectivities and the projective group of a quadric may be used to obtain prop- erties of the quadric analogous to the properties of conic sections studied in Chap. VIII, Vol. I. The discussion is based on Assump- tions A, E, P, HQ, improper points being adjoined to the space whenever this is required for quadratic constructions.
In Chap. VIII, Vol. I, we have seen that any projectivity on a conic determines a unique point, the center of the projectivity, and that the axes of any two involutions into which the projectivity may be resolved pass through its center. If, now, a projectivity F be given on a regulus, any plane TT meets the regulus in a conic C* on which is determined a projectivity F' having a point P as center. This determines a correspondence between the planes TT and points P of space which is a null system (§ 108, Vol. I), and hence the axes of the involutions into which the projectivity F' can be resolved form a linear complex. The formal proof of this statement follows.
THEOREM 37. For any nonidentical projectivity of a regulus there exists a linear complex of lines [1] having the property that if l^ is any line of the complex not tangent to the regulus. there are three lines lf.
INYKKSION GEOMETRY [CHAI-. vi
/3, /4 su,ch that 13 is polar to ^ and /8 to /4 with respect to the regulus, and such that the collineation
effects the given projectivity on the regulus. Moreover, every line l^ ha ring this property belongs to the complex, and so do /2, /8, /4.
Proof. Let A*2 be a regulus and F a projectivity of AJ2. If IJ2 and /8/4 are pairs of polar lines such that {/^2} • {/8/4} effects the given projectivity on R*, let TT be any plane containing lv and not tangent to A*2. The projectivity F on R* is perspective with a projectivity F' on the conic C'2 in which TT meets R\ Moreover, {IJ2} and {I.J.J effect involutions on A!2 which are perspective with involutions I' and I" on C2. Thus on C2 r" — l'\"
But (cf. § 77, Vol. I) ^ is the axis of I' and hence passes through the center of F'. A similar argument shows that lt (i = 2, 3, 4) passes through the center of the projectivity perspective with F on the conic in which A*2 is met by any plane containing lt and not tangent to A?2.
Hence all lines llt 12, 13, 14 defined as above are contained in the set [I] of all lines I such that if TT is any plane on / and not tangent to A*2, 1 is also on a point P defined as follows : Let C2 be the conic in which TT meets A*2 and F' the projectivity on (72 perspective with the projectivity F on R2; then P is the center of F'.
The set [I] obviously contains all lines tangent to R* at points of the double lines (if existent) of F. If /x is any other line of [/] let TT be a plane on ^ and not tangent to R2, let C2 be the conic in which TT meets A'2, and let F' be the projectivity on C2 perspective with F. By § 79, Vol. I, and the definition of [/], F' is a product of two involutions having /t and another line, la, as axes. Let /2 and 14 be the polars of /, and la respectively. Then {l^} - {1314} effects the perspectivity F' on C and hence effects F on A*2. By the first paragraph of the proof lz, l.A, 14 are all lines of [I]. Hence all lines of [/] have the property enunci- ated in the theorem. It remains to prove that [I] is a linear complex.
By definition, if TT is a plane not tangent to A)2 the lines of [/] in TT form a flat pencil. If TT is tangent to A2 let p be the line of A2 on TT and q the line of the conjugate regulus on TT. In case p is a fixed line of F, the lines I on TT are the tangents to A12, i.e. the pencil of lines on TT and the point pq. In case p is not a fixed line of F, q is a tangent to A!2 which meets a fixed line of F and hence is
§107] PROJECTIVITIES OF A QUADRIC 275
a line of [1]. Any other line ^ of [/] in TT must have a polar line /, passing through the point pq. Let F" be the projectivity on q perspec- tive with F. If F is effected by {IJJ • {IJ.}, then F" is the product of two involutions, I' and I", which are perspective with the involu- tions effected on It* by {/^J and {/g/4} respectively. Since l^ must piss through the point pq, the latter is a double point of I'. But when F" is expressed as a product of two involutions, one of these involutions is fully determined by one of its double points in case the latter is not a double point of F" (cf. § 78, Vol. I). Hence the other double point, P, is fixed ; and since /t must pass through it, it follows that all lines of [/] on TT pass through P. Moreover, it is evident that if ^ is any line (except q) on TT and P, ly its polar line, and {/8/J any line reflection effecting an involution on JK2 which is perspective with I", the projectivity F is effected by {IJJ • {IJJ- Hence [7] contains all lines on TT and P. Hence [/] is a linear complex by Theorem 24, Chap. XT, VoL L
THEOREM 38. A direct projectivity F of a quadric surface Q2 which does not leave all lines of either regulus invariant determines a linear conyruence of lines having the property that if al is any line of the congruence not tangent to Q2 there exist lines a2, 6^ b2 of the congruence such that
(33) r = {a1aa}.{616J.
Moreover, each line a^ having this property belongs to the congruence, and so do «2, blf Z>2.
Proof. F effects a projectivity on each regulus of Q*t and each of these reguli by the last theorems determines a linear complex of lines. The two complexes are obviously not identical and hence have a linear congruence in common. Any line «x of this congruence is either tangent to Q2, or such that there exist lines a2, blt by which are in both complexes and such that {a^} • {^2} effects the same projectivity as F on both reguli. Hence {«ta2} • {6,6,} = F. Moreover, any al for which «2, Jj, &2 exist satisfying this condition must, by the last theorem, belong to both complexes and hence belong to this congruence.
COROLLARY 1. TJie congruence referred to in the theorem may be degenerate and consist of all lines on a point of ($* and on a plane tangent to Q? at this point ; or it may be parabolic and have a line of the quadric as directrix ; or it may be hyperbolic and have a pair
276 IN VK us ION GEOMETRY [CHAI- vi
of polar lines as directrices ; or it may be elliptic and have a pair of i>iij>rti/»?r polar lines as directrices.
Proof. Let C denote the congruence referred to in the theorem and let IT be the polarity by which every point is transformed into its polar plane with respect to Q?. This polarity transforms any line a of C into its polar line, and the latter, by the theorem, is in C. Hence IT transforms C into itself.
According to § 107, Vol. I, any congruence is either degenerate, parabolic, hyperbolic, or elliptic. If degenerate, it consists of all lines on a point R or a plane p, R being on p. If II transforms such a con- gruence into itself, it must interchange R and p, and hence R must be on Q2 and p tangent to Q* at R. The congruence C will be of this type if 6j meets al in a point of Q* and does not meet a2.
If C is parabolic, its one directrix must be transformed into itself by II, and hence must be a line of Q*. This case arises if alf a2, blt bt all meet the same line of Q2 and do not meet any other line of Q*.
If C is hyperbolic, II must either leave the two directrices fixed individually or interchange them. In the first case each directrix must be a line of tf, which implies that alt «2, blt £>2 all meet two lines of Q* and hence that all lines of one regulus are left invariant by F, contrary to hypothesis. Hence the second case is the only possible one. It occurs when alt «2, blf b2 do not all meet any line of <f, but are met by a pair of real lines.
If C is elliptic, it has two improper directrices* and the reasoning is the same as for the hyperbolic case.
DEFINITION. A line / is said to meet or to be met by a pair of lines pq if and only if it meets both of them. A pair of lines Im is said to meet or cross a pair pq if both I and m meet pq.
EXERCISES
1. The lines which cross the distinct pairs of an involution on a regains together with the lines tangent to the regulus at points of the double lines (it existent) of the involution form a nondegenerate linear complex.
2. If two pairs of polar lines, rtj«2 and bfa, of a regulus meet each other, the involutions effected by {«i02} and {hfa} are harmonic (commutative) and their double lines form a harmonic set.
* This may be proved as follows : Let lv 1.,, J3, Z4 be lines of C not on the same regulus. Any plane on lt meets the regulus R* containing lv i2, ls in a conic, and Z4 meets this conic in two improper points Pt, P2. The two lines of the regulus conju- gate to IP which pass through Pv P '_, meet J1? f2, la, 14 and hence meet all lines of C.
§§ 107, 108] PROJECTIV1TIES OF A QUADR1C 277
3. Let T be a projectivity on a regains A'-. A variable plane meets A'2 in a conic C3 on which there is a projectivity P i>erspective with F. The axes of the projectivities I" are lines of a linear congruence.
4. Enumerate the types of collineations leaving invariant a quadric (1) in the complex space, (2) in a real space, (3) in various modular spaces.
* 108. Products of pairs of involutoric projectivities.
THEOREM 39. A direct protective collineation of a quadric surface is a line reflection whose axes are polar, if it interchanges two points of the quadric which are not joined by a line of the quadric.
Proof. Denote the collineation by F, the quadric by Q*, the two reguli on it by R* and R^, and the two points which F interchanges by A and B. Let a and b be the lines of Rf on A and B respectively, and a' and b' those of R% on A and B respectively. Since F interchanges a and b it effects an involution on R£, and since it interchanges a' and b' it effects an involution on R%. Let I, m be the double lines of the involution on Rf, and p, q those of the involution on R%. F is evidently the product of {lm} by {pq} and hence is a line reflection whose axes are the line joining the points Ip and mq and the line joining the planes Ip and mq. These two lines are polar with regard to Q2.
THEOREM 40. Two lines which are not on a quadric Q2 and do not meet the same line of Q* are met by one and but one polar pair of lines.
Proof. Let one of the given lines meet the quadric in A and A' and the other meet it in B and B'. By Theorem 35 there is a unique direct projective collineation of the quadric which carries A to A', A' to A, and B to B'. By Theorem 39 this is a line reflection {lm} and / and m are polar with respect to Q3. Since {lm} transforms A to A', I and m both meet the line AA', and since {lm} transforms B to B', I and m both meet the line BB'.
If there were another pair of polar lines I', mf meeting AA' and BB', {I'm'} would interchange A and A' and B and B'. By Theorem 35 {lm} = {I'm1}.
COROLLARY. Two lines which are not on a quadric Q? and do not meet the same line of Q2 are met by two and only two lines which are conjugate to them both with regard to Q2.
Proof. This follows directly from the theorem, because two mutu- ally polar lines a, b meeting two lines / and m are both conjugate to
•J7S INVKKSloN (IKOMKTKY [CHAI-. vi
/ and m and, moreover, if a line a meets and is conjugate to both I and in its polar line also meets and is conjugate to both / and m.
THKOKKM 41. If a simple hexagon is inscribed in a quadric sur- face in such a way that no two of its vertices are on a line of tlie quadric, the three pairs of opposite edges are met each by a polar pair of lines, and these three polar pairs of lines are in the same linear congruence.
Proof. Let A^B^C^B^C^ be the simple hexagon. By the last theorem the pair of opposite edges A^B^, A^Bl is met by a pair of lines Cj, ca which are polar with respect to the quadric. In like manner B9Clt BvCt are met by a polar pair ajf a3, and C^, by a polar pair 6^ b2. Consider the product of line reflections,
The line reflection (fi^aj carries Bl to C2, {bfa} carries Ca to Alt and {CjCj} carries Al to By. Likewise {«j«2} carries B^ to Clt {bfa} carries Cl to A2, and {c^J carries AZ to Bf Hence F interchanges 7?t and B^, and by Theorem 39 it is a line reflection. Denoting F by {d^} we have
By Theorem 38 the axes of the four line reflections in this equation are all lines of the same congruence.
In view of the corollaries of Theorems 38 and 40 this theorem may be restated in the following forms :
COROLLARY I. If a simple hexagon is inscribed in a quadric in such a way that no two of its vertices are on a line of the quadric, the three polar pairs of lines which meet the pairs of opposite edges are met by a polar pair of lines (which may coincide).
COROLLARY 2. If a simple hexagon is inscribed in a quadric sur- face in such a way that no two of its vertices 'are on a line of the quadric, each pair of opposite edges is met by a unique pair of lines conjugate to both edges, and the latter three pairs of lines are met by a pair of lines conjugate to each of them. The lines of the last pair may coincide*
* Bulletin of the American Mathematical Society, Vol. XVI (1909), pp. 65 and 62. A Hit-nrcm of non-Euclidean geometry from which this may be obtained by generalization has been given by F. Klein, Mathematische Annalen, Vol. XXII (1883), p. 248.
§108] PRODUCTS OF INVOLUTIONS 279
This theorem is closely analogous to Pascal's theorem on conic sections (Chap. V, Vol. I). In the Pascal hexagon the pairs of opposite sides deter- iniiu' three points .,-1, B, C which are collinear. In the hexagon inscribed in a ([iiadric they determine three pairs of lines a^, b^, CjC2 which are in a linear congruence. In case the vertices of the hexagon are coplanar, the theorem on the quadric reduces directly to Pascal's.
The Pascal theorem may be proved by precisely the method used above. For let AlBtClAtB1Cz be a hexagon inscribed in a conic and let A be the jKiint (BVCV C\B3), B be (C\A2, A^\)t and C be (A&, B^A^). Let {Aa}, /.'//;, and {Co} be the harmonic homologies effecting the involutions having .1, B, Cas centers. By construction the projectivity effected by {Cc} • {Bb} • {Aa} on the conic carries Bl to B2, and />., to 7>p and hence is an involution. Denoting its center and axis by D and </, we have
{Cc} • {Bb} • {Aa} = {Dd}. This implies {Bb} - {Aa} = {Cc} • {Dd}.
I5y the theorems of Chap. VIII, Vol. I, the line AB is the axis of the projec- tivity effected by {Bb} • {Aa} and must contain C and D. Hence A, B, C are collinear.
Pascal's theorem is thus based on the proposition that the product of three involutions on a conic is itself an involution if and only if the centers of the three involutions are collinear, i.e. if and only if their axes are concurrent. Let us denote an involution whose double points are L and M by {LM}, as in Ex. 11, § 52. If the involution is represented on a conic, the double points are joined by the axis of the involution. -The proposition above then takes the form : The product {L9M3} • {LZMZ} • {Z13/1} is an involution if and only if the lines LlMl, LzMt, L3J\I3 concur. The concurrence of the three lines means either that the three point pairs have a point in common or that they are themselves pairs of an involution. Thus the theorem on involutions may be stated as follows :
THEOREM 42. In any one-dimensional form a product of three involutions i/.pl/j}, {L23/2}, {ZgA/j} itt an involution in cage the pairs of points L1MV I...M.,, £3 A/g have a point in common or are pairs of an involution; and t/n- /innluct is not an involution in any other case.
The double points of the involutions may be either proper or improper (real or imaginary). In order to state the result entirely in terms of proper elements, the involutions may be represented on a conic and the condition stated in terms of the concurrence of their axes, as above; or it may be expressed by saying that they all belong to the same pencil of involutions, or by saying that they are all harmonic to the same projectivity.
This theorem on involutions in a one-dimensional form is fundamental in the theory of those groups of projectivities, in a space of any number of dimensions, which are products of involutoric projectivities. For example,
i2so INVERSION GEOMETRY [CHAF.VI
it is essentially the same as Theorem 8, Chap. IV, which was fundamental in the theory of the parabolic metric group in the plane. Corresponding theo- rems in tin- Kin-liileaii g«-ometry of three dimensions will be found in $§ 114 and 121, Chap. VII. Tin- same principle appears as Theorem 27, Cor. 1, Chap. Ill, in connection witli the equiaffine group.
These L;n>u|>s are all projective and on that account related to the projec- tive group of a one-dimensional form. But the essential feature which they have in common is that ecery transformation of each group is a product of tiro inroliitnrii- transformations of the same group. On this account, even without their common projective basis, the geometries corresponding to these groups must have many features in common. In particular, whenever there is some class of figures such that if two of the figures are interchanged by a trans- formation, the transformation is of period two, there must exist a theorem analogous to Pascal's theorem. As examples of this may be cited Theorem 41 above; Ex. 6, § 80, Vol. I; Ex. 1, § 122, below; and in the list of exercises below, Ex. 4, referring to the group of point reflections and translations, Exs. 5, 6 referring to the Euclidean group in a plane, Ex. 7 referring to the equiaffine group. On this subject in particular and also on the general theory of groups generated by transformations of period two, the reader should consult a series of articles by H. Wiener in the Berichte der Gesell- schaft der Wissenschaften zu Leipzig, Vol. XLTI (1890), pp. 13, 71, 'Jl.~>: Vol. XLIII (1891), pp. 424, 644; and also the article by Wiener referred to in § 45, above. Cf. also § 80, Vol. I.
EXERCISES
1. (Converse of Theorem 41.) If the three pairs of opposite edges of a simple hexagon are met by three pairs of lines «1«2, o^, c^c^ in pairs of points which are harmonically conjugate to the pairs of vertices with which they are collinear, and if the lines av a2, bv 62, cv cz are in the same linear congruence, then the vertices of the hexagon are on a quadric surface with regard to which Ojn,,, ijftj, CjC2 are polar pairs of lines.
2. Two pairs of lines which are polar with regard to the same regulus cannot consist of lines of a common regulus.
3. If two lines / and m are met by two pairs of lines which are polar with respect to a quadric, / and m are polar.
4. In a Euclidean plane let A, B, C be the three points of intersection of pairs of opposite sides of a simple hexagon. If A and B are mid-points of the sides containing them, and C is the mid-point of one side containing it, then C is also a mid-j>oint of the other side containing it.
5. Let AlBtClAtBlG9 be a simple hexagon in a Euclidean plane. If the j>erpendicular bisector of the point pair Alli^ coincides with that of A^B^ and the |>eri«-ndicular bisector of BtCl with that of /^C,,, and the perpendicular bisector of C*2/l t with that of t\Av then the three perpendicular bisectors meet in a point.
§§ioK,i09] CONJUGATE IMAGINARY LINES 281
6. Let a, b, c, a', I/, c' be six concurrent lines of a Euclidean plane. If there is a pair of lines bisecting each of the pairs ah' and nit, and a pair bisecting bS aud l>'c, there is a pair bisecting ca and c'n.
7. If the pairs of opposite sides of a simple hexagon are parallel, the lines
their mid-points are concurrent.
109. Conjugate imaginary lines of the second kind. The theory of antiprojectivities (§ 99) and the extended theory of projectivities of one-dimensional forms (§ 106) will now enable us to complete the theory of conjugate imaginary elements in certain essential details which we were not ready to discuss in § 78. Let S' be a complex projective space aud let S be a three-chain of S', Le. a space related to S' in the manner described in §§ 6 and 70, and let us use the definitions and notations of § 70. The simplest type of antiprojective colluieation of S' is given by the equations
(o4)
The frame of reference is such that the points of S have real coordi- nates. The transformation changes each point
K + #o' «! + #i' «2 + #>' "a + #.)• where the a's and /3's are real, into the point
(«0-*0o» ai -*'#!> a2-*&' «,-*$•)• These two points if distinct are joined by the real line
(«0 + X/3o, «t + \/3lt «2 + X^2, a. + X/33)
and are the double points of the involution determined by the transformation of the parameter X,
Comparing with the definition of conjugate imaginary points in § 7<S, it is clear that (34) is the transformation by which every point of S' goes to its conjugate imaginary point, the points of S being regarded as real
From the fact that the transformation (34) leaves no imaginary point invariant, it follows that it cannot leave any imaginary line or plane invariant. For the real line through an imaginary point P of the given line or plane is left invariant by (34), and hence P would be left invariant by (34). On the other hand, (34) leaves every real
282 INVERSION GEOMETRY [CHAP vi
element invariant and hence leaves every elliptic involution in a real one-dimensional form invariant. Since (34) cannot leave the double elements of such an involution invariant, it must interchange them. Hence (34) interchanges any element of S' with the element which is its conjugate imaginary according to the definition of § 78.
The definition of § 78 defines the notion of conjugate imaginary elements for all one-dimensional forms of the first or second degrees, and the theorems of that section cover all cases except that of a pair of conjugate imaginary lines which are the double lines of an elliptic involution in the lines of a regulus.
DEFINITION. An imaginary line which is a double line of an ellip- tic involution in a flat pencil is said to be of the first kind, and one which is a double line of an elliptic involution in a regulus is said to be of the second kind.
THEOREM 43. Any imaginary line is either of the first or of the second kind.
Proof. Let I be an imaginary line. It cannot contain two real points, else it would be a real line (§ 70). Hence it contains one or no real point. In the first case let 0 be the real point on I, P one of the imaginary points on I, and P the imaginary point conjugate to P. The line PP is real, and hence the plane OPP is real. Hence by § 78 the lines OP and OP are the double lines of an elliptic involution in the pencil of real lines on the point O and the plane OPP.
In the second case let P, Q and R be three points of I and let P, Q and It be their respective conjugate imaginary points. The lines PP, QQ, RR are real and no two of them can intersect, for if they did I would be on a real plane, and we should have the case considered in the last paragraph. Hence these lines determine a regulus R? in S. On the real line PP there is by § 78 an elliptic involution having P and P as its imaginary double points. Hence there is an elliptic involution in the regulus R'*, conjugate to R-, having I as one double line and a line I through P as the other. The lines I and I are conjugate imaginary lines by definition, and satisfy the definition of imaginary lines of the second kind. Since (34) transforms each element into its conjugate element, it is clear that I contains Q and R as well as P.
The system of real Hues obtained by joining each point of I to its conjugate imaginary point on I is, by the reasoning above, a set of
§i«)] CONJUGATE IMAGINARY LINES 283
lines of the real space S, no two of \vliidi intersect. Any four of them determine a linear congruence (§ 107, Vol. 1) C in S and also a linear congruence C of S'. The congruence C has the property that each of its lines is contained in a line of 6', and C evidently is the set of all lines joining points of / to points of L Hence C is an elliptic con- gruence according to the definition of § 107, Vol. I, and consists of all real lines meeting I and L Hence the system of real lines joining points of I to their conjugate imaginary points is an elliptic congruence in S, or in other words :
THEOREM 44. An imaginary line of the second kind is a directrix of an elliptic congruence.
The observation, made in the argument above, that there is one line of a certain elliptic congruence through each point of an imaginary line of the second kind, shows that an elliptic congruence may be taken as a real image of a complex one-dimensional form. This of course implies that the whole of the real inversion geometry can be' carried over into the theory of the elliptic congruence and vice versa. Cf. the exercises below.
The relations between the imaginary lines of the second kind and the regulus and elliptic congruence are fundamental in the von Staudt theory of imaginaries which has been referred to in § 6. In addition to the references given in that place, the reader may consult the Ency- clopedic des Sciences Mathematiques, III 8, § 19, and III 3, §§ 14, 15.
EXERCISES
\
1. An elliptic congruence in a real space has a pair of conjugate imaginary lines of the second kind as directrices.
2. The correspondence by which each point of an imaginary line / corre- sponds to its conjugate imaginary point is an antiprojectivity between / and its conjugate imaginary line.
3. Under the projective group of a real space any imaginary point is trans- formable into any other imaginary point, any imaginary line of the first kind into any imaginary line of the first kind, and any imaginary line of the second kind into any imaginary line of the second kind; an imaginary line of the first kind is not transformable into one of the second kind.
4. There is a one-to-one reciprocal correspondence between the points of a complex line and the lines of an elliptic congruence in a real space in which the points of a chain correspond to the lines of a regulus. By means of this correspondence, make a study of the elliptic congruence and its group.
INYKKSION (JKOMETRY [CHAI-. vi
5. I.et S, !•«• ii three-dimensional complex space. Any five nonroplanar ]M>inU «>f SJ determine a unique three-chain, which is a real Ss. This S:; is related to S» in the manner described in jjjj ti and 70. Through any |Miint /' of S,,' not on S;1, there is (§ 78) a unique line which contains a line of S. (i.e. a chain C,) as a subset. On this chain C, there is a unique elliptic invo- lution having /' as a cloul>l«> point. Let 7* be the other double point of this involution. /' ami 1' are the conjugate imaginary points with regard to Un- real space S,, and the transformation of S3' by which each point P not on S, goes to P, and each point on Sa is left invariant, may be called a reflec- tion in the three-fhain Ss. Any transformation which is a product of an odd number of reflections in three-chains is an antiprojective collineation, and any transformation which is a product of an even number of reflections in three-chains is a projective collineation. Every collineation is expressible in this form.
110. The principle of transference. We have seen how the geometry of the inversion group in the plane, arising initially as an extension of the Euclidean group, is equivalent to the projective geometry of the complex line and also to that of a real quadric which may be specialized as a sphere. We have also seen the equivalence of the projective groups of all one-dimensional forms in any properly pro- jective space. Since the regulus is a one-dimensional form, this gave a hold on the group of the general quadric. The latter group in a complex space has been seen to be isomorphic with the complex inversion group and also with the fundamental group of the function plane.
At each step we have helped ourselves forward by transferring the results of one geometry to another, combining these with easily obtained theorems of the second geometry, and thus extending our knowledge of both. This is one of the characteristic methods of modern geometry. It was perhaps first used with clear understanding by O. Hesse,* and was formulated as a definite geometrical principle (Uebertragungsprinzip) by F. Klein in the article referred to in § 34.
This principle of transference or of carrying over the results of one geometry to another may be stated as follows : Given a set of elements [e] and a group G of permutations of these elements, and a set of theorems [T] which state relations left invariant by G. Let [er] be another set of elements, and G' a group of permutations of [e']. If there is a one-to-one reciprocal correspondence between [e] and \e'~\
* Gesaimuelte Werke, p. 681.
* iio.l PRINCIPLE OF TRANSFERENCE 285
•in irhich G is simply isomorphic with G', the set of theorems [T] deter- mines by a mere change of terminology a set of theorems [T1] which state relations among elements e' which are left invariant by d'.
This principle becomes effective when the method by which [e] and G are defined is such as to make it easy to derive theorems which are not so easily seen for [e'] and G'. This has been abundantly illustrated in the present chapter, but the series of geometries equiv- alent to the projective geometry on a line could be much extended. Some of the possible extensions are mentioned in the exercises below.
From the example of the conic and the quadric surface (§ 107) it is clear that in order to carry results over from the theory of a set [e] and a group G1 to a set [e'] and a group G' it is not necessary that the correspondence be one-to-one. The transference of theorems is, however, no longer a mere translation from one language, as it were, to another, but involves a study of the nature of the correspondence.
DEFINITION. Given a set of elements [«] and a group G of permu- tations of [e], the set of theorems [T] which state relations among the elements of [e] which are left invariant by G and are not left invariant by any group of permutations containing G is called a generalized geometry or a branch of mathematics*
This is, of course, a generalization of the definition of a geometry employed in §§34 and 39. At the time when the role of groups in geometry was outlined by Klein, the only sets [e] under consideration were continuous manifolds, i.e. complex spaces of n dimensions or loci defined by one or more analytic relations among the coordinates of points in such spaces. The older writers restrict the term "geometry" by means of this restriction on the set [e]. But in view of the exist- ence of modular spaces and other sets of elements determining sets of theorems more nearly identical with ordinary geometry than some of those admitted by Klein's original definition, it seems desirable to state the definition in the form adopted above.
In case the set of theorems [T] is arranged deductively, as explained in the introduction to Vol. I, it becomes a mathematical science. The problem of the foundation of such a science is that of determining, if possible, a finite set of assumptions from which [ T] may be deduced.
* The generalized conception of a geometry is discussed very clearly in the article by G. Fano in the Encyclopadie der Math. Wiss. Ill AB 46. A number of special cases are outlined in the latter half of the article.
INVKKSinN < ; Ko.M KTltY ' [CHAP. VI
EXERCISES
1. If a projective collineation interchanges tin- two rcguli on a qnadric, homologous lines of the two refill i meet in ]>oints of a plane.
*2. Let Jl- l>e a regulus, <D a plane not tangent to If-, ami () the pole of w (<i> may conveniently lit- regarded as the plane at infinity of a Euclidean space). A projectivity F of J!' may be effected l>y a collineation I" leaving all lines of the conjugate regains invariant. This collineation multiplied by the harmonic lioinology i <>w, gives a collineation F" interchanging the two regnli. Hy Ex. 1, F" determines a unique plane. Let /' lie the point polar to F" with regard t . /.'-. The correspondence thus determined between the projectivities F of I!- and the jx)ints of space not on It- is one to one and reciprocal. It is such that projertivities which are harmonic (§ 80, Vol. I) correspond to conjugate points with respect to /.'-, and all the involutions correspond to points of o>.
*3. The construction of Ex. '1 sets up a correspondence between the pro- jectivities of a one-dimensional form and the points of a three-dimensional space which are not on a certain quadric. The same correspondence may be
obtained by letting a projectivity
x, _ apx + a,
o-jX + a3
correspond to the point (a0, ap a%, a3). The relations between the one-dimen- sional and three-dimensional projective geometries thus obtained have been studied by C. Stephanos, Mathematische Annalen, Vol. XXII (1883), p. 2!)'.». *4. Develop the theory of the twisted cubic curve in space along the fol- lowing lines: (1) Define it algebraically. (2) (live a, geometric definition. (3) Prove that Definitions (1) and (2) are equivalent. (4) Derive the further theorems on the cubic as far as possible from the geometric definition. It will be found that the properties of this cubic can be obtained largely from those of conic sections and one-dimensional projectivities in view of an isomorphism of the groups in question. The theorems should be classified according to the principle laid down in § 83.
* 5. A rational curve in a space of k dimensions is a locus given paramet- rically as follows :
*0 = fl0(0, *l = ^'l CO'""' *»=^n(0>
where /?„(<), • • •, -Kn(0 are rational functions of /. In case k — n and the locus is not contained in any space of less than n dimensions, the curve is a norm id run-?. Develop the theories of various rational curves along the lines outlined in Ex. 4. For reference cf. § 28 of the encyclopedia article by Fa no referred to above and articles by several authors jn recent volumes of the American Journal of Mathematics.
* 6. The linear dependence of conic sections may be denned by substituting "point conic" or "line conic," as the case may be, for "circle " in the definition given at the end of § 100. Develop the theory of linear families of conies of one, two, three, and four dimensions, using the principle of correspondence whenev.-r pov>ible and clarifying theorems according to the principle laid down in § S3. Cf. Kncyclopedie des Sc. Math. Ill 18,
CHAPTER VII
AFFINE AND EUCLIDEAN GEOMETRY OF THREE DIMENSIONS
111. Affine geometry. DEFINITION. Let TTX be an arbitrary but fixed plane of a projective space S. The set of points of S not on 7r» is called a Euclidean space and IT* is called the plane at infinity of this space. The plane TT*, and the points and lines on TT« are said to be ideal or at infinity, all other points, lines, and planes of S are said to be ordinary. When no other indication is given, a point, line, or plane is understood to be ordinary. Any projective collinea- tion transforming a Euclidean space into itself is said to be affine ; the group of all such collineations is called the affine group of three dimensions, and the corresponding geometry the affine geometry of three dimensions.
DEFINITION. Two ordinary lines which have an ideal point in common are said to be parallel to each other. Two ordinary planes which have an ideal line in common, or an ordinary line and an ordinary plane which have an ideal point in common, are said to be parallel to each other.
In particular, a line or plane is said to be parallel to itself or to any plane or line which it is on. For ordinary points, lines, and planes we have as an obvious consequence of the assumptions and definitions of Chap. I, VoL I, the following theorem :
THEOREM 1. Through a given point there is one and only one line parallel to a given line. Through a given point there is one and only one plane parallel to a given plane. If two lines, I and V, are not in the same plane there is one and only one plane through a given point parallel to I and I'. If I and V are parallel, any plane through I is parallel to I'.
Another obvious though important theorem is the following : THEOREM 2. The transformations effected in an ordinary plane -TT
by the affine group in space constitute the affine group of the Euclidean
plane consisting of the ordinary points of TT.
287
288 AFKINK AM) EUCLIDEAN GEOMETRY [CHAI-.VII
In consequence of this theorem we have the whole affine plane geometry as a part of the affine geometry of three dimensions, and we shall take all the detinitions and theorems of Chap. Ill for granted without further comment.
This discussion is valid for any space satisfying Assumptions A, E. The affine geometry of an ordered space (A, E, S) has already l»-rn considered in § 31, and certain additional theorems are given in Kxs. 5~7 below.
EXERCISES
1. The lines joining the mid-points of the pairs of vertices of a tetrahedron meet in a point
2. Classify the quadric surfaces from the point of view of real affine geometry. Develop the theory of diametral lines and planes. The real projec- tive classification of the nondegenerate quadrics has been given in § 103. The affine classification is given in the Encyclopedic des Sc. Math. Ill 22, § 19.
*3. Classify the linear congruences from the point of view of the real affine geometry. Cf. § 107, Vol. I.
*4. Classify the linear complexes from the point of view of real affine geometry. Cf. § 108, Vol. I.
5. With respect to the coordinate system used in § 31 the points of the line joining A = (av a2, as) and B = (liv b2, &3) are
/«i_+_X&i \ 1 + X '
1 + X 1 + X
B corresponding to X = =e and the point at infinity to X = — 1. The segment - 1 B consists of the points for which A > 0 and its two prolongations of those for which X < — 1 and — 1 < X < 0 respectively.
6. Two points D and U are on the same side of the plane ABC if and
S (A BCD) = S (A BCD').
7. Using the notation of § 101 and dealing with an ordered Euclidean space, { Ota} is an affine collineation which alters sense if O or o> is at infinity and {//'} is an affine collineation which does not alter sense if I or /' is at infinity. In an ordered protective space {II'} is, and {O<o} is not, a direct collineation.
112. Vectors, equivalence of point triads, etc. DEFINITION. An elation having TT. as its plane of fixed points is called a translation. If / is an ordinary line on the center of the translation, the translation is said to be parallel to /.
The properties of the group of translations follow in large part fmm the .following evident theorem.
§iu] AFFINE GEOMETRY 289
THEOREM 3. The transformations effected in an ordinary plane TT by the translations leaving TT invariant constitute the group of trans- lations of the Euclidean plane composed of the ordinary points of ir.
As corollaries of this we have statements about translations in space which are verbally identical with Theorems 3~7, Chap. III. Theorem 8, Chap. Ill, generalizes as follows :
COROLLARY. If OX, OY, and OZ are three noncoplanar lines and T any translation, there exists a unique triad of translations T^, Ty, Tz parallel to OX, OY, OZ respectively and such that
T = TXTJ2.
The theory of congruence under translations generalizes to space without change, and the contents of §§39 and 40 may be taken as applying to the affine geometry in three-space. In like manner the definition of a field of vectors and of addition of vectors is carried over to space if the words " Euclidean plane " be replaced by " Euclidean space." The theorems of § 42 then apply without change.
We arrive at this point on the basis of Assumptions A, E, HQ. Adding Assumption P we take over the theory of the ratio of col- linear vectors from §§ 43, 44. Some of the theorems to which it may be applied without essential modifications of the methods used in the planar case are given in the exercises below.
The definition of equivalence of ordered point triads in § 48 is such that if a plane TT be carried by an affiue collineation to a plane TT', any two equivalent point triads of TT are carried to two equivalent point triads of TT'. Moreover, the definition of measure of ordered point triads in § 49 is such that if two coplanar ordered point triads ABC, DEF are carried by an affine collineation to A'B'C', D>E'F' respectively,
m (ABC) _ m (A'B'C1) m (DEF) ~ m (D'E'F1) '
This result in view of Theorem 39, Chap. Ill, depends on the corre- sponding theorem about the ratios of collinear vectors. In (1) the unit of measure in any plane is regarded as entirely independent of the unit of measure in every other plane, but nevertheless the ratio of the measures is an invariant of the affine group. Certain ratios of ratios of measures are invariants of the protective group (cf. Ex. 1 7 below).
290 A F FINK AND EUCLIDEAN GEOMETRY [CHAP.VII
The notion of equivalence of ordered point triads may be extended as follows:
DEFINITION. Two ordered point triads ABC and A'B'C' are equiva- lent if and only if ABC may be carried by a translation to an onU-n-d triad A"B"C" which is equivalent in the sense of § 48, Chap. Ill, i.. A'li'C9.
The fundamental propositions with regard to equivalence, as devel- oped in § 48, remain valid under the extended definition. Thus if ABC^A1B1C\ and
etc. 111
This extension of the notion of equivalence carries with it a cor- responding restriction of the idea of measure, i.e. measure is now defined as in § 49, with the added proviso that the unit triad in any plane shall be equivalent to the unit triad in any parallel plane.
The method by which the theory of equivalence of ordered point triads was developed in Chap. Ill does not generalize directly to the case of ordered tetrads in three-dimensional space.* We shall there- fore give an algebraic definition of the measures of an ordered set of four points, leaving it to the reader to develop the corresponding synthetic theory (cf. Ex. 13 below).
DEFINITION. By the measure of an ordered tetrad of points Alf A2, AS, A4 relative to an ordered tetrad OPQR as unit is meant the number
(2)
= m
where (a,.,, a,.2, a.8) are nonhomogeneous coordinates of A((i=l, 2, 3, 4) in a coordinate system in which 0, P, Q, R are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively. Two ordered tetrads are said to be equivalent if and only if they have the same measure. In real affine geometry the number ^\m (A^A^A^A^ is called the volume of the tetrahedron A1A2AaAt relative to the unit tetrahedron OPQR and is denoted by v (A^A^A^A^).
The theory of the equivalence of point pairs, triads, tetrads, etc. is the most elementary part of vector analysis and the Grassmann Ausdehnungslehre. This subject in particular, and the affine geometry
• Cf. M. Dehn, Mathematische Annalen, Vol. LV (1902), p. 465.
§n-'] AFFINE GEOMETRY 291
of three dimensions in general, is worthy of a much more extensive treatment than it is receiving here. We have referred only to that part of the subject which is essential to the study of the Euclidean geometry of three dimensions.
In the following exercises the coordinate system is understood to be that which is described in the definition of measure of ordered tetrads above. The vectors Ol', ()(t), (Hi are taken as units of measure for the respectively parallel systems of vectors. The ordered point triads OPQ, OQR, ORP are taken as units of measure for the respec- tively parallel systems of ordered point triads.
DEFINITION. By the projection of a set of points [A"] on the a>axis is meant the set of points in which this axis is met by the planes through the points X and parallel to the plane x = 0 ; and the projection on the y- and z-axes have analogous meanings.
By the projection of a set of points [X] on the plane x = 0 is meant the set of points in which this plane is met by the lines on points X and parallel to the avaxis; and the projections on the planes y = 0 and z = 0 have analogous meanings.
EXERCISES
1. The measures of ordered tetrads of points are unaltered by trans- formations
x' = bnx + fe,2?/ + blsz + 1ilQ, (3) y' = l>2lx + l^y + f>23z + f>20,
subject to the condition A = 1, where
'>n ''i
!,„ ti
l/'31 ^32 ''3
A=
This group is called the e^uiajfine group and also the special linear group. The group for which A2 = 1 leaves volumes invariant.
2. Ratios of measures of ordered tetrads of joints are left invariant by the affine group.
3. In an ordered space two ordered sets of jwints A BCD and A'B'C'D' art- in the same sense or not according as m (AHCJ)) and /« (A'B'C'D") have the same sign or not.
4. The product of two line reflections {//'} and {mm'} (cf. §101) is a translation if /' and //<' are at infinity and / and m are parallel.
5. Determine the subgroups of the group of translations in space.
AFFINE AND EUCLIDEAN GEOMETRY [CMAP.VII
6. The projections of a point pair PtPj on the x-, y-, and : ir- respectively have the measures
and those of the ordered point triad OP1PZ on the planes z = 0, y = 0, 2 = 0 respectively have the measures
/* = These numbers satisfy the relation
oA + ftft + yv = 0.
Any two points PJPa' of the line P^P.^ such that Vect PiP2 = Vect Pj'P^ deter- mine the same six numbers a, ft, y, A, ft, v. These numbers are proportional to the Plucker coordinates (cf. § 109, Vol. I) of the line PjP.,.
7. Using the notations of Ex. 6, \=m(OPPlPi), ft = m (OQ1\I',), v = m (ORPlPt). If a, ft', y', A', ft, v' are the numbers analogous to a, ft, y, A, ft, v determined by an ordered pair P8P4,
m (P1P2P8P4) = oA' + ftft' + yv' + Aa' + ftft' + vy'.
8. The measures of the projections of an ordered point triad PjPjPj on the planes x = 0, y = 0, z = 0 respectively are
|
^i 2i ! |
xi 2i * |
^i y\ i |
||||
|
«1 = |
y2 22 1 |
«2=- |
x2 r2 1 |
U8 = |
x2 y2 i |
|
|
y* z* i |
x3 z3 1 |
X8 #8 ! |
||||
|
The homogeneous coordinates of the plane PjPjPj are (u0, «t, u2, u8), where |
||||||
|
«o = |
X/J "• 2 t/2 *** |
= ».(OPlPsP,). |
9. If Pj, P2, Ps, P4 are four noncoplanar points and Pg, P4' are two points collinear with P8 and 7J4, then Vect (P8'P4') = Vect (P3P4) if and only if m (P,P2P8P4) = m (/WiPi).
10. If Pj, P2, P8, P4 are four noncoplanar points and the lines PiP4, P\P-i, PI'PJ' have a point in common and
Vect (PjPj) = Vect (P{P't) + Vect (Pj'Pr/), then m (^i/WO = m (P{P*P*P<) + »«
•11. Study barycentric coordinates and the barycentric calculus for three- dimensional space. Cf. § 51, § 27, and references to Mdbius in § 49.
•12. Study the measure of n-points in space, generalizing the exercises in §40.
*13. Define two ordered tetrads A BCD and A'B'CD' as equivalent pro- vided that (1) A =A', B = ff, C = C', and the line DU is parallel to the plane ABC, or (2) if there are a finite number of ordered tetrads /,,•••, fw such that A BCD is in relation (1) to tv ^ in a like relation to /2, ta to J4, • • •
§§ii'-Mi:t] ORTHOGONAL LINES AND PLANES 293
and tH to A'B'C'D'. Develop a theory of equivalence as nearly as possible analogous to that of § 48. Show that two tetrads are equivalent in this sense if and only if they are equivalent according to the definition in the text.
*14. An elation whose center is at infinity and whose plane of fixed points is ordinary is called a aim/tie shear. The set of all products of simple shears is the equiafline group. Develop the theory of the equiafnne group on this basis. Is it possible to generalize § 52 to space?
15. If a plane meets the sides A0AV A^^,..., AnAQ of a simple polygon A0A1A., . . . J,, in points B0, Bv . . ., Bn, respectively,
AB AB AR =
A,B0 A,B, A0Bn
16. If a quadric surface (§ 104, Vol. I) meets the lines A0AV A^^, . . ., AnA0 respectively in the pairs of points B0C0, B1CV . . ., BnCn, respectively,
A AC A AC AB AC =
A.B. A,C\ A.2B, A.2C\ AQBn A
*17. Six points of a plane no three of which are collinear satisfy the following identity :
m (123) m (456) - m (124) m (563) + m (125) m (634) - m (126) m (345) = 0.
The ratio of any two terms in this sum is a projective invariant. These propositions are given by W. K. Clifford in the Proceedings of the London Mathematical Society, Vol. II (1866), p. 3, as the foundation of the theory of two-dimensional projectivities. Develop the details of the theory outlined by Clifford. Cf. also Mobius, Der barycentrische Calcul, § 221.
113. The parabolic metric group. Orthogonal lines and planes. DEFINITION. Let 2. be an arbitrary but fixed polar system in the plane at infinity TTX. This polar system shall be called the absolute or orthogonal polar system. The conic whose points lie on their polar lines with respect to 2«, is, if existent, called the circle at infinity. The group of all collineations leaving 2,, invariant is called the parabolic metric group and its transformations are called similarity transformations. Two figures conjugate under this group are said to be similar.
DEFINITION. Two ordinary planes or two ordinary lines are orthog- onal or perpendicular if and only if they meet TT. in conjugate lines or points of the absolute polar system 2.. An ordinary line and plane are orthogonal or perpendicular if and only if they meet TTX in a point and line which are polar with regard to 2.. A line perpendic- ular to itself, Le. a line through a point of the circle at infinity, is
J'.'l \1FINE AND EUCLIDEAN GEOMETRY [CHAI-. vn
called a mini/mil or isotropic line. A plane perpendicular to itself, i.r. a plain1 Hireling irf in a tangent to the circle at infinity, is called a minimal or isotropic plane.
As the analogue of Theorems 2 and 3 we have
THK.DKK.M 4. The similarity transformations which leave an ordi- nary no it minimal plane ir invariant, effect in TT the transformations of a parabolic metric group in the Euclidean plane consisting of the ordinary points of TT.
Generalizing Theorem 1, Chap. IV, we have
THEOREM 5. At every point 0 of a Euclidean space the correspond- ence between the lines and their perpendicular planes is a polar system, the projection of 2,. All the lines through 0 perpendicular to a given line are on the plane perpendicular to the given line at 0; and all the planes through 0 perpendicular to a given plane are on the line through
0 perpendicular to this plane. If existent, the isotropic lines through a point 0 constitute a cone of lines, and the isotropic planes through 0 the cone of planes tangent to this cone of lines.
COROLLARY 1. Two perpendicular nonminimal planes meet in a nonminimal line, and two perpendicular nonminimal lines are par- allel to a nonminimal plane.
COROLLARY 2. If a plane 1 is perpendicular to a plane 2, and 2 is parallel to a plane 3, then 1 is perpendicular to 3. If a plane 1 is perpendicular to a line 2, and 2 is parallel to a line or plane 3, then
1 is perpendicular to 3. If a line 1 is perpendicular to a plane 2, and 2 is parallel to a line or plane 3, then 1 is perpendicular to 3. If a line 1 is perpendicular to a line 2, and 2 is parallel to a line 3, then 1 is perpendicular to 3.
THEOREM 6. Two nonparallel lines not both parallel to the same minimal plane are met by one and only one line perpendicular to them both ; this line is not minimal.
Proof. Let An and J?« be the points in which the given lines meet TT.. By hypothesis A* =£ 7?., and the line A^B* is not tangent to the circle at infinity. Let C» be the pole of the line A^B* with respect to 2.. The required common intersecting perpendicular is the line through C* meeting the two given lilies; this line is obviously unique and not minimal.
5§ in, 114] ORTHOGONAL PLANE REFLECTIONS 295
EXERCISE
The planes perpendicular to the edges of a tetrahedron c,t the mid-points of the pairs of vertices meet in a point O. The line perpendicular to any face of the tetrahedron at the center of the circle through the three vertices in
this face passes through O.
114. Orthogonal plane reflections. DEFINITION. A homology of period two whose center, P, is a point at infinity polar in the absolute polar system to the line at infinity of its plane of fixed points, TT, is called an orthogonal reflection in a plane or an orthogonal plane reflection or a symmetry with respect to a plane, and may be denoted by {TTP}.* The plane of fixed points is called the plane of symmetry of any two figures which correspond in the homology.
Since the center and the line at infinity of the plane of fixed points of an orthogonal reflection in a plane are pole and polar with respect to 2«, we have
THEOREM 7. An orthogonal reflection in a plane is a transforma- tion of the parabolic metric group.
By a direct generalization of Theorems 3 and 4, Chap. IV, we obtain the following :
THEOREM 8. (1) If IT and p are two parallel nonminimal planes, the product {pR} • {"rrP} is a translation parallel to any line per- pendicular to TT and p. (2) If T is a translation parallel to a non- minimal line I, 7T any plane perpendicular to I, and p the plane perpendicular to I passing through the mid-point of the point pair in which TT and T (TT) meet I, then
and if a- is the plane perpendicular to I passing through the mid-point of the pair in which TT and T'^TT) meet I,
T = {TrP} • {<rS}.
(3) A translation parallel to a minimal line I is a product of four orthogonal plane reflections.
THEOREM 9. A product AnA,1_1 . . • Al of orthogonal plane reflec- tions is expressible in the form Af,A^_j • • • AJT or T'A.'HA.'H_l • • • A(, where Aj, A^, • • •, A^ are orthogonal plane reflections whose planes of
« In the rest of this chapter this notation will be used in the sense here denned and not in the more general sense of § 101.
•2M A1-FINE AND EUCLIDEAN GEOMETRY [CHAI-. VH
fixed points all contain an arbitrary point 0, and T and T' are translations. In case 0 is left invariant by AJ1An_1 . • • A,, T and T' reduce to the identity.
Proof. Let A,' (i = 1, 2, . • • n) denote the orthogonal plane reflection whose plane of fixed points is the plane through 0 parallel to the plane of fixed points of A,. Then by Theorem 8, A,.AJ = T,., Tf being a translation. Hence A, = T, A,' and
(5) AA-I • • • AI= TXT.-tA: . . . T\A;.
Ilv the generalization to space of Theorem 11, Cor. 2, Chap. Ill, if 2 is any afime collineation and T a translation, T2 = 2T', where T' is a translation. By repeated application of this proposition, (5) reduces to
A A A — A'A' A'T — T'A'A' A'
idgO,.) • • • iVj — iV,,iVM_1 • • • ^Val 1 2\lll\H_1 • • . l\.lf
where T and T' are translations.
In case 0 is a fixed point for the product AHAn-i • • - A1? since it is also left invariant by each of the reflections A,', it is left invariant by T and T'. Hence in this case T and T' reduce to the identity.
THEOREM 10. If A1? A2, A3 are three orthogonal plane rejections whose planes of fixed points meet in a line I, ordinary or ideal, the product A8A2At is an orthogonal plane reflection whose plane of fixed points contains I.
Proof. One of the chief results obtained in Chap. VIII, VoL I, can be put in the following form : * If T1? T2, T3 are harmonic hoinologies leaving a conic invariant and such that their centers are collinear, TjjTjTj is a harmonic homology leaving the conic invariant. For by Theorem 19 of that chapter, and its corollary, the product T2Tt is expressible in the form TgT, where T is a harmonic homology whose center and axis are polar with respect to the conic, the axis being concurrent with those of T:, T2, and T3 ; and from T2Tt = T8T follows
T T^ T _. rr^ rr^ ^r^ rr*
321—133 '
Now if Aj, A2, A8 are orthogonal plane reflections whose planes of fixed points meet in an ordinary line / their centers are collinear. Hence they effect in TT,, three harmonic hornologies whose centers are the poles of their axes with respect to the absolute polar system and whose centers are collinear. Hence A3A2A1 effects a harmonic homology in the plane at infinity and its axis, m,, passes
• Cf . the fine print in § 108.
§ 114, 115] DISPLAi ' KM KXTS 29V
through the point at infinity of /. Since / and mm are both lines of fixed points of A3A2A1, all points of the plane TT containing / and m» are invariant. Hence A^Aj effects a homology having the pole of ?»„ with respect to 2., as center. Since this homology is of period two in TT. it must be an orthogonal plane reflection.
In case the planes of fixed points of Alt A0, A8 are parallel we have by Theorem 8 (1) that A2Aj is a translation parallel to a line perpen- dicular to these planes, i.e. parallel to a nonminimal line. Hence by Theorem 8 (2) there exists an orthogonal plane reflection, A4, such that
AA=AA
or A8A2A1=A4.
COROLLARY. If {\L^ and {\L^} are two orthogonal plane reflec- tions, and \{ is any ordinary nonminimal plane in the same pencil with Xx and X2, there exists a plane X£ and points L[ and L^ such that
Proof. By the theorem, if L[ is the point at infinity of a line per- pendicular to \'v there exists an orthogonal plane reflection such that {x^} {XA} {x,z,} = {x,z,}>
and hence (X2£J • {X^} = {X^} • {\(L[}.
115. Displacements and symmetries. Congruence. We may now generalize directly from § 57, Chap. IV :
DEFINITION. A product of an even number of orthogonal plane reflections is called a displacement or rigid motion. A product of an odd number of orthogonal plane reflections is called a symmetry.
THEOREM 11. The set of all displacements and symmetries is a self-conjugate subgroup of the parabolic metric group and contains the set of all displacements as a self -conjugate subgroup.
DEFINITION. Two figures such that one can be transformed into the other by a displacement are said to be congruent. Two figures such that one can be transformed into the other by a symmetry are said to be symmetric.
THEOREM 12. If a figure Fl is congruent to a figure Ff and /•', t» a figure Ft, then Fl is congruent to FS. If FV is symmetric with F and Ff with F3, then FI is congruent to F3. If FI is congruent to Ff and Fa symmetric with F3, then FI is symmetric with FS.
298 All INK AND EUCLIDEAN GEOMETRY [CHAF. VH
Tn K< >i: KM 1 .".. Any displacement leaving an ordinary point 0 inva- riant is a product of two orthogonal plane reflections whose planes of fixed points contain O.
Proof. Consider a product of four orthogonal plane reflections, whose planes of fixed points pass through 0.
r={X4L4}.{X8L8}.{X2L3}.{X1i1}.
Let I be the line of intersection of X: and X2, m that of X8 and X4, and let X be a plane containing I and m, where in case l = m, X is chosen so as not to be minimal. If X is nonminimal, by the corollary of Theorem 10 there exist orthogonal plane reflections {p>M}, {vN} such that
{W . {W = {XL} .
and {X4Z4}.{X3L8} = {^}.
Hence T = {vN} - {\L} • {\L}
In case X is minimal* {\L^ transforms X to the other minimal plane through / (i.e. the other plane containing / and a tangent to the circle at infinity), and {\L2} transforms this plane back to X. In like manner the product {\L4} • {\LS} leaves X invariant. Hence X is left invariant by T. On the other hand the line / is obviously not left invariant by F, and therefore F does not leave all points at infin- ity invariant. Hence F leaves at most two tangents to the circle at infinity invariant, and thus leaves at most two minimal planes through O invariant. Let X^ be any plane of the bundle containing X2 and Xg which does not meet Xt in a line of an invariant minimal plane of F. By the corollary of Theorem 10 there exists a plane X, and points L!t and Zg such that
and hence such that
Now let I be the line of intersection of \ and X£, m that of X3 and X4, and X' the plane containing I and m. If X' were minimal it would, as argued above for X, be invariant under F, whereas X.^ was so chosen that / cannot be in such a plane. Hence the argument in the pre- vious paragraph can be applied to the last expression obtained for F.
•This case obviously does not arise in the real Euclidean geometry (§ 116), so that this paragraph may be omitted if one is interested only in that case. It is needed, however, in complex geometry.
§ 115] DISPLACEMENTS 299
Thus, in any case, a product of four orthogonal plane reflections whose planes of fixed points pass through O reduces to a product of two such reflections. By Theorem 9 any displacement leaving 0 invariant is a product of an even number, say 2 n, of orthogonal reflections in planes through O. This may be reduced to a product of two orthogonal reflections in planes through 0 by n — 1 applica- tions of the result proved above.
COROLLARY. An orthogonal plane reflection is not a displacement.
Proof. Let O be a point of the plane of fixed points of an orthog- onal plane reflection A. If A were a displacement it would, by the theorem, be a product of two orthogonal plane reflections containing 0 and hence could only have a single line of fixed points.
DEFINITION. A displacement which is a product of two orthogonal plane reflections whose planes of fixed points have an ordinary line I in common is called a rotation about I, and / is called the axis of the rotation. If the axis is a minimal line the rotation is said to be isotropic or minimal.
THEOREM 14. The product of two orthogonal reflections in perpen- dicular planes is a rotation of period two. It transforms every point P not on its axis to a point P' such that the axis is perpendicular to the line PP' at the mid-point of the pair PP'. It leaves invariant the points of its axis and the points in which any plane perpen- dicular to its axis meets the plane at infinity. Its axis cannot be a minimal line.
Proof. Consider any plane TT perpendicular to the planes of fixed points of the two orthogonal plane reflections At and A2. By the first corollary of Theorem 5 the axis of A2A1 is non minimal and hence TT is nonminimal. In TT the transformations effected by At and A2 are orthogonal line reflections in the sense of Chap. IV, and their product is a point reflection (Theorem 5, Chap. IV) in the plane. From this the theorem follows in an obvious way.
DEFINITION. The product of two orthogonal reflections in perpen- dicular planes is called an involutoric rotation or an orthogonal line reflection or a half turn. If I is its axis and /' the polar with respect to 2. of the point at infinity of /, it may be denoted by {//'}.*
* In the rest of this chapter this notation will be used in the sense here defined and not in the more general sense of § 101.
:>"ii AFF1NE AND Kl CIJDHAN ilKiLMETRY U'HAF. VH
THEOREM 15. DEFINITION. 7Yi« product of the orthogonal plane reflections in t litre perpendicular planes is a transformation carrying «n-h /mint I' to a j>oi/tt /'' suck that the point O of intersection of Ike three planes is the mid-point of the pair PI1'. A transformation of this sort is called a point reflection or symmetry with respect to the point O as center. It is not a displacement. The points P and P' are said to be symmetric with respect to 0.
Proof. In the plane at infinity the three orthogonal plane reflec- tions effect the three harmonic homologies whose centers and axes are the vertices and respectively opposite sides of a triangle. The product therefore leaves all points at infinity invariant. It also leaves 0 invariant and is evidently of period two on the line of intersection of any two of the planes of fixed points of the orthogonal plane reflec- tions. Hence it is a homology of period two with O as center and TT. as plane of fixed points. It is not a displacement, since by Theorem 13 a displacement leaving 0 invariant would have a line of fixed points passing through 0.
THEOREM 16. The transformations effected in a nonminimal jtlane •JT by the displacements leaving TT invariant constitute the group of displacements and symmetries of the parabolic metric group whose absolute involution is that determined by 2*. on the line at infinity of IT.
Proof. Let F be any displacement leaving TT invariant, <) an arbitrary point of TT, and T the translation carrying 0 to F (O). Then T"1 F (0) = 0, and hence, by Theorem 13, T"1 F is a rotation. Moreover, T"1 F leaves TT invariant.
It is obvious from the definition of a rotation that it can leave TT invariant only in case its axis is perpendicular to TT or in case it is of period two and its axis is a line of TT. If T"1 F falls under the first of these cases, it effects a rotation in TT according to the definition of rotation in Chap. IV, and thus F effects a displacement in TT. If T"1 F falls under the second of these cases it effects, and therefore F also effects, a symmetry hi TT according to the definition in Chap. I V.
COROLLARY 1. The transformations effected in a nonminimal plane TT by the displacements and symmetries leaving IT invariant constitute the group of displacements and symmetries of the parabolic metric group whose absolute involution is that determined by £„ on the line at infinity of IT.
§§ 1 1.".. 1 1«] EUCLIDEAN c; K< >.M KTKY 301
COKOLI.ARY 2. If O is an arbitrary point, any displacement T is expressible in the forms
F = TP and F = P'T,
where T, T' are translations and P, P' rotations leaving O invariant.
Proof. As in the proof of the theorem above, let T be the translation carrying 0 to F (0). Then T'1 T (0) = 0 and hence, by Theorem 13, T-' F is a rotation, P. Hence T = TP. If T' is the translation carry- ing 0 to F-^O), it follows in like manner that FT'(O) is a rotation P' and hence that F = P'T'-1.
COROLLARY 3. The transformations effected on a nonminimal line p by the displacements leaving p invariant constitute the group com- posed of all parabolic transformations and involutions leaving the point at infinity of p invariant.
EXERCISES
1. Two point pairs are congruent if they are symmetric.
2. The set of all point reflections and translations forms a group which, unlike the analogous group in the plane (§ 45), is not a subgroup of the group of displacements. The product of two point reflections is a translation, and any translation is expressible as a product of two point reflections, one of which is arbitrary.
3. Study the theory of congruence in a minimal plane.
4. A rotation leaves no point invariant which is not on its axis. It leaves invariant all planes perpendicular to its axis and no others unless it is of period two, when it is an orthogonal line reflection.
116. Euclidean geometry of three dimensions. The last theorem may be regarded as the fundamental theorem of the parabolic metric geometry in space, for by means of it all the results of the two-dimensional parabolic metric geometry become immediately applicable.
Suppose now that we consider a three-space satisfying Assumptions A, E, H, C, R (or A, E, K), i.e. a real projective space. Suppose also that 2, be taken to be an elliptic polar system,* i.e. the polar system of an imaginary ellipse (§ 79). Then in any plane the parabolic metric geometry reduces to the Euclidean geometry and the displacements which leave this plane invariant are Euclidean displacements.
« The existence and properties of an elliptic polar system may be determined without recourse to iinaginaries (in fact, on the basis A, K, 1', S), as in § 89.
302 A I FINE AND EUCLIDEAN GEOMETRY [CHAP.V*
A set of assumptions for the Euclidean geometry of three dimen- sions is composed of I-XVI, given in §§ 29 and 66. We have M in
>; '_".» that I-IX are satisfied by a Euclidean space of three diiiiriisimis. Assumption XI is a consequence of Theorem 12, and Assumptions X, XII -XVI of Theorems 11 and 16. Hence in a real three-space, if 2, is an elliptic polar system the parabolic metric geometry is the lidean geometry.
The general remarks in § 66 are applicable to the three-dimensional case as well as to the two-dimensional one.
It was stated in § 66 that the congruence assumptions are no longer strictly independent when a full continuity assumption is added, because by intro- ducing ideal elements and an arbitrary 2» (as in the present chapter) a relation of congmence may be denned for which the statements in X-XVI are theorems which can easily be proved. This view is not accepted by certain well-known mathematicians, who hold that the arbitrariness in the definition of the absolute involution somehow conceals a new assumption.* It may, therefore, be well to restate the matter here.f
Assumptions I-IX, XVII are categorical for the Euclidean space ; i.e. if two sets of objects [P] and [Q] satisfy the conditions laid down for points in the assumptions, there is a one-to-one reciprocal correspondence between [P] and [Q] such that the subsets called lines of [P] correspond to the subsets called lines of [Q]. Thus the internal structure of a Euclidean space is fully determined by Assumptions I-IX, XVII. The group leaving invariant the relations described in these assumptions is the affine group, and all the theorems of the affine geometry are consequences of these assumptions. The latter may therefore be characterized as the assumptions of affine geometry.
Among the theorems of the affine geometry is one which states that there is an infinity of subgroups, each one conjugate to all the rest and such that the set of theorems belonging to it constitutes the Euclidean geometry. Each of these groups is capable of being called the Euclidean group, and there is no theorem about one of them which is not true about all of them. The set of theorems stating relations invariant under any one of these groups is the Euclidean geometry. This set of theorems is the same whichever Euclidean group be selected, i.e. the Euclidean geometry is a unit/ue body of theorems.
Each Euclidean group has a self-conjugate subgroup of displacements which defines a relation called congruence having the properties stat<-<l in
* Cf . the remarks on a paper by the writer in the article by Enriques, Encyclo- pe"die des Sc. Math. Ill 1, § 12.
t This discussion should be read in connection with the remarks on foundations of geometry in the introduction to Vol. I and in § 18 of this volume ; also in con- nection with the remarks on the geometry corresponding to a group, §§ 34, 39, 110.
§116] FOUNDATIONS OF GEOMETRY 303
Assumptions X-XVI. Moreover, any relation which satisfies these asHuiiij>- tions is associated with a group of displacements which is self-conjugate under a Euclidean group.
Thus Assumptions X-XVI characterize the relation of congruence as com- pletely as possible, i.e. any relation satisfying these assumptions must be that determined l>y one of the infinitely many groups of displacements. The set of theorems about congruence is unique and is the Euclidean geometry.
The relation between the affine geometry and the Euclidean geometry is analogous to that between the Euclidean geometry and the geometry belong- ing to any non-self-conjugate subgroup of a Euclidean group. Consider, for example, the subgroup obtained by leaving a particular point O invariant. A relation which is left invariant by this group may be defined as follows :
DEFINITION. A point P is nearer than a point Q if and only if Dist (OP) < Dist (OQ). P and Q are equally near if Dist (OP) = Dist (OQ).
There is an element of arbitrary choice in this definition, just as there is in the choice of an absolute involution to define the notion of congruence. Moreover, the geometry of nearness is just as truly a geometry as is the Euclidean geometry.* It woujd be easy to put down a set of assumptions (XVIII-JV) in terms of near regarded as an undefined relation, which would state the abstract properties of this relation, just as X-XVI state the abstract properties of congruence.
Another non-self-conjugate subgroup of the Euclidean group which gives rise to an interesting geometry is the group leaving invariant a line and a plane on this line. In terms of this group the notions of forward and back- iranl and up and down can be defined, and the geometry corresponding to this group is a set of propositions embodying the abstract theory of this set of relations.
It is a theorem of Euclidean geometry that the Euclidean group has subgroups with the properties involved in these geometries, just as it is a theorem of affine geometry that the affine group has Euclidean subgroups and a theorem of protective geometry that the projective group has affine subgroups.
Assumptions I-IX, XVII have a different role from X-XVI or XVIII-A", in that they determine the set of objects (points and lines, etc.) which are presupposed by all the other assumptions. The choice of these assumptions is logically arbitrary. The choice of such sets of "assumptions" as X-XVI is not arbitrary; it must correspond to a properly chosen group of jH-rmu- tations of the objects determined by I-IX, XVII. When indei>endence proofs are given for Assumptions X-XVI, it is done by giving new interpretations to the term "congruence," not to "point" or "line."
* It is even possible to give a psychological significance to this geometry. The normal individual has a certain place, say liomc. in terms of nearness tn which other places are thought of ; here O is the central point of home. In astronomy stars are regarded as near or the contrary, according to their distance from the sun ; here O is the center of the sun.
304 AFFLNE AND EUCLIDEAN GEOMETRY [CHAP, vn
The point of view of the writer is that if X-XVI or XVIII-JV are to be regarded as independent assumptions, their indej>endence is of a lower grade than that of I-IX, XVII. They constitute a definition by postulates of a relation (congruence or nearness) among objects (points, lines, etc.) already fully determined. Their significance is that they characterize that subset of the theorems deducible from I-IX, XVII which corresponds to any Euclidean group and which therefore is the Euclidean geometry.
EXERCISES
* 1. Develop the geometry corresponding to some non-self-conjugate sub- group of the Euclidean group. Determine a set of mutually independent assumptions characterizing this geometry.
2. The identity is the only transformation of the Euclidean group which leaves fixed two points A and B and two rays (cf. definition in § 16) A C and AD orthogonal to each other and to the line AB.
3. If a and b are any two rays having a common origin, O, and on different lines, there is a unique orthogonal line reflection and a unique orthogonal plane reflection transforming a into 6.
4. If A, B, C, D are any four points no three of which are collinear, there exists a unique rotation leaving the line AB invariant and transforming C into a point of the plane ABD on the same side of AB with D.
5. Any transformation of the Euclidean group which leaves a line point- wise invariant and preserves sense is a rotation.
6. Any transformation of the Euclidean group which leaves a line point- wise invariant and alters sense is an orthogonal reflection in a plane containing this line.
7. There is one and only one displacement which transforms three mutually orthogonal rays OA, OB, OC into three mutually orthogonal rays (FA', O'B',
\ provided that S (OABC) = S^A'B'C).
*117. Generalization to n dimensions. The discussion of the Euclid- ean and affine geometries in §§ 111-116 is so arranged that it will generalize at once to any number of dimensions. It is recommended to the reader to carry out this generalization in detail, at least in the four-dimensional case.
The elementary theorems of alignment for four dimensions are given in § 12, Vol. I. The definition of a Euclidean four-space is given in § 28, Vol. II. The generalization of §111 is obvious on comparing these two sections. A four-dimensional translation may be denned as a protective collineation leaving invariant all points of the three-space at infinity and also all lines through one of these points. The generalization of § 112 then follows at once.
§§iiT,ii8l GENERALIZATION TO N DIMENSIONS 305
A three-dimensional polar system may be defined as the polar system of a proper or improper regulus (Chap. XI, Vol. I ; cf. also §§ 100-108, Vol. II), or it may be studied ab initio by generalizing Chap. X, Vol. I. The notion of perpendicular lines, planes, and three- spaces then follows at once and also the theorems generalizing those of § 113. An orthogonal reflection in an Sg is next defined as a projec- tive collineation of period two, leaving invariant a point P at infinity and each point of a three-space whose plane at infinity is polar to P in the absolute polar system. All the theorems of §§ 114, 115 up to Theorem 13 then generalize at once. Theorems 13-15 must be modi- fied, in view of the fact that there are more than one type of four- dimensional displacements leaving a point invariant. Theorem 16 holds unchanged.
Finally, it can be proved as in § 1 1 6 that in case of a real space and an elliptic polar system the parabolic metric geometry satisfies a set of axioms for Euclidean geometry of four dimensions. This set differs from the one used above, in that VIII is replaced by
VIII'. If A, B, C, D are four noncoplanar points, there exists a point E not in the same S8 with A, B, C, D, and such that every point is in the same S4 with A, B, C, D, E,
The introduction of nonhomogeneous coordinates in a space of n dimensions may be made by direct generalizations of § 69, Vol. I. The formulas for the affine group, the group of translations, the Euclidean group, and the group of displacements are then easily seen to be identical with those given in the sections below, except that the summations from 0 or 1 to 3 must in each case be replaced by summations from 0 or 1 to n.
118. Equations of the affine and Euclidean groups. With respect to a nonhomogeneous coordinate system in which TT*, is the singular plane, the affine group is evidently the set of all projectivities of
the form
*'=au*
(6) y1 = anx
where A = a,, «„ aM =£ 0,
306 AFFINE AND EUCLIDEAN GEOMETRY [CHA, vn
and the variables and coefficients are elements of the geometric number system.
In the system of homogeneous plane coordinates in which the plane at infinity is represented by [1, 0, 0, 0], this group takes the form
{
u =
In an ordered space the affine group has a subgroup consisting of all transformations for which A is positive. This group has been considered in § 31. It also has obvious subgroups consisting of all transformations for which A2 = 1 and for which A = 1.
The equations of a translation parallel to the x-axis are evidently x1 = x + a, y' = y, z' = z, and similar expressions represent a transla- tion parallel to any other axis. Hence by the corollary of Theorem 3 the equations of the group of translations are
x' = x + a,
(8) y' = y + &,
z' = z + c.
If the coordinates are so chosen that the planes x = 0, y = 0, z=0 are mutually orthogonal, the equations of the circle at infinity in terms of the corresponding homogeneous coordinates are
axl + bx% + cx\ =0, x0 = 0. These are reducible by the transformation
(9) #0 = ^o> a^Vaz,, x.1 = Vbx.y «3 = Vcx8 to
(10) as? + a£ + a£ = 0, x9 = 0.
In the real geometry a, b, c are positive if the polar system is elliptic (§ 85), and the transformation (9) carries real points to real points. The formulas (9) are the only ones in the present section in which irrational expressions appear. Hence the rest of the discussion holds for any space satisfying Assumptions A, E, P, H0. In any such space it is easily seen that (10) repre- sents a conic whose polar system may be taken, as 2«, but it does not follow, as in the real case, that any improper conic can be reduced to this form. The situation here is entirely analogous to that obtaining in § 62.
§HH] ORTHOGONAL TRANSFORMATION 307
In the three-dimensional homogeneous plane coordinates, TT. and the i 'lanes tangent to the circle at infinity (10) satisfy the equation
(11) af+X + ttf-O.
Any plane
(12) u0 + ulx' + ug' + uj/=Q
is the transform under a collineation of the form (6) of the plane
( 1 3) (MO + ai(A + anut 4-
Hence (11) is the transform of
(14) « 4- a*, + a*) u* + K + a*2 + «£) u* + « + «, «, + «£) w,2 + 2 (auan + altaa + a-uan) u^ + 2 (ana81 + a12«82 + a18aM) up,
+ 2 (a2l«31 + "A + VJ S«, = °'
In order that (11) and (14) shall represent the same locus, we must have
(15) «n + «,2a + «£ = < + < + a» = < + «i + ««, aiian+ °Maa+ a18a23= «n% +«««,«+ «„«»
These conditions are equivalent to the equation (cf. § 95, Chap. X,
/«11 «ia «18\ /«!! «21 «31\ //> 0 0\
(16) Ia21 a^ a28l-la12 a22 a82) = IO /) 0),
\«81 a82 a88/ \«18 «28 a88/ \° ° P/
where p = a^ 4- a^ + af,.
If the matrix (alla^a33)=^ be interpreted as the matrix of a planar collineation, as in § 95, Vol. I, this states that the product of the collineation by the collineation represented by the transposed matrix is the identity. Hence the product of the two matrices in the reverse order is a matrix representing the identity. This means that
and anau + «„«„ + anan = anow 4- aslaB + «31«M
= a,2a,3+a22a23+a»aM=°-
Since the determinants of a matrix and of its transposed matrix are equal, we have
308 AFFINE AND EUCLIDEAN GEOMETRY [CHAP.VII
DEFINITION. A matrix such that its product by a given matrix A is the identical matrix (§ 95, Vol. I) is called the inverse of A and is denoted by A~l. A square matrix whose transposed matrix is equal to its inverse is called orthogonal. A linear transformation,
x' = anx + a^ + al3z, (17) y' = anx + *^y + avp,
whose matrix (O'lla^a^ is orthogonal, is said to be orthogonal.
The results at which we have arrived may now be expressed in part as follows :
THEOREM 17. The transformations of the parabolic metric group can be written in the form
x' = p (anx + a^
(18) y' = p (anx + a^ + a23z + kj, z' = p (a9lx + a^ + a33z + kj,
where the matrix (aua'Kas^ is orthogonal.
From the form of these equations we obtain the following corollaries :
COROLLARY 1. Any transformation (18) of the Euclidean group is the product of an orthogonal transformation, a translation, and a homology of the form
x' = px,
(19) yf = py,
z' = pz.
COROLLARY 2. A homology (19) is commutative with any collinea- tion leaving the origin invariant.
Since an orthogonal matrix is any matrix satisfying (16) with p = 1, we have
COROLLARY 3. The product of two orthogonal transformations is orthogonal. The determinant of an orthogonal transformation is + I or - 1.
§118] ORTHOGONAL TRANSFORMATION :JO'.»
In view of the formula for the inverse of a matrix (§ 95, Vol. I), we have
COROLLARY 4. A matrix (au«MaM) is orthogonal if and only if (20) .',=Aa,(, (i=l, 2,3; j= 1,2, 3)
where A is the determinant of the matrix and AfJ the cof actor of ay.
The matrix of an orthogonal transformation of period two is its own inverse and hence its own transposed. Hence
COROLLARY 5. An orthogonal transformation is of period two if and only if aij= aj{.
The double points of any orthogonal transformation (17) must satisfy the equations
The determinant of the coefficients of these equations is
But since the transformation is orthogonal, Ait = Aa.,.. Hence the determinant of (21) reduces to
Another determinant which is of importance in the theory of orthogonal transformations is that of the equations
(22)
Any point satisfying these equations is transformed into its symmet- ric point with respect to the origin. The orthogonal transformation therefore transforms the line joining these points into itself and effects an involution with the origin as center on this line. The determinant of the equations (22) is
Dt = A + (An + An + AJ + (au + an + aj 4- 1, which reduces to
310 All INK AND KICIJDKAN GEOMETRY [CHAI-. vn
Let us now consider an orthogonal transformation (17) which we shall denote by 2. If A = — 1 for 2, />2= 0, and hence there is at least one point which is carried by 2 into its symmetric point with respect to the origin. The plane through the origin perpendicular to the line joining these points is left invariant by 2. On the other hand, I\=tQ unless
(23) «11 + «22+a33=1>
and hence 2 leaves no other point than the origin invariant unless (23) is satisfied. Suppose now that (23) is satisfied. A cofactor of an element ati of the main diagonal of Dl is
^,--K- + «U-) + l>
where i =£/ =£ k =£ i. By (20) this reduces to
-K, + ««+a.,) + 1.
which vanishes. The cofactor of an element a- (i ^ j) of Dl is
4»+Ap
and by (20) this vanishes when A = — 1. Thus we have that if A = — 1 and (23) is satisfied, 2 has a plane of fixed points. Since it transforms one point into its symmetric point with respect to the ori- gin, it must be an orthogonal plane reflection. Thus we have proved
THEOREM 18. An orthogonal transformation for which A = — 1 always has an invariant plane. It either leaves no point except the origin invariant or it is an orthogonal plane reflection. The latter case occurs if and only if an + a^ + ag3 = 1.
By comparison with Corollary 5 above we have
COROLLARY. An orthogonal transformation for which A = — 1 is an orthogonal plane reflection if and only if «12=«21, a^=a^ ana ai8=a3r
Let us now consider an orthogonal transformation 2 for which A= 1. In this case D^= 0, and hence there is always a line of fixed points passing through the origin. Let At be an orthogonal plane reflection containing a line of fixed points of 2. Then 2At is an orthogonal transformation for which A = — 1 and for which there are other fixed points than the origin. By the last theorem, therefore, it is an orthogonal plane reflection A2. From 2At = A2 follows 2 = A2Aj. We therefore have
THEOREM 19. An orthogonal transformation for which A = l is a rotation.
§§118,11-.] OKTIKKJOXAL TKANSFOKMATION 311
COKOLLARY 1. An orthogonal transformation for which A = — 1 is a a//mmetry.
Any transformation (18) for which p = 1 is a product of an orthog- onal transformation and a translation. It is therefore either a dis- placement or a symmetry. By Theorem 16, Cor. 1, a homology (19) for which p~=£ 1 is not a displacement or a symmetry. Hence we have
COROLLARY 2. The subgroup of (18) for which p = l and A=l in (he group of displacements.
COROLLARY 3. The subgroup of (18) for which p = 1 and A2= 1 is the group of displacements and symmetries.
The coordinate system which has been employed above is such that the planes x=Q,y=Q,z=Q are mutually orthogonal. Moreover, the displacement x, = y> y, = z> z, = x> ^
leaves (0, 0, 0) invariant and transforms (1, 0, 0) to (0, 1, 0) and (0, 1, 0) to (0, 0, 1). Hence the pairs (0, 0, 0) (1, 0, 0), (0, 0, 0) (0, 1, 0), and (0, 0, 0) (0, 0, 1) are congruent. Coordinates satisfying these conditions are said to be rectangular.
EXERCISES
1. The group of displacements and symmetries leaves the quadratic fonn
Mf + «22 + «82
absolutely invariant.
2. Two point pairs (a, b, c)(a', b', c") and (x, y, z)(x', y', z") are congruent if and only if (a - a')2 + (b - &')2 + (c - c')2 = (x - a^)2 + (y - y')2 + (z - z')*.
3. Two planes MI:C + «rf + «,z + «0 = 0,
v^ + r2y + c3c + r0 = 0 are orthogonal if and only if Ujt^ + u2i-2 + «si?8 = 0.
4. Three planes ^ + ^ f ^ + ^ = 0> (. = ^ 2> 8)
the coefficients being such that «,.f + u,-2, + w,.2, = 1, (i = 1, 2, 3)
an- mutually perpendicular if and only if the matrix (wn«22Mg8) is orthogonal.
5. The three ordered triads of numbers (</,-,, a/.,, «,-g), i = 1, 2, 3, are direction cosines of mutually perpendicular vectors if and only if the matrix ("n"..^^ is orthogonal.
119. Distance, area, volume, angular measure. The definition (§67) of distance between two points extends without modification to the three-dimensional case. The distance between a point 0 and a plane TT is the distance between O and the point P in which
BIS A I TINK AND EUCLIDEAN GEOMETRY [CHAP.VII
•TT is met by the line through 0 perpendicular to TT. The distance between two lines 7,^ is Dist (/?/£), where Pl and Pi are the points in which the common intersecting perpendicular line meets l^ and ls respectively.
If the notion of equivalence of ordered point triads (§ 112) be extended by regarding two ordered triads as equivalent whenever they are congruent, it is obvious that any triad is equivalent to triads in any plane whatever and not merely, as in $ 1 1 '2, to triads in a system of parallel planes. Moreover,. if ABC are noncollinear points such that AB is congruent to AC, the ordered triad ABC is congruent and therefore equivalent to the ordered triad ACB. Hence
^ BCA ^ CAB^ACB^CBA^BAC,
i.e. according to the extended definition, any ordered triad is equivalent to any permutation of itself.
Since m(ABC} = — m(ACB), the definition of measure (§ 49) can- not be extended to correspond to the new conception of equivalence. On the other hand, the notion of area (§ 68) of a triangle is directly applicable. The situation here is entirely analogous to that described in § 67 with regard to the measure of a vector and the distance between two points. The formal definition may be made as follows :
DEFINITION. Let OPQ be a triangle (called the unit triangle) which is such that the lines OP and OQ are orthogonal and the point pairs OP and OQ are congruent to the unit of distance. Then if A'B'C' is a triangle coplauar with OPQ and congruent to ABC, the positive
number I m(A'B'C'}\ = a(ABC),
where m (A'B'C') is the measure (§ 49) of the ordered triad A'B'C' rela- tive to the ordered triad OPQ, is called the area of the triangle ABC.
The definition of the measure of an ordered tetrad and of the vol- ume of a tetrahedron may be taken from § 112, with the proviso that the unit tetrad OPQR is such that the lines OP, OQ, OR are mutually orthogonal and the point pairs OP, OQ, OR congruent to the unit of distance.
The definition of the measure of angle may be taken over literally from § 69. Since, however, any symmetry in a plane can be effected by a three-dimensional displacement, the indetermination in the meas- ure of an angle is such that any angle whose measure is /3 also has the measure brr + fi, where k is a positive or negative integer. The
§119] ANGULAR MEASURE 313
of an angle may therefore be subjected to the condition 0 ^ & < ir or - 7T/2 < £ ^ tr/'l.
I >KFINITION. The angular measure of a pair of intersecting lines ab is the smallest value between 0 and 2 TT, inclusive, of the measures of the four angles 4.aJ\ formed by a ray al of a and a ray 6l of b. It is denoted by m(ab). If a and b do not intersect, m(ab) denotes m (a'b), where a' is a line having a point in common with b and parallel to <(. The angular measure of two planes TT, TT' is the angular measure of two lines /, V perpendicular to TT and TT' respectively.
The following statements are easily proved and will be left to the reader as exercises (cf. § 72) : In the case where a and b do not inter- sect, the value of m (ab) is independent of the choice of a'. Although in Euclidean plane geometry 0 ^ m (ab) < TT, in the three-dimensional case
0^m(db)<*-
If /t and la are any two lines parallel to a and b respectively, and il and i2 are the minimal lines through the intersection of ^ and /2, m(ab) is the smaller of the two numbers
ei=-\ log R* (1A > v2) and 0a = - \ lo§ fcx (* A, \\}>
that determination of the logarithm in each case being chosen for which 0 ^ 6l < TT and 0 ^ 02< TT.
The numbers which we have been denning in this section are some of the simplest absolute invariants of the group of displacements. The algebraic formulas for these invariants and some others are stated in the exercises below. In every case the radical sign indicates a positive root. By the angle between two vectors OA and OB is meant the measure of &AOB.
The orthogonal projection of a set of points [P] on a plane IT is the set of points in which the lines perpendicular to TT through the points P meet TT. The orthogonal projection of a set of points [P] on a line / is the set of points in which the planes perpendicular to / through the points P meet I.
The exercises refer to four distinct noncoplanar points Pl = (xv ylt z,),
K= (*v y*> «2>. J?=fo, y»> *»)» J?=(*4> y*> z^ n° two of which are
< ollinear with the origin. The coordinate system is rectangular, and 0, P, Q, R denote the points (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively, as in § 112.
AFFINK AND P]UCLIDEAN GEOMETRY [CHAP vu EXERCISES
2. The cosines of the angles between a vector Of\ and the x-, y-, and z-axes respectively are
v *\ T Jfl T *f • *f T #t -T «1 ' *j T yt T ~!
These are referred to as the direction cosines of the vector OPr If r = Dist (P,P2), the direction cosines of the vector PtP2 are
z-2 - *i y-i - y\ 2-2 - 2i — , — *, — •
r r r
3. The equation of a plane perpendicular to the line OI11 is
4. The distance from the point Pt to the plane or + fy + yz = 8 is
PI + &i + y^i - 8
Va2 + /8* + f 5. If Qt is the orthogonal projection of P2 on the line OPV then
is Dist(OQj) in case Qt and Pl are on the same side of O, and — Dist(OQ1) in case Qt and Pt are not on the same side of O.
6. 7n(P1P2P8P4) = Dist(P1P2) • Dist(P8P4) • r • sin0, where r is the dis- tance between the lines P1P2 and P8P4, and 6 the angle between the vectors PjP2 and P8P4.
7. If 0 denotes the measure of 4 P\OPf> and /, ?n, n the direction cosines of a vector OK perpendicular to the plane OPiP2 and such that S(
Jl I1 = Dist (OPX)- Dist (OP2). sin 0 • I,
y% *2
= Dist(OP1) . Dist(OP2) • sin 0 • m,
= Dist (OPX) • Dist(OP2) • sin 6 • n*
8. With respect to the coordinate system employed in § 118, the angle between two lines which meet TT« in (0, av a^, Og) and (0, (3V /32, /88) is
5+/8I)
9. If four planes a, /3, y, 8 meet on a line,
where (ay) denotes the angular measure of the ordered pair of planes ay.
*Cf. Ex.6, §112.
§i--"j THE SPHERE 315
120. The sphere and other quadrics. DEFINITION. A sphere is the set of all points [/'] such that the point pairs OP, where 0 is a fixed point, are all congruent to a fixed point pair OP0. In case the line OP0 is minimal, the sphere is said to be degenerate ; otherwise it is nonde- generute. The point 0 is called the center of the sphere.
By comparison with the definition in § 60 it is clear that any sec- tion of a nondegenerate sphere by a nonminimal plane is a circle. In case the circle at infinity exists, two perpendicular sections Cf and C* of a sphere S by nonminimal planes constitute with the circle at infinity three conic sections intersecting one another in pairs of dis- tinct points. By § 105, VoL I, there is one and but one quadric surface containing them. A nonminimal plane TT through the center of the sphere meets this quadric in a conic section which contains at least two points of the circles C* and C\ and two points of the circle at infinity. This conic is therefore a circle containing the points of the sphere S which are in IT. Hence the sphere S is identical with the set of all ordinary points of the quadric surface containing C'f, C£, and the circle at infinity. Since 0 is the center of each circle in which S is met by a nonminimal plane through 0, 0 is the pole of the plane at infinity with regard to the quadric. Since a circle in a non- minimal plane contains the ordinary points of a nondegenerate conic, it follows that the quadric surface is nondegenerate, i.e. is a quadric which contains two proper or improper reguli.
In case the circle at infinity does not exist, improper elements may be adjoined as explained in § 85, Vol. I, so that the circle at infinity exists in the resulting improper space. The argument in the para- graph above thus applies to any space whatever which satisfies Assumptions A, E, P, HQ. Thus we have
THEOREM 20. A nondegenerate sphere consists of the ordinary points of a nondegenerate quadric surface *S'2 such that all pairs of points in tin' plane at infinity conjugate with regard to S* are conjugate unth regard to the absolute polar system. The center of the sphere is the pole of the plane at infinity relative to this quadric.
Comparing the definition above with Theorem 7, Chap. IV, we have
COROLLARY. A degenerate sphere unth a point 0 as center consists of all ordinary points on the cone of minimal lines through O, except 0 itself.
316 AFFIX K AND KICLIDEAN GEOMETRY [CHAI-. vn
Had a degenerate circle in the plane been denned in the same way that a degenerate splu-rr is dctiiicd alwve, it would have been found to r<m>:- I-oints on only one minimal lim* through O, since in the plane the group «»i displacements h-avrs «-arh minimal Hue invariant.
The Euclidean classification of quadric surfaces may now be made in a manner entirely analogous to the Euclidean classification of conic sections in Chap. V. After completing the projective classification (§ 103) and the aftine classification (§ 111, Ex. 2) and obtaining the properties of diameters and diametral planes, the principal remaining problem is that of determining the axes, an axis being defined as a line through the center of the quadric perpendicular to its conjugate planes.
A line / and a plane TT meet the plane at infinity in a point L* and a line pm respectively. If / and IT are perpendicular, Lx and p*> are polar with respect to 2,,,. If / and IT are conjugate with regard to a quadric Q2, L* and px are polar with respect to the conic (real, imaginary, or degenerate) in which Q2 meets IT*. Hence the problem of finding the axes is reduced to that of finding the points which have the same polar lines with respect to two conies. This problem has been treated in § 101, Vol. I, for the case where both conies are nondegenerate. In general the two conies have one and but one common self-polar triangle. Hence, in general, a quadric surface has three axes which are mutually orthogonal The determination of the other cases which may arise is a problem (Ex. 5, below) requiring a comparatively simple application of methods and theorems which we have already explained.
The classification of point quadrics includes that of cones and conic sections, the properties of cones and conies in three-dimensional Euclidean geometry being by no means dual to each other. In con- nection with this it is of interest to prove the following theorem, which embodies perhaps the oldest definition of a conic.
THEOREM 21. Any nondegenerate real conic is perspective with a circle.
Proof. Let C2 be a given conic and K* a circle in a different plane having a common tangent and point of contact with (72. By Theorem 11, Chap. VIII, VoL I, <72 and K* are sections of the same cone.
COROLLARY. Any cone of lines is a projection of a circle from a point.
§§ i*), 121] QUADRIC SURFACES 317
EXERCISES
1. The equation of a sphrrr nf center (a, b, <•) in rectangular coordinates is
(x - «)2 + (y - /0s + 0 - c)» = L
2. The set of points on the lines of intersection of homologous planes in the corresponding pencils,
is a sphere.
3. A riff/il circular cone is a projection of a circle from a point from which the extremities of any diameter are projected by a pair of perpendicular lines. Any conic may be regarded as the plane section of a right circular cone.
*4. Develop the theory of stereographic projection of a sphere on a plane (cf. § 100).
*5. Classify the quadric surfaces from the point of view of Euclidean geometry. Having made the classification geometrically, find normal forms for the equations of the quadrics1 of the different classes and the criteria to determine to which class a given quadric belongs. This is analogous to the work in Chap. V.
*6. Classify the linear complexes from the point of view of Euclidean geometry.
*7. Starting with a definition of an inversion with respect to a sphere analo- gous to that of an inversion with respect to a circle (§ 71), develop the theqry of the inversion group of three-dimensions. This should be done both in the real and complex cases and the real and complex inversion spaces studied.
121. Resolution of a displacement into orthogonal line reflections. The properties of the group of displacements are closely bound up with the theorem that any displacement is a product of two orthogonal line reflections. In proving this theorem we shall place no restriction on the absolute polar system 2^, except that it be nondegenerate, and shall base our reasoning on Assumptions A, E, H0 only. We are therefore obliged to consider transformations which do not exist in the Euclidean geometry, namely those with minimal lines as axes.
DEFINITION. The line at infinity polar in 2. to the center of a trans- lation is called the axis of the translation. If the axis is tangent to the circle at infinity, the translation is said to be isotropic or minimal.
THEOREM 22. A product of two orthogonal line reflections whose axes I and m are parallel is a translation whose axis is the line at infinity of any plane perpendicular to the plane of I and m and par- allel to 1. Conversely, let T be any translation and I any nonminimal line meeting its axis ; then if m is the line containing the mid-points
318 AFFINE AND KICLIDKAN (iKo.MKTKY [CHAP, vn
of every pair of points, L and T (L), for which L is on I, and if V is the pole in 2, of the point at infinity of I,
Proof. If the axes I and m of two orthogonal line reflections {II'} and {mm1} are parallel, they meet TT. in a point H,. Each of the orthogonal line reflections effects in irm a harmonic homology whose axis / is the polar of P* in 2*. Hence the product leaves all points at infinity invariant. In the plane of I and m the product {ml'} • {IV} effects a planar translation parallel to any line perpendicular to /. Therefore the product {ml'} • {IV} is a translation in space parallel to this line. Its axis, therefore, is the line at infinity of any plane perpendicular to the plane of I and m and parallel to /.
The converse follows directly in the same manner as the analogous statement in Theorem 4, Chap. IV.
THEOREM 23. Any displacement is a product of two orthogonal line reflections.
Proof. In case the displacement, which we shall denote by A, is a translation the theorem reduces to Theorem 22. In any other case A is a product of a rotation and a translation (Theorem 1 6, Cor. 2), i.e.
A = (^ p}. {£„•}• T,
where T is a translation which may be the identity. Thus A effects in the plane at infinity a product of two harmonic homologies whose centers and axes are Px,px and R*,, rx respectively, where px is the line at infinity of TT and r*, that of p.
Let Q be an arbitrary ordinary point and Q' = A (Q). Let / be the line of intersection of the planes joining Q to px and Q' to rx. These planes cannot be parallel, because p* and rx do not coincide ; and I cannot contain Px or Rx, because Pv is not on p* and R*, is not on rx.
Let O be an ordinary point of / such that neither of the lines OQ and OQ' contains P* or R*. (If the lines OQ and OQ' coincide, they coincide with I.) Let P be the mid-point of OQ, R the mid-point of OQ', and let p and r be the lines PPX and RR*. respectively. Then p and r are such that there exist orthogonal line reflections and {rr*} such that (
{pp»}(0)=Q. Hence {pp*} . {rr.} • (<?')= Q.
§121]
DISPLACEMENTS
319
Moreover, {pp*,} • {rr,} effects the inverse of the transformation effected in the plane at infinity by A. Hence {/>/?»} • {rr,} • A leaves invariant all points at infinity as well as Q, and hence
or
FIG. 75
It is now very easy to enumerate the possible types of displace meuts. A displacement A being expressed in the form {II1} • {mm1}, the following cases can arise : *
I. The lines I and m intersect in an ordinary point O. A is a rota- tion which is the product of the orthogonal reflections in the planes perpendicular to / and ra respectively at 0. Two subcases must be distinguished :
(a) The plane containing I and m is not minimal. A is a rotation about the common intersecting perpendicular of / and m.
(b) The plane containing / and ra is minimal. A is an isotropic rotation about the line joining 0 to the point in which the plane of
* It is to be remembered that neither I nor m can be minimal.
AITIM: AND KIVLIDEAN <;K<»MKTUY [CHAP.VU
/ and /// touches the circle at infinity. It evidently effects a parabolic transformation in tin- pencil <>f planes meeting its axis and also effecta
an elation in the tixeil plane nn the axis.
II. The lines / and //* are jiarallel. If we denote their common point at infinity by 11, and its polar line with respect to £„ by //,; 'I'heoreiu -'- states that A is a translation whose axis is the line polar in Sx to the point in which the plane of I and m meets px. The latter I x lint is the center of the translation. Two cases arise:
' The axis of the translation is not tangent to the circle at infinity. (b) The axis of the translation is tangent to the circle at infinity, and the translation is isotropic.
III. The lines I and m do not intersect. Again two cases arise : (a) The lines / and m have a common intersecting perpendicular
line a (Theorem 6) which is not minimal. Let p be the line parallel to m and passing through the point of intersection of / with a. Then
Thus A is the product of a rotation {IV} • {pp'} about a by a translation {pp'} ' {mm1} parallel to a.
(b) The lines I and m have no common intersecting perpendicular. In this case they are (Theorem 6) both parallel to the same minimal plane a. Let a., be the line at infinity of a, and An its point of con- tact with the circle at infinity. Then / and m pass through points of an distinct from each other and from Ax, and I' and m' pass through Ax. Therefore A effects a transformation of Type III (§ 40, Vol. I) in the jilane at infinity, with A*, as its fixed point and «„ as its fixed line. It also effects a parabolic transformation in the pencil of planes with ax as axis. Thus its only fixed point is Ax, its only fixed line ax, and its only fixed plane TTX.
DEFINITION. A displacement of Type Ilia, i.e. a product of a non- isotropic rotation by a translation parallel to its axis, is called a twist or screw motion. The axis of the rotation is called the axis of the twist.
THK.OKKM 24. A displacement which interchanges two distinct ordi- nary points is an orthogonal line reflection.
Proof. Denote the given points by A and B. The given displace- ment A cannot be a translation, because a translation carrying a point A to a point B would carry B to a point C such that B is the
§§ i-.-i, i±-'] DISPLACEMENTS 321
mid-point of the pair AC. Nor can A be a twist or a transformation of Type Illb, because either of these types effects the same transforma- tion as a translation on a certain system of parallel planes, and hence no ] •» tint can be transformed involutorically. And A cannot be an isotropic rotation, because in this case it would effect a parabolic trans- formation in the planes on its axis and an elation in the one fixed plane on the axis. Hence A is a nonisotropic rotation. By reference to § 115 it follows that A must be an orthogonal line reflection.
THEOREM 25. If A1? A2, A3 are three, orthogonal line reflections whose axes are parallel or have a common intersecting perpendicular I, the product AsA2Aj is an orthogonal line reflection whose axis is parallel to the other three axes in the Jirst case and is an inter- secting perpendicular of I in the second case.
Proof. In case the three axes are parallel, by Theorem 22, A8Aj is a translation which is also expressible as the product of A8 by another orthogonal line reflection A4, so that
AA = A8A4, and hence AaA.A, = A ..
o Z L 4
In case the three axes have a common intersecting perpendicular I, the orthogonal line reflections effect involutions on I having the point at infinity of I as a common double point. Hence (§ 108, Theorem 42) the product A.( A2 At effects an involution on I whose double points are the point at infinity and an ordinary point P. Hence, by Theorem 24; A^Aj is an orthogonal line reflection A4. Since P is left invariant by A4, it is on the axis of A4 ; and this axis is perpendicular to / because
A4 leaves / invariant.
EXERCISE
The product of an isotropic rotation by a translation parallel to its axis is an isotropic rotation about an axis in the same minimal plane.
122. Rotation, translation, twist. Let us now require the absolute polar system to be elliptic, as in the real Euclidean geometry. In this case there are no minimal lines, and hence the possible types of dis- placement are reduced to la, Ha, Ilia. Thus we have
THEOREM 26. In case the absolute polar system is elliptic any dis- placement is a rotation or a translation or a twist.
3-2'2 AFF1NE AND EUCLIDEAN GEOMETRY [CHA.MI
Witli this assumption about the absolute polar system we have a particularly simple method for the combination of displa • •incuts which depends on Theorem 25. Suppose that we wish to combine two displacements {/a/2'} • {l^} and {ltl[} • {(,/,}. Let a be a coninion intersecting perpendicular of l^ and l^ and b of /8 and /4, and let m be a common intersecting perpendicular of a and b. Then the product A of the two displacements satisfies the following conditions :
A =im -tub
By the theorem just proved there exist two orthogonal line reflections (PP'}> to') 8ucn
(24) '
and
(25)
Hence A = {qq1} • {pp'}.
Another way of phrasing this argument is as follows :
By (24), {WMWH to'} •(*"»'},
and, by (25), {l£} - {1&} = {mm'} . {pp'}.
Hence A = {qq'} • {mm1} - {mm'} - {pp'} = {qq'} • {pp'}.
The analogy of this process with that of the composition of vectors is very striking. A vector is denoted by two points. A displacement is denoted l>v Aj . At where A,, and At are the orthogonal line reflections of which it is the 'product. In order to add two vectors AB and CD we choose an arbitrary point O and determine points P and Q such that
AB = PO and CD = OQ. Then we have AB + CD = PO+OQ = PQ.
In the case of two displacements A2At and A4A3 we find an orthogonal line reflection A (which is not arbitrary but is determined according to Theorem 25), for which there are two others, A6 and A0, such that
A2A1 = AA6 and A4A3 = A0A. Hence A^A^jAj = AeAAA6 = AgA,;.
Similar remarks can be made with regard to any group of transformations which are products of pairs of involutoric transformations. See § 108 and, particularly, the series of articles by H. Wiener which are there referred to.
The resolution of a general displacement into a product of two rotations of period two is a special solution of the problem to express
§i±.'] DISPLACEMENTS 323
a given displacement A as a product PA where P and A are rotations, A l>eing of period two. The general solution of this problem may be found very simply in terms of the special one as follows :
Let P be any point of space, and let a be any line through P such that A={M'}.{oa'}.
Let p be the line through P perpendicular to a and intersecting b, and let TT be the plane through P perpendicular to p. Then any line I on P and TT may be taken as the axis of A. This is obvious if / = a. If / =£ a, the product {aa'} • {//'} is a rotation about p, because I and a are perpendicular to p at P. Hence
A . {U1} = {&&'} • {aa1} • {II'} = P
is a rotation about an axis through the point of intersection of b and p. Hence
(26) A=PA
where A ={«'}.
Moreover, if I be any line through P and not in TT, {aa'} - {IV} is a rotation about a line q perpendicular to a and / and hence distinct from p. Since q is perpendicular to a and not identical with p, it does not meet b. Hence the displacement
A • {IV} = {bb'} • {aa'} . {IV}
is not a rotation. Hence the pencil of lines on P and TT is the set of all lines on P which are axes of the rotations A of period two such that A = PA where P is a rotation.
This argument applies to any ordinary point P. There is no diffi- culty in seeing that any point at infinity is also the center of a flat pencil of lines any one of which may be chosen as the axis of A in (26). From this it follows by Theorem 24, Chap. XI, Vol. I, that the set of all lines which are axes of A's satisfying (26) form a linear complex. The argument for the case when P is at infinity is left as an exercise for the reader (Ex. 7). By another application of Theorem 24, Chap. XI, Vol. I, it is easy to prove that the axes of the rotations P which satisfy (26) are the lines of another linear complex. This is also left as an exercise (Ex. 8). Other instances of the resolution of a general displacement into displacements of special types are given in Hxs. 9-11. These exercises all connect closely with those given in the next section
AFFINE AND EUCLIDEAN GEOMETRY [CHAI-.VII
DKHNITIOX. A twist F such that F2 is a translation is called a
half tirint.
An orthogonal line reflection is a special case of a half twist, and any half twist is a product of two orthogonal line reflections wlmsc axes are perpendicular.
EXERCISES
1. If the three common intersecting perpendiculars of the pairs of oppo- site edges of a simple hexagon are also the lines joining the mid-points of the pairs of vertices on opposite edges, they have a common intersecting perpendicular.
2. If the product of three orthogonal line reflections is another line reflec- tion, the three axes are parallel or are all met by a common perpendicular.
3. For any three congruent figures Fv F2, F3 there exists a figure F and three lines lv /„, /8 such that
*'i = { Vi} F, F2 = {//3} F, Fs = {lsl'3} F.
(See the note by G. Darboux on p. 351 of Legons de Cinematique, Paris, 1897, byCJ. Koenigs, where the theorem is credited in part to Stephanos.)
4. The axes of two harmonic orthogonal line reflections meet and are j»erpendicular.
5. For any pair of orthogonal line reflections there is a third which is harmonic to both.
6. Under what conditions are two displacements commutative ?
7. For any displacement A there exists a linear complex C of lines such that every ordinary line of C is an axis of a rotation A of period two such that
A = PA
where P is a rotation. No line not in C is an axis of such a A.
8. If A is a displacement which is not of period two, the axes of the rotations P determined in Ex. 7 form a linear complex Cl which has in common with C all the lines perpendicular to the axis of A.
9. Any displacement A can be put in the form
A = AP
where A and P are rotations and A is of period two. The axes of the A's satisfying this condition constitute the ordinary lines of the complex C*(Ex. 7) and those of the P's the ordinary lines of Cl (Ex. 8). 10. Any displacement A can be put in the form
(27) A = P2.P,
where P, and P2 are rotations or translations. If A is not a rotation or trans- lation, the axis of Pj or of P2 can be chosen arbitrarily. The axes of the Pt's which satisfy (27) are carried into the axes of- the corresponding P2's by a correlation F.
§§ r_-2,
DISPLACEMENTS
325
11. Any displacement A can be put in the form (28) A = PH
where P is a rotation or translation and H a half twist. The axis either of P or of H can be chosen arbitrarily. For any P and H satisfying (28) there exists a rotation or translation P' and a half twist H' such that
A = HP' and A = H'P.
12. Every symmetry is expressible as a product in either order of an orthogonal reflection in a plane ir and a rotation about a line / perpendicular
tOTT.
13. The mid-points of pairs of points which correspond under a symmetry are the points of the plane TT (Ex. 12) or else coincide with the point lir. The planes perpendicular to the lines joining these pairs at their mid-points pass through the point ITT.
14. Every symmetry transformation is expressible as a product in either order of an orthogonal plane reflection and an orthogonal line reflection.
15. Determine the types of symmetry transformations which are distinct under the Euclidean group.
123. Properties of displacements. The main properties of displace- ments which we have found may be stated as follows for the real Euclidean geometry : B,
Any displacement A has a unique axis a which is a line at infinity only in case A is a translation. The displace- ment is a product of two orthogonal line reflections, i.e.
The lines ^ and /2 meet a in two points
Al and A^ and are perpendicular to it.
Let the measure of the angle between
Jj and 12 be 0 and the distance between
Av and A2 be d. Then A is the result-
ant of a translation T parallel to a which carries every point X
to a point A" such that
FIG. 76
and a rotation P with a as axis which carries each plane TT on d to a plane TT' such that the angular measure of TT and -TT' is 2 6.
DEFINITION. The numbers 2 6 and 2 d respectively are called the angle of rotation and distance of translation respectively of A.
326 AFFINE AND EUCLIDEAN GEOMETRY [CHAI-. vn
The rotation P such that A= TP= PT is
win-re /a is the line through Al parallel to J2. Let Bl and BZ be two points of /j and /2 respectively, so chosen that the measure oi&B^AJi^ is 0(and not -rr — 8).* Let one of the two sense-classes (§ 31) in the Euclidean space be designated as positive.
If 0 ^ 6 ^ — , there are two points Ba, B^ on a such that Dist (A&) = Dist (A&) = tan 0.
These points are on opposite sides of the plane AJS^B^ and hence S(AlBlB.1Bt) =£ S(A1B1BZB^). Let Bz be that one of these points for which S(AlBlBiBs) is positive. If 0 = 0, let B^ = AV It is easily seen that this determination of Ba is the same for any choice of BI and B3 subject to the conditions imposed above. Hence any displacement A
7T
for which 0 =£ — determines uniquely a line a and two vectors A^A^
£
and A^Bt, which are parallel to a if a is ordinary. If a is ideal, A is a translation and A. II, zero.
1 o
Conversely, an ordinary line a and two vectors parallel to a deter- mine a unique displacement A. For let Al be any point of a, and ^ any line through Al and perpendicular to a. Then the first vector determines a unique point Az and the second a unique point Bt. There are two lines 12, ly through A^ perpendicular to a and such that m (1~12) = m (1~12) = 6 where tan 6 = Dist B^^. Let B^ be an arbitrary point of lv and B2, B^ points of 12, lz respectively, such that 0 is the measure of 4.B1A1BZ and 4-B^A^B^. Then let BZ be that one of B^ and Bt such that S(A1B1B^BS) is positive, and let ln be the line through At parallel to A^Bf The displacement determined is
Hence any displacement A which is not a half twist determines and is determined by a line a and two vectors A^t and A^B^. From this it is plain that if it be desired to specify a displacement by means of parameters or coordinates, it is necessary to give a set of numbers which will determine the line a (e.g. the Pliicker coordinates of
• The meamire of any pair of lines in three-dimensional Euclidean geometry
Katiufies the condition 0 = 6 ^ -. Cf. § 119.
2
§ i£»] DISPLACEMENTS 327
t
the line) and two additional numbers which will specify the vectors JrJa and A^B^. This question is considered from various points of vif\v in the following sections.
For a treatment of the general problem of parameter representations of displacements and, indeed, of the whole theory of displacements, see the articles by E. Study, Mathematische Aunalen, Vol. XXXIX (1891), p. 441, and Sitzungsberichte der Berliner Mathematischen Gesellschaft, Vol. XII (1913), p. 36. The exercises in this section and the last one are largely drawn from the first of these articles and from the articles by Wiener, referred to above.
EXERCISES
1. Let / be the axis of a twist, a any ray perpendicular to and intersecting /, and b the ray into which a is displaced. Let c be the ray with origin at the mid-point of the segment joining the origin of a and b and bisecting the angle between the rays through this point parallel to a and b respectively. (Two rays are parallel if they are on parallel lines and on the same side of the line joining their origins.) The given twist is the product of the line reflection whose axis contains a by the line reflection whose axis contains c.
2. The product of three rotations whose axes have a point in common and whose angles of rotation are respectively double the angles between the ordered pairs of planes determined by the pairs of axes in a definite order, is the identity.
3. The rotations P and P' described in Ex. 11, § 122, have the same angle of rotation, and the half twists H and H' described in the same exercise have the same distance of translation.
4. There exists an orthogonal line reflection interchanging two congruent ordered pairs of points A^B^ and A2B2 if and only if A^^ is congruent to AzBr
5. There is a unique orthogonal line reflection carrying a given sense-class on a line / to a given sense-class on a line I'. The axes of the two orthogonal line reflections carrying a line I to a line /' are perpendicular to each other and to the common intersecting perpendicular of / and /' at the mid-point of the pair of points in which the latter meets / and /'.
6. If an ordered triad of noncollinear points A1B1C1 is congruent to an ordered triad A2B2C2, the axis of the displacement carrying Av Bv Cl to A», B2, C2 respectively meets orthogonally the axis of the orthogonal line reflection which carries AI and B1 to two points A{ and B{ of the line ,42/?2 such that S(A{Bi) * S(A.,B.2).
7. If three noncollinear points Av At, As are displaced into Aa, At, At respectively, the axis of the displacement is the common intersecting per- |»Midicular of the line joining .12 to the mid-point of A^-ls and the line joining Aa to the mid-point of AtAt.
8. Show how to construct the axis of the displacement carrying an ordered point triad A^B^C^ to a congruent ordered triad rl2#2C,.
328 AFFINE AND EUCLIDEAN GEOMETRY [CHAP.VII
9. If s line / l>e displaced to a line /', the mid-points of pairs of congruent jNiints an- tlu- joints ..f a line /or are identical; the planes perpendicular to the lines joining the pairs of congruent points at their mid-points meet on a line / or are parallel or coincide. Under what circumstances do the different cases arise?
10. If a plane a be displaced to a plane a, the mid-points of the pairs of congruent points are the points of a plane u or the points of a line or coincide ; the planes jx>r|>endicular to the lines joining the pairs of congruent points at their mid-points pass through a point .1 or all meet on a line or coincide. Under what circumstances do the different cases arise ?
11. Let A be a displacement, P a variable point of space, P" = A(/3), 1' the raid-point of the pair PP", and ir the plane through P perpendicular to the line PIy if P ^ ly. Then if A is not a half twist, the transformations Tt such that Tj (P) = P and T2 such that T2 (P) = P/ are affine collineations and
T T — A — T T 121i — ** — AiA2-
If A is not a rotation, the transformation F such that F(P) = TT is a projective correlation such that F(ir) = P'; i.e. such that
T2=A.
If A is not a rotation or a half twist, the transformation N such that N (P) = ir is a projective correlation, and in fact is the null-system of the complex ( ' referred to in Ex. 7, § 122. These transformations also satisfy the equations
Tx = NF, T2 = TN, NA = AN.
12. Using the notations of Ex. 11, if a is any plane, A (a) = a', and T2 (a) = a, then a bisects the pair of planes a and a', and Tl (d) = a.
13. In the correlation N the lines / and / denned by Ex. 9 correspond. The plane a and the point A defined in Ex. 10 also correspond in N.
14. The linear complex C(Ex. 7, § 122) contains every line / which coin- cides with the line / determined by the same line /(Ex. 9). Hence it is the set of those lines /which are perpendicular to the lines joining corresponding points of / and /', and it is also the set of lines / which intersect the lines join- ing corresponding points of / and /'.
15. The affine collineation Tt (Ex. 11) carries the axis of P (Ex. 11, § 122) to that of H'.
16. The correlation T (Ex. 11) carries the axis of Pl (Ex. 10, § 122) to that of P2.
17. The transformations T-1, T2, T, T~l all carry C (Ex. 7, § 122) into Ct (Ex. 8, § 122).
124. Correspondence between the rotations and the points of space. If we confine attention to the rotations leaving a point () invariant,*
* By the reasoning in § 90 it is clear that this amounts to considering the effect of all displacements on the field of vectors.
§ i-J4] ROTATIONS 329
the considerations of the last section simplify considerably. The points Av and A^ may be taken as coincident with 0, and the point Ba shall be denoted by R. Then every uoninvolutoric rotation P corresponds to a definite point R on its axis. An involutoric rotation (orthogonal line reflection) may be taken to correspond to the point at infinity of its axis. Hence the rotations leaving 0 invariant cor- respond in a one-to-one and reciprocal way to the points of the real projective space consisting of the given Euclidean space and its points at infinity.
Let OX, 0 Y, OZ be axes of a rectangular coordinate system with 0 as center such that S(OXYZ) is the positive sense-class. Whenever R is distinct from the origin, denote the measures of 4.ROX, 4.ROY, £ROZ by a, ft, 7 respectively. Then the coordinates of R are
x = tan 6 cos a, y = tan 6 cos ft, z — tan 6 cos 7.
Let (afl, cclf a2, ag) be the homogeneous coordinates of R, so chosen that if R is ordinary,
Y— 1 V 2 7 "3 .
.,1= > -T = > j&= ,
«0 a0 «0
and if R is at infinity, «0 = 0. In either case we may take
«o = cos 0, al = sin B cos a, «2 = sin 6 cos ft, ag = sin 0 cos 7.
According to Theorem 23 any rotation («0, at, «2, «3) is expressible as a product of two involutoric rotations (0, \t, X2, Xg) and (0, /^ /A2, /*8). According to the convention just introduced, the X's and /A'S may be regarded as direction cosines. Hence, by Exs. 5 and 7, § 119,
(29) «0=
X.
8 1
Two fundamental problems now arise: (1) to express the coordi- nates of the point representing the resultant of two rotations in terms of the coordinates of the points representing the rotations, and (2) to write the equations of a rotation in terms of the parameters
(«0» °v aa> «,)•
The formulas (29) are a special case of the formulas which furnish
the solution of the first of these problems. The formulas for the general case may be found by an application of the method for
880
Ail INK AND EUCLIDEAN GEOMETRY [CHAI-. vn
compounding rotations described in § 122. Let the two rotations correspond to A = (a0, a,, «2, «3) and # = (£0, /3V, £2, £t) respectively. Let /A , /A,, /AS be direction cosines of a line perpendicular to OA and OB. Then the rotation («0, «lf «„, a3) is expressible by means of the formulas (29) and (£0, 0lt &, £3) by the following:
(30) /S^/V
According to the principle explained in § 122, the point (y0, 7^ 72, 78) wliich represents the product of («0, at, «2, «3) followed by (£0, /8lf y32, /S3) is
\\
X. X.
(31) 70 = Vi The result of eliminating the X's, ps, and v's from these equations is
(32)
This is most easily verified by substituting (29), (30), and (31) in (32). The rotation (70, 7^ 72, 73) which is the product of («0, a}, a2, «3) and (y3o, fi^ y32, /98) must be that given by (32) ; for if not, there would be some case in which (32) would not be satisfied by the values of «o, 00, 7o, etc. given by (29), (30), and (31).
The formulas (32), which are due to 0. Rodrigues, Journal de Mathe"matiques, VoL V (1840), p. 380, are the same as those for the multiplication of quaternions. Cf. § 127.
The problem (2) of expressing the coefficients of the equations (17) of a rotation in terms of the coordinates of the corresponding point («0, alt af, a3) may be solved very easily by the formulas and theorems of § 118, in the case of rotations of period two. The involutoric rotation corresponding to (0, \lt X2, X3) is, in fact,
x' = (2 \- - 1) x + 2 \\y + 2 X, V,
(33) y' = 2 \\x + (2 X| - 1) y + 2 \,\z,
z'=2 X,X8ic + 2
This is easily verified, because (1) the ma'trix is orthogonal and its determinant is +1, (2) the transformation leaves the point (Xt, X.,, X.()
§124] ROTATIONS 331
invariant, (3) the matrix is symmetric and hence corresponds to a transformation of period two.
To obtain the equations of the transformation corresponding to (aQ> ffj, «2, aa) it would be sufficient to take the product of (33) and the corresponding transformation in terms of f^, /*.,, p3 and compare with equations (29). The algebraic computations involved would, however, be more complicated than in the following method, which is based on a simple observation with regard to collineations whose equations are of the form
(34)
If P = (x, y, z) and P=(x,y,z), then the vector OP is perpendicular to the vector PP, because
(35) x(x-x) + y(y-y) + z(z-z)=0.
The transformation (34) also has the obvious property of leaving invariant all points on the line joining the origin to (at, «a, «8). Conversely, if a collineation
(36) py = anx + a^ + aj, pz = a3ly + aj + aj,
has the property that whenever P = (x, y, z) is distinct from P = (x, y, z), OP is perpendicular to PP, the relation (35) requires that a,y = — «,, whenever i=£j and that p= «u= a^= a9!f If. moreover, (36) leaves all points of the line joining the origin to (at, «2, «a) invariant, it must be either of the form (34) or of the form
a0x = aox — asy -f a^z, (34') ajj = a3x + ajj - a^
a0z = -azx + aly + aoz.
It is also to be observed that the determinant of Transformations (34) and (34') is a0A, where
(37) A = < + a,2 -f al + a2.
This determinant can vanish for real a's only if «0= 0.
332 All INK AM) EUCLIDEAN GEOMETRY [CHA.. vn
Now consider an orthogonal transformation (17) representing a rotation P which is not <>l period two. \j&\, I' be an arbitrary point, p' = l*(/'), and /' the mid-point of P and P'. The relation between P and /' is given by the equations*
(38) 2 y = a2lx + (un+ l)y + a^z,
The line PP is perpendicular to OP and (38) must have the same invariant points as P. Hence if P is the rotation corresponding to (a , a , a2, «8), the equations of the transformation from P to P must be'Vthe form (34) or (34').
Forming the determinants analogous to (19) in § 31, we see that S(OPPR), where R = («0, alf «2, «8), is positive if P is given by (34) and negative if P is given by (34'). Hence (38) must be the inverse of (34). Solving the equations (34) we have
- = , i o3 . , 0a
^ A A
(39) - = «!«, «o^8 g + y + «
a
Z.
Since (38) and (39) must be the same transformation, we have
a =2 1 a =oi*-«* „
-4 .4 .4
(40) a2i = 2
>
8i M M .
These are the formulas, due to Euler, for expressing the coefficients of an orthogonal transformation in terms of the homogeneous parameters
•The transfonnation from /' to P is tliat denoted by T, in Ex. 11. § 123.
§§ i-M, 125] ROTATIONS 333
The formulas for the a's in terms of the ay's may be obtained by taking linear combinations of Equations (40) :
•A a . 'it.
From this it follows that
/ J. 1 \ n • n • ft • n 1 _L n _i_ r* _i_ n • n n • n n • n n
125. Algebra of matrices. The algebra of the last section may be put in a most compact form by means of matrix notation. This requires one or two new definitions. The sum of two matrices is defined by means o£ the following equation :
/bU &12 '
This operation obviously satisfies the associative and commutative laws, namely A + (B + C} = (A +B) + C,
where A, B, C stand for matrices.
Multiplication of matrices has been defined in § 95, Vol. I, i.e.
(43) K) •<&<,)=('<,),
o
where c;j = V aa\,-. Under this definition it is clear that
A (B +C) = AB + A C
and (B + C)A = BA+ CA.
Also it has already been proved that
(AB)C=A(BC).
It is now easy to see that, under these definitions, matrices have most of the properties of a noncommutative number system in the sense of Chap. VI, Vol. I, the matrices
'0 0 0\ /I 0
000 and (010 0 O/ \0 0 It
taking the roles of 0 and 1 respectively. The matrices of the form
fx 0 0x0
.0 0 xl
334
\ I 11 M<: AND EUCLIDEAN GEOMETRY [CHA.VII
form by themselves a number system which is isoinorphic with the number system of the geometry. Such a matrix may be called a scalar and be denoted by x.
Now let us denote the orthogonal matrix of the equations of a rotation (17) by R, and let the skew symmetric matrix
-a.\
_± « o acn an
be denoted by S. Then the matrix of the transformation (34) is 1 -I- S and the matrix of the transformation (38) is 1 (1 + R). The comparing of coefficients of (38) and of (39) amounts to writing
This equation may be transformed as follows :
The last equation, however, states a relation which is obvious from the point of view of matrices. For if S be any skew symmetric matrix, the transposed of S is — S. Since the product of the trans- posed matrices of the two given matrices is the transposed of the product, the transposed of
is (1 + ,<?)(!-, <?)-',
which is also its inverse. Hence, whenever
is orthogonal.
This equation may be solved as follows :
§§ us, i-'"J ROTATIONS 335
which gives the formula for a skew symmetric matrix in terms of an orthogonal matrix.
The operation of taking the inverse of a matrix is defined (of. § 95, Vol. I) in case the determinant of the matrix is distinct from zero. In the operations above, this is a restriction on the matrix 1 + R and, by comparison with Equations (22), is seen to mean that no point must be transformed by the rotation corresponding to R into its symmetric point with respect to the origin.
The generalization from three-rowed to ?i-rowed matrices is obvious, and we thus have the skew symmetric and orthogonal matrices of n rows connected by the relations
(44) Z = (l
(45) S=(l
The equations between the corresponding elements in the matrices which enter in the first of these two matrix equations are the formu- las given by Cayley (Collected Works, Cambridge, 1889, Vol. I, p. 332), expressing the nz coefficients of an orthogonal transformation as
rational functions of — ^ — - parameters.
126. Rotations of an imaginary sphere. The group of rotations leaving a point invariant may be regarded as a subgroup of the collineations of a sphere having this point as center. Let us consider the imaginary sphere
(46) z02 + #12 + z22 + <=0
and apply some of the results obtained in § 102. If a collineation
X0 = CQOX0 + CQIX1 + £02*^2 «" co»x»>
xl = cwx0 -f cu#i + Ci2#a + cwxt,
X<i = CmX0 •+• C.ftX^ + CyfE% •* ^28*^3' X» = C80X0 ~t~ C81Xl ~^~ C31X* ~f~ C3SX»>
carries each line of one regulus on the sphere into itself, any point (x0, xv x2, xs) satisfying the condition (46) must be carried into a point (xl, x[t «2', #,') satisfying the condition
(48) a^ + ^ + a^ + ^O, which states that it is on the sphere, and the condition
(49) XQX'Q + #X + #X + xtx't = 0,
336 AFFINE AND EUCLIDEAN GEOMETRY [CHAP.VII
which states that it is on the plane tangent at (x0, xv xv x9). Substi- tuting (47) in (48) we have, as in § 118,
888
i-O i=0 t = 0 i=0
coicoj + CHCU + c.2ic.2j + c8l.c8, = 0 if * * j.
Substituting (47) in (49) we have
The matrix of the equations (47) must therefore be of the form
/ «o ai «2 «-
I _ yv yv /v
(50) or (51)
On multiplying together two matrices of one of these forms, the product is seen to be of the same form ; whereas if two matrices of different forms are multiplied together, the product does not satisfy the condition c,7 = — cjit i j=j. Hence the matrices of the form (50) must represent the projective collineations leaving all lines of one regulus on (46) invariant, and those of the form (51) must represent the projective collineations leaving all lines of the other regulus invariant. Hence, by § 102, any direct projective collineation leaving the sphere invariant is represented by a product of a matrix of type (50) by one of type (51).
A rotation is a direct collineation leaving invariant both the sphere and the plane at infinity XQ = 0. A collineation (47) leaves x0 = 0 invariant if and only if c01 = c^ = c03 = 0. But on multiplying (50) and (51) it is clear that this can happen only if «0 = pj3Q, al = — pft^ «j = — ^/82, <*„ = — /3/93, p being any number except zero. Hence the matrix representing a rotation is A A, where
A i — a, an — a, a \
A- 2 i and A =?
,— a.— a, a, aj \a. — «.,
§§ i-'., 127] QUATERNIONS 337
The matrix of the product A A is
,-r-j+as 000
0 at+al-al-al 2 («,«,- «0ag) 2 (a,at + «0ag
0 2(a,a8-a0<ri) 2 (a,a, + «„«,) a*
which agrees with (40) of § 124.
Hence the parameters (a0, «„ a.lt a,) in the Euler formulas may be regarded as the elements of a matrix of the form (50) which represents the projectivity effected on one of the reguli of (46) by the rotation.
If two rotations effect pro jectivi ties A and B respectively on* a regulus, the product of the rotations effects the projectivity BA on the regulus (§ 102). Hence the product of two rotations whose parameters are («0, alt #„, a3) and (£0, 0lt /92, j33) respectively has the parameters (yQ, ylt 72, 7,), where
70 7, 72 73\ / ft ft ft ft -7, 70 7, -7, Li -ft ft -ft ft
-72-73 70 7, -ft ft ft -ft
7, ~7i 7o/ \~ft -ft ft ft
This yields the same formulas as (32) in § 124.
EXERCISE
A parameter representation for the sphere (46) is
-a. -a., a. an,
where i2 = — 1. The two reguli on the sphere are the sets of lines for which AJ/AO and p-i/pv respectively are constant. The transformation whose matrix is (50) is given by the projectivity
A£ = (a0 + laj A,, + (a3 - /a.,) AP ^i = ~ (as
127. Quaternions. The definitions of sum and product of matrices in § 125 for three-rowed matrices clearly apply to matrices of any number of rows. With this understanding the sura of two matrices of the form (50) is obviously a matrix of the same form. The same has been seen in the last section to be true of the products of two
338 All INK AND EUCLIDEAN GEOMETRY [CHAP. VII
such matrices. Hence the set of all such matrices is carried into itself by the operations of addition and multiplication of matrices defined in § !-•">.
Let us introduce the notation
/I 0 0 0\ / 0 1 0 0
i/O 1 0 0\ ,-_|-l 000
10 010' 00 0-1
*0 0 0 I/ \0010
Io o i o\ / o o o i'
0 0 0 1 7/0 0-1 0
-1000' 0100
0-1 0 O/ \-l 000;
Then any matrix of the sort we are considering is expressible in
the form
a0l + aj, + aj+ajc.
The matrices i,j, k satisfy the following multiplication table:
(52)
It has been seen in § 125 that matrices satisfy the associative and commutative laws of addition, the associative laws of multiplication, and the distributive laws. They obviously do not, in the present case, satisfy the commutative law of multiplication. Addition is performed by the rule
(53) («ol + aj + aj + «/•) + (/3(1 + ftj + &j + £,/•)
= (ao + £0) 1 + (ttl + £) i + K + 0t)j + (a, + £,) k,
and multiplication by the rule
(54) («o 1+ aj + aj + ajc) . (/3Q1 + /8,i 4- PJ +
•"%l+7t*' +.«*/+ 7&
where
7o = «o^o ~ a A ~ a A ~ "A
(55) 7l = a^1 + a^° + a^3 ~ a^2'
|
i |
J |
k |
|
|
t |
_ i |
k |
-j |
|
J |
-k |
_ i |
i |
|
k |
j |
— i |
-1 |
S§i->7, 128] QUATERNIONS 339
From (53) it is clear that the operation of subtraction can be per- formed on any two matrices of this form. From (55) it is clear that (J3J. + ftj + flj + J3jc)~l exists whenever the determinant
/•? vQ Q Q
01 _0l ft-0!s(/fi
& 0* -0i 0o
is different from zero. This condition is satisfied whenever £o, 0lt $2, $3 are real.
Hence when a(), a^ a2, as are real, the matrices of the form (50) constitute a noncommutative number system in the sense of Chap. VI, Vol. I. This number system is, in fact, the Hamiltonian system of quaternions. Compare the references at the end of the next section, particularly p. 178 of the article in the Encyclopadie and the article by Dickson in the Bulletin of the American Mathematical Society.
EXERCISE
A system of quaternions may be defined as a set of objects [<?] such that (1) for every ordered pair of vectors a, b there is a q, which we shall denote by ("I; (2) for every q there is at least one pair of vectors; (3) two pairs of
vectors OA, OB and OA', OB' correspond to the same q if and only if the ordered triads OA B and OA'B' are coplanar and directly similar in their common plane; (4) the q's are subject to operations of addition and multi- plication defined by the equations
(I) * 0=0-
Prove that a system of q's satisfies the fundamental theorems of a number system with the exception of the commutative law of multiplication. See G. Koenigs, Lecons de Cine'matique (Paris, 1897), p. 4(54.
128. Quaternions and the one-dimensional projective group. On comparing (32) and (55) it is clear that there is a correspondence between quaternions, taken homogeneously, and the rotations leaving a point invariant in which if two quaternions qlt qz correspond to the rotations PI} P2 respectively, the product q.iql corresponds to PaPr The group of rotations is isomorphic with the group of projective trans- formations of the circle at infinity and hence with the projective group
AI i IM: AND EUCLIDKAN GEOMETRY [CHA,- . \u
i
of any complex oiie-dimensional form. There must, therefore, be a relation between quaternions and the one-dimensional projeetivities,
The simplest way to obtain a number system corresponding to these transformations is to apply the operations of addition and multiplication as defined above to two-rowed matrices, i.e.
V .
« A + #
If we write
„_ 0\ /O 1\ /O 0\ /O 0
". —
we have ( a ^\ = ael + fie2 + ye3 •+- 8et.
The units elf ez, ea, ef satisfy the multiplication table
e, *2 0 0 0 0 el e^
€3 e* ° ° 0 0 e3 e4
Although these matrices satisfy the associative and distributive laws of addition and multiplication and the commutative law of addition, it is clear that they do not constitute a number system, because it is possible to have ab = 0 when a =£ 0 and b =£ 0. Never- theless, if we write
/\
^1 is expressible linearly in 1, i, j, k; and
1 2 = y2 = k* = — 1 , V = — ji = k, jk = — kj = i, ki = — ik = j. Hence the system of two-rowed matrices
"
where a, /8, 7, 8 are complex numbers, is equivalent to the set of elements
(56) al-\-bi + cj+dk,
§ 128] QUATERNIONS 341
where 1, i, j, k satisfy the multiplication table (52) of quaternions. The elements (56) are quaternions, properly so called, only when a, l>, r, (I are real. When a, b, c, d are ordinary complex numbers, the elements (56) do not form a number system in the sense of Chap. VI, Vol. I, because there can be elements x, y both different from 0 such that xy=Q.
It is interesting to note that 1, i,j, k are the matrices
/i o\ /V3T o \ / o i\ / o V=I\ V> i/' Vo-vTi/' l-i.oj' Vvri o/'
which represent the identity, and three mutually harmonic involutions
x' = — x, x' = , x' = —
X X
If the projectivities are represented on a conic, these three involutions have the vertices of a self-polar triangle as centers. The matrix represented by
is
«2 + VEla8\ <*„ ~ V~ 1 V '
and its determinant is
< + a? + a * + a*.
The geometric significance of this remark is obvious on comparison with the exercise in § 126.
The relation between quaternions and the one-dimensional projec- tive group was discovered by B. Peirce (cf. Chap. VI by A. Cayley in Tait's Quaternions, 3d edition* Cambridge, 1890). It is an instance of a general relation, noted by H. Poincare, between any linear associative algebra and a corresponding linear group. On this subject see E. Study, Mathematical Papers from the Chicago Congress (New York, 1896), p. 376, and Encyclopadie der Math. Wiss., I A 4, § 12; Lie-Sheffers, Kontinuierliche Gruppen (Leipzig, 1893), Chap. XXI ; and L. E. Dick- son, Bulletin of the American Mathematical Society, Vol. XXII (1915), p. 53. On the general subject of linear associative algebra see L. K. Dickson, Linear Algebras, Cambridge Tracts in Mathematics, No. 16, 1914; and the article by E Study and R Cartan in the Encyclopedic des Sciences Mathematiques, I 5.
342 All INK AND EUCLIDEAN GEOMETRY [CHAP.VII
* 129. Representation of rotations and one-dimensional projectivities by points. The parameter representation of the rotations altout a point which we based iu § 124 on a Euclidean construction lias now been seen to be connected in the closest way with the theory of the one-dimensional protective group. It is therefore of interest to set up the correspondence between the points of space and the rotations about a point in a form which puts in evidence also the correspond- ence between the points of space and the one-dimensional projec- tivities. This has been studied in detail in the memoir by Stephanos referred to in Ex. 3, §110. Ifc will be merely outlined here, because the proofs are all simple applications of theorems which should by this time be familiar to the reader. The construction given below has the advantage over the one given in § 123 of being valid in a general projective space.
Let S* be an arbitrary sphere. (In order to connect with our pre- vious work $2 may be taken as the imaginary sphere xz + y2 + £ •+• 1 = 0). Let R$ and R$ be the two reguli on S2, 0 the center of S*, and C'J the circle at infinity.
An arbitrary rotation P leaving O invariant determines and is fully determined by a projectivity F of C£, and hence is fully determined by its effect on three points Plt Plt P^ of C£. If lv l^ la are the lines of RI on Pv P.it Pa respectively, and m1? ra2, m3 the lines of R% on the points P (P^, P (P2), P (Pa) respectively, the planes Ijn,^ ^m^ ^ms meet in a point R. Let R correspond to P and to F (cf. Ex. 2, § 110).
The following propositions are now easily established by reference to theorems on one-dimensional forms :
The point R is on the axis of P and is independent of the choice of Plt P2, Pz.
If the line OR meets S* in two points Q^, ft (Q^, OR) is the cross ratio of F.
The involutions correspond to points of the plane at infinity.
Pairs of inverse projectivities correspond to pairs of points having O as mid-point.
Harmonic projectivities (§ 80, Vol. I) of Cl correspond to points which are conjugate with respect to S*.
The projectivities of C* harmonic to a given projectivity corre- spond to the points of a plane. Such a set of projectivities may be called a bundle of projectivities.
ROTATIONS REPRESENTED BY POINTS 343
The projectivities common to two bundles correspond to the joints of a Hue and may be called a pencil of projectivities.
A pencil of involutions according to this definition is the same as a pencil of involutions according to the definition in §78, Vol. I.
The product of the projectivities corresponding to points 7^ and Rt, not collinear with 0, corresponds to a point Ra obtained by the fol- lowing construction : Let /', I" be the lines of R2 through the points in which ORl meets S2, and let m', m" be the lines of Rf. through the points in which OR^ meets S'2. The line through A't meeting m' and m" intersects the line through R^ meeting V and /" in the point RS. If V and I" coincide, the line meeting them is understood to be tangent to S2, and a similar convention is adopted in case m' and m" coincide.
If Rl be regarded as fixed and R^ as variable, R3 is connected with R by the relation
where A is a projective collineation leaving the lines I', I" pointwise invariant. In case /' = I", A is a colliueation of the type in which all points and planes on V are invariant and each plane on V is trans- formed by an elation whose center is the point of contact of this plane with S2.
If 7?3 be regarded as fixed and Rl as variable, the transformation
defined by the relation
>
is a collineation interchanging the reguli R2 and R~, and carrying each line /of R2 into the line m of R2 in the plane R.J, and each line m of R2 into the line I of Rf in the plane Om.
The propositions above are derivable from Assumptions A, E, P. In a real space we have
The rotations represented by points of a line all carry a certain ray with 0 as origin to a certain other ray with O as origin. Con- versely, all rotations carrying a given ray with O as origin to a second ray with 0 as origin are represented by points of a line.
The necessary and sufficient condition that two rotations Pt, P2 be harmonic is that there exists a ray r such that P^r) is opposite to P2(r).
The representation of rotations by points given in § 124 is identical with the one given in this section, in case fP is imaginary. In case S2 is real, the real points of space represent imaginary rotations.
344 AFFINE AND EUCLIDEAN GEOMETRY [CHAP.VII
If S* is a ruled quadric and Cl a real conic, the construction above gives a representation of the real projectivities of a one-dimensional form by the points of space not on S3. The sets of points [Z>] and [0] representing the direct and opposite projectivities respectively are such that any two points of the same set can be joined by a segment consisting of points of this set, whereas any segment joining a D to an O contains a point of $2. The sets [D] and [O] are called the two sides of S*.
EXERCISES
1. Study the configuration formed by the points representing the rotations which carry into itself (a) a regular tetrahedron; (b~) a cube; (c) a regular icosahedron. (Cf. Stephanos, loc. cit., p. 348.)
2. A real quadric (ruled or not) determines two sets of points, its W>x, such that two points of the same side can be joined by a segment consisting entirely of points of this side and such that any segment joining two points of different sides contains one point of the quadric. If the quadric is not ruled, one and only one of its sides contains all points of a plane. This side is called the outside or exterior, and the other the inside or interior.
130. Parameter representation of displacements. Simple algebraic considerations will enable us to extend the parameter representation of rotations considered in the sections above so as to cover the case of displacements in general. We will suppose the general displace- ment given in the form
x' =
(57) o «, , .'
< = Vo + Vi + V* + «»*,»
<= «»*« + Vl + «**,+ W
where the matrix (aua22«88) is orthogonal. According to § 126, if aw= aa>= as>= 0» the matrix of (57) is expressible in the form AA, A and A being defined at the bottom of page 336.
Now observe that if
0 0
2£8 0 0 O/
and C is any four-rowed matrix, C • B is a matrix in which all elements except those of the tirst column are zero. From this it
DISPLACEMENTS
follows that A (A — Ji) will be of the form (57). In fact, if we require also that
we have
1 9
. — a.
— a.
Hence the coefficients of (57) are given in terms of two sets of homo- geneous parameters «0, at, as, a3 ; /80, /S^ /82, /33 by the equations (40), together with aw=l and
(60)
= 2 («2/3o- «A- a«
provided that the a's and /S's are connected by the relation (59). Conversely, the «'s and /S's are determined by the coefficients of (57) according to the equations (41) and the following:
(61) /30:/91:^:^8=a10(a82-a28)-fa20(a13-a81) + a30(«2l-ai2):
The last equations are obtained by solving (59) and (60) simultaneously for the £'s and substituting the values of the a's given by (41).
It remains to find the formulas for the parameters (a'0't a", a'J, a'3'; 0", ft', ft', ft) of a displacement A" which is such that A" = A' • A, where A has the parameters («0, «,, «2»aa; ft, £,, yS2, /9a) and A' the parameters (aj, «{, a!2, a't; ft, ft, ft, ft).
34G All INK AM) EUCLIDEAN GEOMETRY [CHAF. VH
We have seen that the matrix of A is of the form A(A — B), where A and A are of the form given at the bottom of page 336 and />' is given by (58). In like manner A' can be expressed in the analogous form A'(A'-B') and A" in the form A"(A"-B"). Since the /^s do not enter into any coefficients of (57) except a10, a2o, a^, it is clear that a", a", a,', a,' are given by the formulas (32), or, in other words, that A"=A'A. By definition,
A"(A"-B") = A'(A'-B')A(A-B)
= A' A' A A - A'A'AB - A'B'AA + A'B'AB.
In view of (59), the elements of the first row of AB are all zero. Hence all the elements of B 'AB are zeros. Hence
A'B'AB=Q.
Since A and A' are the matrices of transformations of two conjugate reguli, each transformation leaving all the lines of the other regulus invariant, they are commutative. Hence
A" (A" - B") = A'AA'A- A'AA'B - A'AA^B'AA.
But A-* = A*-- -r^— >
«0- 4- a- + a* + «l
/!° where
and B'AA = B' - « + «? + a~ -f <rs2).
Hence
(62) A" (A" + B") =A'A (A' A- A'B - A*B'). Since A" = A'A and A" = A'A, it follows that
(63) B"= Hence
(64)
§!••«>] BIQUATERNIONS 347
Rewriting (32) in our present notation, we also have
<= «X- «i'«i- «X~ «'»<**> a'' = « -|- aa a> - a'a
The formulas (64) and (65) can be put into a very convenient form by means of the notation of biquaternions.* Let us define a bi- quaternion as any element of a number system whose elements are expressions of the form
(66) s = (aQ+ali + aJ + alc) + e(^ + Pii + PJ + f3Jc},
where the a's and /S's are numbers of the geometric number system, i, j, k are subject to the multiplication table (52), and € is subject to
therules e2 = 0, ex = x€,
where x is any other element, and where the elements (66) are added and multiplied according to the usual rules for addition and multipli- cation of polynomials.
If the product of s and s', where
s' = « + a[i + a'J + a'Jc) + e (fl + fti + ffj + #*), be denoted by
s" = s'-s = « + <t + <y + <*) + € (%' + fi'i + ffij + file),
the a'J, • • •, /Sg' are given by the formulas (64) and (65).
For a more complete study of the parameter representation of dis- placements, see E. Study, Geometric der Dynamen (particularly II, § 21), Leipzig, 1903.
EXERCISES
1. The parameters of a twist may be taken so that av av at are the direction cosines of the axis of the twist ; aa = cot 0, where 2 0 is the angle of rotation ; and )80 = <l, where 2 d is the distance of translation.
2. Find the equations of Tp T2, F, N, etc. as denned in the exercises of § 123.
*3. Find a parameter representation for the displacements in a plane which is analogous to the one studied above (cf. Study, Leipziger Berichte, Vol. XL I (1889), p. 222).
* W. K. Clifford, Preliminary Sketch of Biquaternions, Mathematical Papers (London, 1882), p. 181. The system of biquaternions here used is one of the three systems of hypercomplex numbers known by this name. See § 146, below.
M I INK AND EUCLIDEAN CKoMKTKV [CHAI-. vn
GENERAL EXERCISES
Classify «it-h t/ixin/n In this list of exercises according to the type of project n-,' • tit n-liiffi it aid i/ lie mini and according to the geometry to icli it-It it l»li>ntjs.
1. A homology whose plane of fixed ]>oints is ideal is called a ifilntinn or expansion. Any traiist'.irmation of the Euclidean group is either a displace- ment or a dilation or the jiroduct of a rotation by a dilation.
2. Anv transformation of the Euclidean group leaves at least one line invariant.
3. Any transformation of the Euclidean group is either a displacement <>r a dilation or the product of a displacement by a dilation, whose center is <m a fixed line of the displacement.
4. Let / be a line which is invariant under a transformation F of the Kurlidean group, and let k be the characteristic cross ratio (§ 73, Vol. I) of the projectivity effected by F on /. F is a displacement or symmetry if and only if k = ± 1.
5. Any transformation of the Euclidean group which alters sense can be expressed as a product APA, where A is a dilation or the identity, P an orthogonal plane reflection, A an orthogonal line reflection or the identity.
6. If two triangles in different planes are perspective, and the plane of one be rotated about the axis of perspectivity, the center of perspectivity will de- scribe a circle in a plane perpendicular to the axis of perspectivity (Cremona, I'rojective Geometry, Chap. XI).
7. The planes tangent to the circle at infinity constitute a degenerate plane quadric. With any real nondegenerate quadric this determines a range of quadrics, i.e. a family of quadrics of the form
/(«i» «s» «s) + M"i + M22 + «s2) = 0,
where f(vv «2, us) is the equation in plane coordinates of the given quadric. This is called a confocal system of ijuadrics. Besides the circle at infinity this range contains three other degenerate quadrics, an. imaginary ellipse, a real ellipse, and a hyperbola. There is one quadric of the range tangent to any plane of space. There are three quadrics of the range through any point of space, and their tangent planes at this ixnnt are mutually orthogonal.
8. Let [/] and [/«] be two bundles of lines related by a projective trans- formation F. There is one and, in general, only one set of three mutually perpendicular lines lv /2, /8 transformed by F to three mutually perpendicular lines //i,, 7«2, my There are two real pencils of lines in [7] which are transformed by F into congruent j*encils of [/»]. What special cases arise ? Cf. Encyclopedic des Sc. Math., Ill, 8, § 9.
9. Let F be a collineation of space. The planes F(7roc) and F-^TT^) are called the vanishing planes of F. Through each point of space there is a pair of lines each of which is transformed by F into a congruent line (i.e. pairs of points go into congruent pairs). These lines are all parallel to F-^TT,,).
§ i*>] EXERCISES 349
10. A collineation F which does not leave the plane at infinity invariant determines two systems of confocal quadrics such that tin- nut- system is carried by F into the other. Cf. § 84 and the references given there.
11. Let T be a direct-similarity transformation of a plane, .Ij a variable point of this plane, A., = T(.lj), and „•!, a point such that the variable triangle A j. 1 ... 1 3 is directly similar to a fixed triangle l^Ii^By Then the transformations fnnu .^ to As and from J2 to At are direct-similarity transformations. Both of these transformations have the same finite fixed elements as T.*
12. Let T be an affine transformation, Al a variable point, .42 = T(.11), and J0 a point such that the ratio J0. 1 J . (0,12 is constant. The transformation P from A1 to A0 is directly similar and has the same fixed elements as T. If T is a similarity transformation, so is P.
13. If T! and T2 are affine transformations,.!,, a variable point, Ar = Tt (/10), At = T2 (--!0), and .43 a point such that .11.10.12.13 is a parallelogram, the trans- formation from -10 to As is affine.
» On this and the following exercises cf . Encyclopadie der Math. Wiss. Ill AB 9, pp. 914-915.
CHAPTER VIII
NON-EUCLIDEAN GEOMETRIES
131. Hyperbolic metric geometry in the plane. According to the point of view explained in § 34 there must be a geometry corre- sponding to the projective group of a conic section. The case of a real conic in a real plane is one of extreme interest because of its close analogy with the Euclidean geometry, as will be seen at once.
DEFINITION. An arbitrary but fixed conic of a plane TT is called the, absolute conic or the absolute. The interior of this conic is called the hyperbolic plane. Points interior to the conic are called ordinary points or hyperbolic points, and those on the conic or exterior to it are called ideal points. A line consisting entirely of ideal points is called an ideal line, and the set of ordinary points on any other line is called an ordinary line or a hyperbolic line. The group of all projective collineations leaving the absolute conic invariant is called the hyper- bolic (metric) group of the plane, and the corresponding geometry is called the hyperbolic plane geometry.
Let us at first assume only that the plane TT is ordered (A, E, S, P). On this basis we have as a consequence the theorems in §§ 74, 75 on the interior of a conic, that the points of an ordinary line satisfy the definition in § 23 of a linear convex region. This determines the meaning of the terms " segment," " ray," " between," " precede," etc. as applied to collinear ordinary points and sets of points in the hyper- bolic plane. The ordinal properties of the hyperbolic plane may be summarized as follows:
TIIKORKM 1. The hyperbolic plane satisfies Assumptions I— VI given for the Euclidean plane in § 29.
Proof. Assumptions I, II, III, V are direct consequences of the proposition that the points of an ordinary line constitute a linear convex region. Assumption VI, that the interior of a conic con- tains at least three noncollinear points, i» an obvious consequence of §§ 74, 75.
350
§wi] HYPERBOLIC GEOMETRY 351
The hypothesis of Assumption IV is that three points A, B, C are noncollinear and that two other points D and E satisfy the order relations {BCD} and {CEA}. The conclusion is that there exists a point F on the line DE and between A and B. To prove this it is necessary to show (1) that the point of intersection F of the protective lines DE and AB is interior to the absolute conic and (2) that F is between A and B. Let I be a line exterior to the conic, and let its points of intersection with the lines AB, BC, CA respectively be F., !)„, Ex. By hypothesis and §75, the pair DDX is not separated by BC and the pair EE<* is separated by AC. Hence, by § 26, the pair FFn is separated by AB. Since F* is exterior to the conic, F is interior (§ 75) and between A and B.
THEOREM 2. The hyperbolic plane does not satisfy Assumption IX, §29. On the contrary, if a is any line and A any point not on a there are infinitely many lines on A and coplanar with a which do not meet a.
Proof. By § 75 the projective line containing a also contains an infinity of points exterior to the absolute. Any line of the hyperbolic plane contained in the projective line joining A to one of these points fails to meet a.
DEFINITION. If a projective line containing a line a of a hyperbolic plane meets the absolute conic in two points B^, €„, and A is any ordinary point not on a, the ordinary lines contained in the projective lines ABX and AC,, are said to be parallel to a. The segments AS, and ACX, consisting entirely of points interior to the absolute, consti- tute, together with A, two rays which are also said to be parallel to a.
If the projective plane TT be supposed real, the points Bx and Cm exist for every line a, and hence we have
THEOREM 3. In the real hyperbolic plane there are two and only two lines which pass through any point A and which are parallel to a line a not on A. There are two and only two rays with A as end parallel to a.
3.VJ NON-EUCLIDEAN GEOMETRIES [CHAP.VIII
This theorem of course does not require full use of continuity assumptions. It would also be valid if we assumed merely that any line through an interior point of a conic meets the conic (cf. § 70).
PKKIMTION. The points on the absolute are sometimes called point*. nt infinity or in finite pointy; and the points exterior to the absolute, ultra-i n finite j><> in /*.
132. Orthogonal lines, displacements, and congruence.
DKKIXITION. Two lines (or two points) are said to be orthogonal or perpendicular to each other if they are conjugate with respect to the absolute.
Of two perpendicular points one is, of course, always ultra-infinite, but no analogous statement holds for perpendicular lines. From the corresponding theorems on conies we deduce at once
THEOREM 4. The pairs of perpendicular lines on an ordinary point are pairs of a direct involution. Through an ordinary point there is one and but one line perpendicular to a given ordinary line,
DEFINITION. A transformation of TT which effects an involution on the absolute conic whose axis contains ordinary points is called an orthogonal line reflection. A transformation of TT which effects an involution on the absolute conic whose center is an ordinary point is called a point reflection. A product of two orthogonal line reflections is called a displacement. A product of an odd number of orthogonal line reflections is called a symmetry. Two figures such that one can be carried to the other by a displacement are said to be congruent, and two figures such that one can be carried to the other by a symmetry are said to be symmetric.
An orthogonal line reflection is a harmonic homology whose center and axis are pole and polar with respect to the absolute conic. Since the axis contains an interior point, the center is exterior and the involution effected on the absolute alters sense (§ 74). Conversely, it follows from § 74 that an involution on the absolute conic which alters sense is effected by a harmonic homology whose center is exterior to the absolute conic, — i.e. by an orthogonal line reflection.
Since any direct projectivity is a product of two opposite involu- tions (§ 74), the displacements as defined above are identical with the projective collineations which transform the absolute conic into it- self with preservation of sense. In particular, a point reflection is a
§i.;-'J HYPERBOLIC GEOMETRY
displacement. On the other hand, the symmetries are the projective collineations which carry the absolute into itself and interchange the two sense-classes on the absolute.
From these remarks it is evident that the theory of displacements can be obtained from the theorems on projectivities of a conic in Chap. VIII, Vol. I, and in Chap. V, Vol. II. Some of the theorems may also be obtained very easily as projective generalizations of simple Euclidean theorems.
In proving these theorems we shall suppose that we are dealing with the real projective plane and not merely with an ordered plane as in Theorem I. It would be sufficient, however, to assume merely that every opposite involution is hyperbolic (i.e. that every line through an interior point of a conic meets it), for this proposition is the only consequence of the continuity of the real plane which we use in our arguments.
Let us first prove that Assumption X (§ 66) of the Euclidean geometry holds for the hyperbolic geometry. It is to be shown that if A, B are two distinct points, then on any ray c with an end C there is a unique point D such that AB is congruent to CD. The points A and C are the centers of elliptic involutions on the absolute. It is shown in § 76 that one such involution can be trans- formed into any other by either a direct or an opposite involution. Hence there is a displacement A carrying A to C.
The absolute conic may be regarded as a circle (72 in a Euclidean plane whose line at infinity is the pole of C with regard to the absolute. In this case C is the center of the Euclidean circle, and the hyperbolic displacements are the Euclidean rotations leaving C invariant. The required theorem now follows from the Euclidean proposition that there is one and only one rotation carrying B to a point D of a ray having C as end. The point D is interior to C2 because B is.
Assumption XI, § 66, holds good in the hyperbolic geometry because the displacements form a group. Assumption XII may be proved for the hyperbolic geometry by the argument used in § 66 for the Euclid- ean case. The same is true of Assumption XIII if we understand by the mid-point of a pair AB the ordinary point which is harmonically separated by the pair AB from a point conjugate to it with respect to the absolute.
354 N"N EUCLIDEAN CKoMKTRIES [CHAI-. vm
DEFINITION. A circle is the set [P] of all points such that the point pairs Of1 where 0 is a fixed point are all congruent to a fixt-il point
If the absolute be identified, as in the proof of Assumption X above, with a Euclidean circle C2, and O with its center, it is obvious that the circles of the hyperbolic plane having O as center are identical with the Euclidean circles interior to and concentric with (72. Hence we obtain from the properties of a pencil of concentric Euclidean circles (§ 71)
THEOREM 5. DEFINITION. A circle in the hyperbolic plane is a conic entirely interior to tJie absolute. It touches the absolute in two conjugate imaginary points A, B, and the tangents at these points pass through the center of the circle. The polar of the center passes through A and B and is called the axis of the circle. All its real points are exterior to the absolute conic.
It will be proved in § 134 (Theorem 7, Cor. 1) that two circles can have at most two real points in common. Once this is established, the proof of Assumption XIV in § 66 applies without change to the hyperbolic geometry.
Assumption XV is proved in § 134 as Cor. 2 of Theorem 7.
Assumption XVI may be proved as follows: Let A, B, C be three points in the order {ABC}, and let P* and Q*, be the points in which the line AB meets the absolute conic, the notation being assigned so that we have {P*,ABCQ*}. Let Blt J52, B8, ... be points in the order {J^ABB1B2B3 • • •} such that AB is congruent to each of the pairs BBV, B^B^ etc. Choose a scale (Chap. VI, Vol. I) in which P«AQ<» correspond to 0, 1, oo respectively, and let b be the coordinate of B. By the hypothesis about the order relations, b>l. The displacement carrying AB to BBl is a projectivity of the line AB which leaves P* and Q. respectively invariant and transforms A to B. Hence it has the equation
x' = bx
with respect to the scale R, A, Q*. The coordinates of Blt Bz, /?,, « • « are therefore J2, &8, b4, • • • respectively. The coordinate of C is, by the hypothesis that {ABC}, "some positive number c greater than b. There are at most a finite number of values of
§§i:fcz,i33] HYPERBOLIC GEOMETRY 355
l"(n = 1, 2, • • •) between l> and c. Hence there are at most a finite number of the points .1^, B^ • - • between B and 6'. This is what is stated in Assumption XVI.
We have now seen, taking for granted two results which will be proved in § 134, that all the assumptions (cf. §§ 29 and 66) of Euclidean plane geometry except the assumption about paral- lel lines are satisfied in the real hyperbolic plane, and that the parallel-line assumption is not satisfied.
EXERCISES
1. If corresponding angles of two triangles are congruent, the correspond- ing sides are congruent.
2. The absence of a theory of similar triangles in hyj>erbolic geometry is due to what fact about the group of the geometry ?
3. The perpendiculars at the mid-points of the sides of a triangle meet in a point (which may be ideal).
* 4. Classify the conic sections from the }K>int of view of hyperbolic geometry.
133. Types of hyperbolic displacements. According to § 77, Vol. I, any displacement has a center and an axis which it leaves invariant. If the center is interior, the axis meets the absolute in two conjugate imaginary points, and the displacement effects an elliptic transfor- mation on the absolute. If the center is exterior, the axis meets the absolute in two real points, and the displacement effects a hyper- bolic transformation on the absolute. If the center is on the absolute, the axis is tangent, and the displacement effects a parabolic trans- formation on the absolute.
In the first case, the points into which a displacement and its powers carry a point distinct from its center are, by definition, on a circle which is transformed into itself by the given rotation.
In the second case, since the displacement is a product of two orthogonal line reflections whose axes pass through the center, it is obvious that the displacement leaves invariant any conic C2 which touches the absolute in the two points in which it is met by the axis of the displacement. Such a come is obtained from the absolute by a hornology whose center and axis are the center and axis of the displacement in question. From this it follows in an obvious way that C2 is entirely interior or entirely exterior to the absolute. We are interested in the case in which T2 is interior.
V\ KtVLIDEAN GEOMETRIES
[CHAP.YIH
Let the points of contact of Ca with the absolute A'2 be 1' and Q respectively. Since the center of the displacement (> and the lim- l'tt> are polar with respect to C*, P and Q are the ends of two segments <r, T of points of <'* which are (in the hyperbolic plane) on opposite sides of the line PQ. Any line through 0 and a point of the hyper- bolic plane is perpendicular to PQ and meets <r, PQ, and r in three points S, M, T respectively. If S', M', T' are Jhe points analogously determined by another line through 0, let M be the mid-point of the pair MM'. Then the displacement which is the product of the orthogonal line reflection with OM as axis by that with OM ' as axis carries S, M, T to S', M', T' respectively.
This result may be expressed by saying that <r is the locus of a point S't on a given side of PQ, such that if M1 is the foot of the perpendicular from S' to PQ, S'M' is congruent to SM. For this reason a and T are called equidistantial curves of PQ.
A point A can be carried into a point B by a displacement leaving a given line /, not on A, invariant, if and only if the two points are on the same equidistantial curve of I. The equidistantial curves have some of the properties of parallel lines in the Euclidean geometry.
A displacement which effects a parabolic transformation on the absolute is a product of two orthogonal line reflections whose axes intersect in the center 0 of the displacement. Hence the displace- ment leaves invariant any conic which has contact of the third order (see § 47, Vol. I) with the absolute at 0. And by the same reasoning as employed in the second case, a point P can be transformed into a point P1 by a displacement which is parabolic on the absolute with a fixed point at 0 if and only if P and P1 are on a conic having contact of the third order witli the absolute at 0.
DKFINITIOX. A conic interior to the absolute and having contact of the third order with it is called a horocycle.
The circles, equidistantial curves, and horocycles are all path curves of one-parameter groups of rotations.
FIG. 78
§§i:-, i:w] HYPERBOLIC GEOMETRY :'.:>7
134. Interpretation of hyperbolic geometry in the inversion plane. Although the theory of conies touching a fixed conic in pairs of points has not been taken up explicitly in this book, we have in the inversion geometry a body of theorems from which the part of it needed for our present purpose can be obtained by the principle of transference.
It has been seen in § 94, Theorem 16, that any transformation of the inversion group which carries a circle A'2 into itself effects a protective transformation of this circle into itself. Moreover, there is one and only one direct circular transformation which effects a given projectivity on A'2. Hence the group of direct circular trans- formations leaving a circle of the inversion plane invariant is simply isomorphic witli the hyperbolic metric group, and the geometry of this subgroup of the inversion group is the hyperbolic geometry.
The circles orthogonal to A2 have the property that there is one and only one such circle through each pair of distinct points interior to A'2. Since they also are transformed into themselves by the group which is here in question, it is to be expected that they correspond to the lines of the hyperbolic plane. This may be proved as follows :
Let the inversion plane be represented by a sphere £2 in a Euclid- ean three-space. Let A2 be the circle in which S* is met by a plane TT through its center, and let us regard the points of TT interior to A*2 as a hyperbolic plane. The circles of 8* orthogonal to A2 are those in which S2 is met by planes perpendicular to TT. Hence if we let each point P of S2 on one side of A'2 correspond to the point P' of TT such that the line PP' is perpendicular to TT, a correspondence F is established between the hyperbolic plane and the points on one side of a circle A'2 in the inversion plane in such a way that the lines of the hyperbolic plane correspond to the circles orthogonal to A'2. Moreover, since the direct circular transformations of the inversion plane are effected by three-dimensional collineations leaving S* in- variant, the direct circular transformations leaving K* invariant cor- respond under F to displacements and symmetries of the hyperbolic plane. Thus we have
THEOREM 6. There is a one-to-one reciprocal correspondence F be- tween the points of a hyperbolic plane as defined in §131 and the points on one side of a circle K* in an inversion plane (or insist « circle of the Euclidean plane] in which sets of collinear points of the
358 NON i:i rui)EAN GEOMETRIES [CHAP.VIH
• rbolic plane correspond to sets of points on circles orthogonal (<> K*, ami in which displacements and symmetries of the hyperbolic plane curn-fijHHut to direct circular transformations leaving Kl invariant.
THEOREM 7. In the correspondence F the circles of the hyperbolic plane correspond to circles of the inversion plane which are entirely on one side of A
Proof. Let C* be any circle entirely on one side of K*, and let O and (J be the two points which are inverse with respect to both JK* and C'1, Le. the limiting points of the pencil of circles containing C* and K* (§§ 71, 96). In the Euclidean plane obtained by omitting <y from the inversion plane, 0 is the center of both A"2 and C2, and hence the direct circular transformations leaving K2 and C2 invariant are the rotations about 0 and the orthogonal line reflections whose axes are on 0. These correspond under F to the displacements and symmetries of the hyperbolic plane which leave 0 invariant. Hence the points of C2 correspond to a circle of the hyperbolic plane.
Since any circle of the hyperbolic plane may be displaced into one whose center corresponds under F to 0, the argument just made shows that every circle of the hyperbolic plane may be obtained as the correspondent under F of a circle of the inversion plane which is interior to K3.
This theorem enables us to carry over a large body of theorems on circles from the Euclidean geometry to the hyperbolic. For example, we have at once the following corollaries :
COROLLARY 1. Two circles in the hyperbolic plane can have at most two real points in common.
COROLLARY 2. If the line joining the centers of two circles in the hyperbolic plane meets them in pairs of points which separate each other, the circles 'meet in two points, one on each side of the line.
The first of these corollaries, on comparison with Theorem 5, yields the following projective theorem: Two conies interior to a real conic and touching it in pairs of conjugate imaginary points can have at most two real points in common, and always have two conjugate imaginary points in common.
THEOREM 8. In the correspondence F equidistantial curves of the hyperbolic plane correspond to those portions of circles intersecting K*, not orthogonally, which are on one side of K2. Two equidistantial
«I:M] HYPERBOLIC (JK< »M KTK Y 359
cut' res which are parts of o tie conic in the hyperbolic plane are part* <>/ <• ire />; s- inverse to each other with respect to A'J.
Proof. A circle A2 of S* wliich intersects A'2 iu two points P, Q without being perpendicular to it is a section of S* by a plane not perpendicular to rr. The correspondence F transforms this circle into a conic section (72 in TT which is the projection of A,2 from the point at infinity of a line perpendicular to IT. The tangents to A~j2 at P and (t> are transformed into tangents to A'2. Hence C* touches A2 at P and V.
The portions of K* on the two sides of A2 on Sz correspond to the two segments of C2 having P and Q as ends ; but only one of these portions of A2 is on the side of A2 which is in correspondence with the hyperbolic plane by means of F. The segment of C* which is not in correspondence with this portion of A2 is evidently in corre- spondence with a portion of the circle into which A'2 is transformed by the three-dimensional orthogonal reflection with TT as plane of fixed points.
This proves that the part of any circle A2 of the inversion plane which is on one side of A'2 corresponds under F to an equidistantial curve E^ and that that part of the circle inverse to A"2 with respect to A'2 wliich is on the same side of A2 corresponds to the equidistantial curve E^ wliich is part of the same conic with E^. That any equi- distantial curve is in correspondence with a portion of some circle of the inversion plane is easily proved by an argument like that used in the last theorem.
COROLLARY 1. In the correspondence F a circle touching A2 corre- sponds to a liorocycle of the hyperbolic plane.
Since each equidistantial curve corresponds to a portion of a circle of the inversion plane, it follows that two equidistantial curves can have at most two real points in common. It must be noted that two conies containing each an equidistantial curve can have four real points in common, since each conic accounts for two equidistantinl curves.
Tu like manner two horocycles can have at most two real points in common, and, still more generally,
COROLLARY 2. Two loci each of which is a circle, horocycle, or equir distantial curve can have at most two points in common.
NON-EUCLIDEAN GEOMETRIES [CHAP.VIII
EXERCISES
1. Show that T may be extended so that the ultra-infinite lines of the hyper- bolic plane correspond to imaginary circles of the inversion plane which are orthogonal to A"*.
2. Study the theory of pencils of circles, equidistantial curves, and horo- cvcles in the hyperbolic plane by means of the correspondence F. (A list of the theorems will be found in an article by E. Ricordi, Giornale di Matematiche, Vol. XVIII (1880), p. 255, and in Chap. XI of Non-Euclidean Geometry by J. L. Coolid-e. Oxford, l!l<>!».)
3. Develop the theory of conies touching a fixed conic in pairs of points.
135. Significance and history of non-Euclidean geometry. In proving the two corollaries of Theorem 7 we have completed the proof (§ 132) that the congruence assumptions of § 66 are satisfied in the hyperbolic plane. Combining this result with Theorems 1 and 2, we have
THEOREM 9. In the real hyperbolic plane geometry, Assumptions I- VI, VII, X~ XV I of the assumptions for Euclidean plane geometry in §§ 29 and 66 are true, and Assumption IX is false.
COROLLARY. Assumption XVII of § 29 is true in the hyperbolic plane geometry. .
The existence of the hyperbolic geometry therefore furnishes a proof of the independence* of Assumption IX as an assumption of Euclidean geometry. This assumption is equivalent to, though not identical in form with, Euclid's parallel postulate.! And it is the interest in the parallel postulate which has been the chief historical reason for the development of the hyperbolic geometry.
The question whether the postulate of Euclid was independent or not was raised very early. In fact, the arrangement of propositions in Euclid's Elements shows that he had worked on the question himself. The effort to prove the postulate as a theorem continued for centuries, and in the course of time a considerable number of theorems were shown to be independent of this assumption. Eventually the question arose, what sort of theorems could be proved by taking the contrary of Euclid's assumption as a new assumption.
•Cf . § 2, Vol. I, and § 18, Vol. II.
tCf. Vol. I, p. 202, of Heath, The Thirteen Books of Euclid's Elements, Cambridge, 1908.
§ I'M] HISTORY 361
Tliis question seems to have been taken up systematically for the tirst time by G. Saccheri,* who obtained a large body of theorems on this basis, but seems to have been restrained from drawing, or at least publishing, more radical conclusions by the weight of religious disapproval. The credit for having propounded the body of theorems based on a contradiction of the parallel postulate as a self-consistent mathematical science, i.e. as a non-Euclidean geometry, belongs to J. Bolyait (1832) and N. I. Lobachevski t (1829), although many of the ideas involved seem to have been already in the possession of C. F. Gauss. § It was not, however, until it had been shown by Beltramill that the hyperbolic plane geometry could be regarded as the geometry of a pseudospherical surface in Euclidean space, that an independence proof (cf. Introduction, Vol. I) for the parallel assump- tion could be said to have been given. The work of Beltrami depends on the investigation by EiemannlF of the differential geometry ideas at the basis of geometry (1854). Kiemann seems to deserve the credit for the discovery of the elliptic geometry (§§ 141-143 below), though it is not clear that he distinguished between the two types of elliptic geometry.**
The proof of the existence of a non-Euclidean geometry was made capable of a simpler form by the discovery of A. Cay ley tt (1859) that a metric geometry can be built up, using a conic as absolute. The relation of Cayley's work to other branches of geometry and the pre- vious studies of non-Euclidean geometry was made plain by F. Klein tt in connection with his elucidation of the r61e of groups in geometry. The representation of the hyperbolic plane by means of the interior
* Euclides ab omni naevo vindicates, Milan, 1733. German translation in "Die Theorie der Parallellinien von Euklid bis auf Gauss," by F. Engel and P. Staeckel, Leipzig, 1896.
t English translation by G. B. Halsted, under the title " The Science Absolute of Space," 4th ed., Austin, Texas, 1896.
t German translation by Engel, under the title " Zwei geometrische Abhand- lungen,"' Leipzig, 1898. Cf. also a translation by Halsted of another work entitled "The Theory of Parallels," Austin, Texas, 1892.
§ \Verke, Vol. VIII, pp. 157-268.
II Saggio di interpretazione della geometria non-euclidea, Giornale di Matema- tiche. Vol. VI (1868), p. 284.
1 English translation by W. K. Clifford, in Nature, Vol. VIII (1878), and in Clifford's "Mathematical Papers" (London, 1882), p. 66.
«*Cf. F. Klein, Autographierte Vorlesungen tiber nicht-euklidische Geometric, Vol. I (Gottingen, 1892), p. 287.
tt Collected Works, Vol. II (Cambridge. 188!»). p. 583.
Jt Mathematische Annalen, Vol. IV (1871), p. 573.
NON-EUCLIDEAN GEOMETRIES [CHAP.VHI
of a circle (§ 134), and the representation of the elliptic plane given in Kx. 12, § 141, are due to R. De Paolis* and H. Poincar&t
For the history of non-Euclidean geometry and an exposition of ]>;uts of it, the reader is referred to R. Bonola, Non-Euclidean Geometry, English translation by H. S. Carslaw, Chicago, 1912. ( Hher texts in English are J. L Coolidge, Non-Euclidean Geometry, Oxford, 1909; Maiming, Non-Euclidean Geometry, Boston, 1901; D. M. Y. Sommerville, The Elements of Non-Euclidean Geometry, London, 1914; H. S. Carslaw, The Elements of Non-Euclidean Plane Geometry and Trigonometry, London, 1916. Besides these we may mention D. M. Y. Sommerville's Bibliography of Non-Euclidean Geometry, London, 1911.
There are numerous other geometries closely related to the non- Euclidean geometries touched on in this chapter. Of particular interest are the geometries associated with Hermitian forms in- vestigated by G. Fubini (Atti del Reale Istituto Veneto, Vol. LXIII (1904), p. 501) and E. Study,* and the geometry of the Physical Theory of Relativity. §
136. Angular measure. The measure of angles may be denned precisely as in the Euclidean geometry, and we carry over the defi- nitions and theorems of § 69 without modification. If we represent the absolute and an arbitrary point 0 by a Euclidean circle C2 and its center, the Euclidean rotations about O are identical with the hyperbolic rotations about O, and hence the two angular measures as determined by the method of § 69 are identical. By § 72, if a and b are two lines intersecting in O, and 0 is the measure of the smallest angle 4. AOB for which A is a point of a and B a point of b,
(I) 0 = -llogR(«M>'2),
where il and iy are the minimal lines through 0. Since il and i2 are the tangents to C'2 through O, it follows that (1) may be taken as the formula for the measure of any ordered pair of lines a, b in the
* Atti della R. Accademia dei Lincei, Ser. 3, Vol. II (1877-1878), p. 31.
t Acta Mathematica, Vol. I (1882), p. 8, and Bulletin de la Soci^te" mathe'matique de France, Vol. XV (1887), p. 203.
I Mathematische Annalen, Vol. LX (1905), p. 321.
§Cf. F.Klein, JahresberichtderDeutschenMathematiker-VerfiiiiuMitii,'. Vol. XIX (1910), p. 281, and the article by Wilson and Lewis referred to in § 48 above.
§§i:«,i:w] HYPERBOLIC CKo.MKTKY 363
hyperbolic plane if il and t's are understood to be the tangents to the absolute through the point of intersection of a and b.
If the hyperbolic plane is represented as in § 134 by the interior of a circle C2, the angular measure of any two hyperbolic lines is identical with the Euclidean measure of the angle (§ 93) between the two circles orthogonal to C'2 which represent them. This has just IKHMI seen for the case where the two circles are lines through the center of C'2. In the general case a point A of intersection of the two circles orthogonal to <72 may be transformed to the center of C3 by a direct circular transformation A. The transformation A as a direct circular transformation leaves Euclidean angular measure invariant (§ 93), and as a displacement of the hyperbolic plane leaves hyper- bolic angular measure invariant. Since the two measures are identical at the center of C'2, they must also be identical at A.
As an application of this result we may prove the following remarkable theorem:
THEOREM 10. The sum of the angles of a triangle is less than TT.
Proof. Let the triangle be ABC, and let the absolute and the point A be represented by a Euclidean circle C*2 and its center. Then the hyperbolic lines AS and A C are represented by Euclidean lines through the center of (72, and the hyperbolic line BC is represented by a circle A'2 through .B and C orthogonal to C2 (fig. 79).
The hyperbolic measures of the angles at A, B, and C respectively are equal to the Euclidean measures of 4. BA C and two angles formed by AB and AC with the tangents to K* at B and C FIG 79
respectively. The sum of these three angles is easily seen to be less than that of the angles of the Euclidean (rectilineal) triangle ABC. Hence it is less than rr.
The theorem that the sum of the angles of a triangle is TT may be substituted for Assumption IX as an assumption of Euclidean geometry;* the proposition just proved can be taken as the corre- sponding assumption of hyperbolic geometry ; and the proposition that the sum of the angles of a triangle is greater than TT can be taken as an assumption for elliptic geometry.
* On the history of this theorem cf. Bonola, loc. cit.. Chap. II. This reference will also be found useful in connection with the exercises.
364 NON-EUCLIDEAN GEOMETRIES
EXERCISES
•1. l'rov«- from Assumptions J-VI^ X-XVI that if the sum of the angles of one triangle is greater than, equal to, or less than v, the corresponding statement also holds for all other triangles.
• 2. Prove from Assumptions I- VI, X-XVI that the sum of the angles of a triangle is less than or equal to IT.
137. Distance. Since the conic section is a self-dual figure, it is to be expected that the formula for the measure of point-pairs is analo- gous to (1). As a matter of fact, we shall only modify the factor - t'/2. If A and B are two ordinary points, let A*, Bx be the points in which the line AB meets the absolute, the notation being assigned so that the points are in the order {A^ABB,,}. Then Tk(AB,AmB«) is positive (§ 24), and hence log B (AB, A*>B<*) has a real value. We define the distance between A and B by means of the equation
(2) Dist (AB) = 7 log B (AB, AnBx),
where 7 is an arbitrary constant and the real determination of the logarithm is taken. It is seen at once that
Dist (AB)= Dist (BA),
because B (AB, A* Bx) = B (BA, Bx AK),
and that if A, B, C are collinear points in the order {ABC},
Dist (AB) + Dist (BC)= Dist (AC), because K(AB, A^B*) . K(BC, A»BX)=K(AC, AXB»).
Moreover, it is evident from the properties of the collineations transforming a conic into itself that a necessary and sufficient con- dition for the congruence of two point-pairs AB, CD is
K(AB, A»B»)= ft (CD, C.D.),
where A*, B«, are chosen as above and C«, Z>, are chosen analogously. Hence a necessary and sufficient condition for the congruence of AB and CD is I)ist(AB) = Diet (CD).
Hence the distance function defined above is fully analogous to that used in Euclidean geometry (§ 67). The constant 7 may be determined by choosing a fixed point-pair OP as the unit of distance. We then have
(3) i
§§ i.".7. i:w; DISTANCE AND ANGLE 365
138. Algebraic formulas for distance and angle. Let us consider the symmetric bilinear form
f(X, A") = aoo*X + a01
and the covariant form F(u, u') =
where the A^s are defined as in § 85. With respect to homogeneous coordinates, f(X, X) = 0 is the equation of a point conic, and F(u, u)=Q of the line conic composed of the tangents to f(X, X) = 0. Let us take this conic as the absolute and derive the formulas for the measure of distance and of angle.
Let Y=(yQ, y^ y^ and Z = (zQ, zlf z2) be two distinct points. The points of the line joining them are
\Y + fiZ = (\y0 + pz9, X^ + M*V *#2 4- M«2),
and the points in which this line meets /( X, X) = 0 are determined by the values of A//* satisfying the equation
0 =/(XF+ fiZ, \Y+ fjiZ) = X2/(F, Y) 4- 2 X/»/( Y, Z) + ^f(Z, Z). These values are
X,= -f(Y, Z) + f(Y, Z)-f(Y, Y)f(Z, Z} ^ f(Y, Y) _ '
\^ -f(Y, Z}-^f(Y, Z)-f(Y, Y)f(Z, Z) ^ f(Y, Y)
Let us denote the two points of the absolute corresponding to (\l} fi^)
and (\2, /*„) by Ir and /2 respectively. Then
Dist(F^) = 7 log R (YZ, /,/,).
Since (X, /A) is (1, 0) for Y and (0, 1) for Z, we have (§ 65, VoL I)
Hence (4)
..
NON-EUCLIDEAN GEOMETRIES [CHAH. vm
l'>\ precisely the same reasoning applied to the dual case we have for tin- MHMSUIV of a pair uf lines u = (MO, w,, uj, V = (VQ, v^ vj.
(5) ni(.f.r) !
i_. (F(u, v) + ^/F*(u, v) -,F(u, u) F(v, ~v) ~2 °g F(u, u)F(v, v)
F(u, v) — ^F'2 (u, v) — F(u, u) F(v, v)
|logtoMiH_
Denoting Dist(y, Z) by d, we obtain and hence
f(Y>Y)f(Z,Z)
and
d rf
, Z)-f(Y, Y)f(Z, Z)
In like manner, if 0 — m (uv),
,e _ F(u, v) + F*(u, v)-F(u, u)F(v, v) e — — —
I F*(u, v) (10) cos# = xl - — — - --
VF(u, u)F(v, v)
(11)
F(u,u)F(v,v)
\F(u, u)F(v, v)-F2(u, v) ^ F(u, u)F(v, v)
For a further discussion of these formulas see Clebsch-Lindemann, Vorlesungen iiber Geometric, Vol. II, Part III, Leipzig, 1891.
*139. Differential of arc. The homogeneous coordinates of all points not on the absolute,
(12) f(X,X) = 0, may be subjected to the relation
(13) f(X,X)=C,
where C is a constant. Since f(X, X] is quadratic, this determines two sets of coordinates (x0, xlt xj for each point of the hyperbolic
§ i-»»] DIFFERENTIAL OF ARC 367
plane instead of an infinity of sets as in unrestricted homogeneous coordinates.*
Some definite determination of the values of each of the homogeneous coordinates is manifestly necessary in order to apply the processes of differen- tial calculus to formulas in homogeneous coordinates. The particular relation / \.\ ', A') = C has the advantage, among others, of not being singular for any point not on the absolute.
Suppose now that (#0, xlt #2) describes a locus determined by the condition that #0, xlt #2 are functions .of a parameter t. Then, in the familiar notation,!
ds__ Dist(A; AT+AA')
('/ A.' " At
2 7 sinh -i- Dist (AT, X+ AX)
= L 27
A.-O A* \ • f(X, A)/(A-f AA', A'-f AA')
by (8). Since f(Y+ Y', Z) =f(Y, Z) +f(Y', Z), this reduces to
T
dt ~
f(X,X) (f(X, X) + 2/(A', AA') + /(AA', AA')) AAT\
= L 472
/(A, AT) (/(AT, AT) + 2/(A', AAT) +/(AA, AA'))
* If (x0, xt, x2) are interpreted as rectangular coordinates in a Euclidean space of three dimensions, f(X, X) = C is the equation of a quadric surface, and we have a correspondence in which each point of the hyperbolic plane corresponds to a pair of points of the quadric surface. By properly choosing f(X, X), this correspond- ence can be reduced to that given in § 134 between the hyperbolic plane and the surface of a sphere.
t We are applying theorems of calculus here on the same basis that we have employed algebraic theorems in other parts of the work.
368 NON-EUCLIDEAN GEOMETRIES [CHAP.VIII
in which — represents ( — a» — r1' -r2)- I" differential notation this dt \dt dt dt/
formula is
, a/2(*. dX)-f(Xt X)f(dX, dX)
*T r(X,X)
By duality we have a corresponding" formula for the differential of
angle,
a F*(u, du) - F(u, u) F(du, du)
(15) dV = — r
F (u, u)
V . ' '
These formulas are independent of the particular determination of our coordinates by means of the relation (13). If we differentiate (13)
weobtain f(X,dX)=0,
so that for this particular determination of coordinates
(16) ds*=-
us now choose the homogeneous .coordinate system so that
and choose C = — 4 y2 so that, for points not on the absolute,
(17) ^ + ^_47^ = _472.
If 7 is real and not zero, we are dealing with hyperbolic geometry, and
(18) dsi=f(dX, dX)
= dx? + dxl - 4 72efo02.
2 x 2 x
If we substitute u = — — > v = - 2—
in the value for ds* given in (18), we obtain (19) ds>-
Regarding u and v as parameters of a surface in a Euclidean space, ( 1 !») gives the linear element of the surface (cf. Eisenhart, Differential Geometry, § 30). This is a surface for which, in the usual notation of differential geometry, E= G and ^=0. The curvature of this surface is constant and equal to — l/472(cf. Clebsch-Lindemann, loc. cit., Vol. II, p. 525). From this it follows that the hyperbolic plane
§§i:«M40] HYPERBOLIC GEOMETRY 369
geometry in the neigh Uu-lu tod of any point is equivalent to the gi-ometry on a portion of a surface of constant negative curvature. If we suhstitute u = XI/XQ and v = #,A'0 in (18), we obtain
dgi= 4 , (4 7'- 0Vtta + 2 uvdudv + (4 7*- u*)dv*
(47* — u*— v*f
This is the form of linear element used by Beltrami in the paper cited above. This form is such that geodesies are given by linear equations in u and v. Hence geodesies of the surface correspond to lines of the hyperbolic plane.
It is to be noted that the curvature of a surface, while often defined in terms of a Euclidean space in which the surface is supposed to be situated, is a function of E, F, and G and therefore an internal property of the surface, i.e. a property stated in terms of curves (u = c and v = c) in the surface and entirely independent of its being situated in a space.
Another remark which may save misunderstanding by a beginner is that the geometries corresponding to real values of 7 are identical. The choice of 7 amounts to a determination of the unit of length, as was shown in § 137.
EXERCISES
1. Express the differential of angle in terms of (z0, xv r2) and their deriva- tives (cf. Clebsch-Lindemann, loc. cit., Vol. II, p. 477).
*2. Develop the theory of areas in the hyperbolic plane. For a treatment by differential geometry cf. Clebsch-Lindemann, loc. cit., p. 489. For a develop- ment by elementary geometry of a theory of areas of polygons which is equally available in hyperbolic, parabolic, and elliptic geometry, see A. Finzel, Mathe- matische Annalen, Vol. LXXII (1912), p. 262.
140. Hyperbolic geometry of three dimensions. A hyperbolic space of three dimensions is the interior (cf. Ex. 2, § 129) of a nouruled quadric surface, called the absolute quadric, and the hyperbolic geom- etry of three dimensions is the set of theorems stating properties of this space which, are not disturbed by the projective collineations leav- ing the quadric invariant. The definitions of the terms " displacement," " congruent," " perpendicular," etc. are obtained by direct generali/a- tion of the definition in § 132 and the corresponding definitions in the chapters 011 Euclidean geometry. They will be taken for granted in what follows, without being formally written down.
370 NON-EUCLIDEAN GEOMETR1KS [CHAI-.VIII
Tlie fundamental theorems on congruence may be obtained from the observations (1) that any displacement of space leaving a plane invariant effects in this plane a displacement or a symmetry in the sense of § 132, and (2) that no two displacements of space leaving a plane invariant effect the same displacement or symmetry in this plane. From this we infer, by reference to § 135,
Tm.oiiKM 11. /// the real three-dimensional hyperbolic geometry Assumptions I-XVI of §§ 29 and 66 are all true except Assump- tion IX, which is false.
By § 100 there is a«imple isomorphism between the displacements of a hyperbolic space and the direct circular transformations of the inversion plane. Hence the theorems of inversion geometry or of the theory of projectivities of complex one-dimensional forms can all be translated into theorems of hyperbolic geometry. The reader who carries this out in detail will find that many of the theorems of Chap. VI assume very interesting forms when carried over into the hyperbolic geometry.
In particular, if an orthogonal line reflection, or half turn, is defined as a line reflection (§ 101) whose directrices are polar with respect to the absolute, it follows at once that every displacement is a product of two orthogonal line reflections. With this basis the theory of dis- placements is very similar to the corresponding theory in Euclidean geometry, but many of the proofs are simpler.
The formulas for distance arid angle are identical with those of § 138, and the differential formulas with those of § 139 if f(X, X'} be understood to be a bilinear form in (#0, xlt xz, x3) and (x'0, x(, x^, #,).
EXERCISES
1. The product of three half turns is a half turn if and only if their three ordinary directrices have a common intersecting perpendicular line.
2. If a simple hexagon be inscribed in the absolute, the common inter- secting perpendicular lines of pairs of opposite edges are met by a common intersecting perj^ndicular line (cf. § 108).
3. I)«'t«-rmine the projectively distinct types of displacements.
* 4. Defining a fmrasphere as a real quadric interior to the absolute and trans- formable into the absolute by means of an elation.whose center is on the absolute and whose plane of fixed points is tangent to the absolute, prove that the hyper- bolic geometry of a horosphere is equivalent to the Euclidean plane gemm-try.
§§140,141] ELLIPTIC GEOMKTKY 371
*5. Classify the quadric surfaces from the }>oint of view of hyperbolic geometry.
*6. (iiven the existence of a hyperbolic space, define a set of ideal }>oints such that the extended space is projective. Cf. R. Bonola, (Jiornale di Mate- matiche, Vol. XXXVIII (1900), p. 105, and F. W. Owens, Transactions of the American Mathematical Society, Vol. XI (1910), p. 140.
* 7. Obtain theorems analogous to those in the exercises of §§ 122, ItM with regard to the hyperbolic displacements.
*8. Study the theory of volumes in hyi»erbolic geometry by methods of differential geometry.
141. Elliptic plane geometry. Definition. The geometry corre- sponding to the group of projective collineations in a real* projective plane TT which leave an imaginary ellipse E* invariant is called the two-dimensional elliptic geometry or elliptic plane geometry. The im- aginary conic E* is called the absolute conic or the absolute. The projective plane TT is sometimes referred to as the elliptic plane.
The order relations in this geometry are of course identical with those of the projective plane (Chap. II). The congruence relations are defined as in § 132, with suitable modifications corresponding to the fact that E* is imaginary. Some of the theorems which run par- allel to the corresponding theorems of hyperbolic geometry are put down in the following list of exercises.
The formula for the measure of angle used in hyperbolic geometry may be taken over without change, i.e.
where l^ and l_2 are intersecting lines and t\ and i'2 are tangents to the absolute in the same flat pencil with ^ and /2. The formula for distance may also be taken from hyperbolic geometry :
In order that this shall give a real value for the distance between two real points, 7 must be a pure imaginary. So we write
ki •
•This geometry can in large part be developed on the basis of Assumptions A, E. S. I' alone, the imaginary conic being replaced by the corresponding elliptic polar system, the existence and properties of which are studied in § 89. As a matter of fact there is considerable interest attached to the elliptic geometry in a modular plane, but the point of view which we are taking in this chapter puts order relations in the foreground.
NON-EUCLIDEAN GEOMETRIES [CH.U -. vm
and in order to have formulas in the simplest possible form, we may
choose k = 1 , so that
rf = ~logR(n>,7i<2,).
The discussion in § 138 is applicable at once to elliptic geometry if /(.V. .V') be taken to be a bilinear form in three variables such that f(X, X) = 0 is the equation of the absolute of elliptic geometry. Thus we have
<22>
EXERCISES
1. The principle of duality holds good in the elliptic geometry.
2. The elliptic geometry is identical with the set of theorems about the geometry of the plane at infinity in three-dimensional Euclidean geometry.
3. The pairs of perpendicular lines at any point are pairs of an elliptic involution.
4. The lines perpendicular to a line / all meet in the pole of I with respect to the absolute. Through any point except the pole of / there is one and but one line perpendicular to /.
5. Defining a ray as a segment whose ends are conjugate with respect to the absolute, prove that Assumption X, § 66, holds in the single elliptic geometry if the restrictions be added that A and B are on the same ray.
6. Assumptions XI and XIII of §66 hold for single elliptic geometry.
7. How may Assumptions XII, XIV, and XV be modified so as to lie valid for single elliptic geometry?
8. A circle is a conic touching the absolute in two conjugate imaginary points.
9. A circle is the locus of a point at a fixed distance from a fixed line.
10. If A, B, C are three collinear points,
Dist (A, B) + Dist (BC) + I>ist(r,l ) = *-.
In other words, the total length of a line is ir.
11. The sum of the angles of a triangle is less than te.
§§i4i,M2] ELLIPTIC GEOMETRY 373
12. Let K' be a circle in a Euclidean plane, and let [C'2] be the set of circles which meet K3 in pairs of j>oiuts on its diameters. An elliptic plain- is determined by denning as "elliptic points" all the Euclidean ]>oints interior t.<> /\"- and all the pairs of Euclidean points in which A'2 is met by its diameters, and denning as collinear any set of elliptic joints on a circle C*.
142. Elliptic geometry of three dimensions. The three-dimensional elliptic geometry is the set of theorems about a three-dimensional projective space which state properties undisturbed by the projective collineations leaving invariant an arbitrary but fixed projective polar system, called the absolute polar system, iu wh'ich no point is on its polar plane. It is a direct generalization of the elliptic geometry of the plane and may be based on a similar set of assumptions.
In a real space this polar system is that of an imaginary quadric (called the absolute quadric) with respect to which each real point has a real polar plane, and the equation of. the absolute quadric may
be taken to be
atf+ajf+aj'+a^O. •
A displacement is defined as a direct* projective collineation (cf. § 32) which leaves the absolute polar system invariant ; a sym- metry is defined as a nondirect projective collineation leaving the absolute polar system invariant. The definitions of congruence, per- pendicularity, distance, etc. follow the pattern of the hyperbolic and parabolic geometries, and the same method may be used, as in those geometries, to extend the theorems on congruence from the plane to space.
It can easily be proved by means of the theorems on the quadric in Chap. VI that any displacement is a product of two line reflections whose axes are polar with regard to the absolute. From this proposi- tion a series of theorems on displacements can be derived, just as in the parabolic and hyperbolic geometries.
Through a given point not on a given line I there is no line parallel to / in the sense in which the term is used in parabolic or hyperbolic geometry. There is, however, a generalization of the Euclidean notion of parallelism to elliptic three-dimensional space which preserves many of the properties of Euclidean parallelism and is, if possible, more interesting.
* Without appealing to order relations, the direct collineations may be charac- terized as those which do not interchange the reguli on the absolute quadric.
374 NON-EUCLIDEAN GEOMETKIKS [CHA.- . vm
Any real line / meets the absolute in two conjugate imaginary points, and through these points there are two lines plt pt of one regulus and two lines qv qt of the other regulus. The lines pf p^ are conjugate imaginary Hues of the second kind (§ 109), and I is one line of an elliptic congruence of which plt pz are directrices. A similar remark applies to the conjugate imaginary lines qlt q^ Any line of the elliptic congruences having plt p2 or qlf qz as directrices is called a Clifford parallel* of I or a paratactic^ of /. Thus there are two Clifford parallels to I through any point not on /, and / is a Clifford parallel to itself.
The two Clifford parallels to any line through any point not on it may be distinguished as follows : Let It* arid R'£ be the two reguli on the absolute. Two real lines I, m meeting two conjugate imaginary lines plt p2 of R? are right-handed Clifford parallels, or paratactics ; and two real lines I', m' meeting two conjugate imaginary lines qlt q2 of R% are left-handed Clifford parallels, or paratactics.
The distinction between right-handed and left-handed Clifford paral- lels may be drawn entirely in terms of real elements by means of the notion of sense-class (§ 32), and thus connected with the intui- tive distinction between right and left. This matter will be taken up again in the next chapter. In the meantime it may be remarked that the definition in terms of the two reguli on the absolute is inde- pendent of all question of order relations and is based on Assumptions A, E, P alone.
From the definition it follows immediately that if 2 is a right-handed Clifford parallel to m, m is a right-handed Clifford parallel to / ; that if m is also a right-handed Clifford parallel to n, I is a right-handed Clifford parallel to n. In general, two lines have one and only one common intersecting perpendicular; but if they are right-handed Clifford parallels, there is a regulus of common intersecting perpen- diculars, and the latter are all left-handed Clifford parallels.
The product of two orthogonal line reflections whose axes are Clifford parallels leaves each line of the congruence of Clifford parallels perpendicular to the axes invariant, and is called a translation. A
•Cf. Clifford, A Preliminary Sketch of Biquaternions, Mathematical Paprrs (I,<im Ion. 1882), p. 181, and Klein, Autographierte Vorlesungen tibernicht-euklidische Geometric, Vol. II (Gottingen, 1892), p. 245.
t K. Study, Jahresbericht der Deutschen Mathematikervereinigung, Vol. XI (1903), p. 819.
§§ 14-.M4S, 144] ELLIPTIC GEOMETRY 375
translation is right-handed or left-handed according as the congruence of its invariant lines is right-handed or left-handed. Any displace- ment can be expressed as a product of two translations.
For a discussion of Clifford parallels and related questions see Appendix II of the book by Bonola referred to above, V. Klein, Mathematische Annalen, Vol. XXXVII (1890), p. 544, and the other references given above in this section.
143. Double elliptic geometry. The geometry corresponding to the group of projective collineatious transforming a sphere .S'2 in a Euclidean three-space into itself is called spherical or double elliptic plane geometry. The sphere S is called the double elliptic plane. The circles in which S2 is met by planes through its center are called lines, and two figures are said to be congruent if conjugate under the group of direct projective collineations transforming the sphere into itself.
The plane which is called elliptic in § 141 is sometimes called single elliptic to distinguish it from the double elliptic plane here described. Since the plane at infinity TT» of a Euclidean space is a single elliptic plane, and since each line through the center of S* meets S2 in two points and TT^ in one point, there is a correspondence between a single elliptic plane and a double elliptic plane, in which each point of the first corresponds tp a pair of points of the latter. By means of this correspondence any result of either geometry can be carried over into the other geometry.
These remarks can all be generalized to w-dimensions. For a set of assumptions for double elliptic geometry as a separate science, see J. R Kline, Annals of Mathematics, 2d Ser., Vol. XIX (1916), p. 31.
144. Euclidean geometry as a limiting case of non-Euclidean. In the two-dimensional case we have seen that the equation of the abso- lute may be taken as
(24) ^+^-47X2=0,
or in line coordinates, as
-•*« -ft
The formulas of hyperbolic geometry arise if 7 is real and not zero, and of elliptic geometry if 7 is imaginary. If we set c = -— ^=0, may be regarded as the equation of the circle at infinity of the
376 N"N KIVUDKAN GEOMETRIES [CHAP.VIH
Euclidean geometry in the form used in § 72. Moreover, if we set c = 0 in the formulas of §§ 138 and 141, we obtain
uv +uv cos 6 = — — l l — 2 * -
and
which agree with the formulas of Euclidean geometry given in § 72. In like manner, if we set c = 0 in the formula for the differential of distance in § 139, we obtain ds* = du* + dv*. The generalization of these remarks to three or n dimensions is of course obvious.
If c changes by continuous variation from a positive to a negative value, it must pass through zero. Since the corresponding geometry is elliptic while c is positive, parabolic when c is zero, and hyperbolic while c is negative, the parabolic geometry is often spoken of as a limiting case both of elliptic and of hyperbolic geometry.
This point of view is reenforced by observing that the formula (10) makes the measure of a fixed angle a continuous function of c, so that fjr a small variation of c the value given by (10) for 0 suffers a correspondingly small variation, A like remark can be made about the distance between a fixed pair of points.
This has the consequence that for a given figure F consisting of a finite number of points and lines, and for a given number e, a num- ber 8 can be found such that if c varies between — 8 and 8, the dis- tance of point-pairs and the angular measure of line-pairs of F do not vary more than €. Nevertheless, in this interval of variation of c the geometry according to which the distances and angles are measured changes from elliptic through parabolic to hyperbolic.
For example, if F were a triangle, and the sum of the angles were found by physical measurement to be between TT + e and TT — e, the geometry according to which the measurements were made might be either parabolic, hyperbolic, or elliptic. Further refinements, of experimental methods might decrease e, but according to current physical doctrine could not reduce it to zero. Hence, while exjx'ii- ment might conceivably prove that the geometry at the bottom of the system of measurements was elliptic or hyperbolic, it could not prove it to be parabolic.
§§144,145] ELLIPTIC GEOMETRY 377
For the details of showing that the Euclidean formula for distance is a limiting case of the non-Euclidean formula, see Clebsch-Lindemann, loc. cit., VoL II, p. 530.
145. Parameter representation of elliptic displacements. Suppose the coordinate system so chosen that the equation of the absolute is
The protective collineations which leave the lines of a regulus on the absolute invariant have been proved to have matrices of the form (50) or (51) in § 126. Let R$ be the regulus on the absolute left in- variant by the transformations of type (50), and R% that left invariant by those of type (51). The transformations of type (50) are the translations leaving systems of right-handed Clifford parallels in- variant, and those of type (51) the translations leaving systems of left-handed Clifford parallels invariant.
Since any transformation leaving the quadric invariant is a product of one leaving the lines of R* invariant by one leaving the lines of R% invariant, any displacement is a product of a transformation of type (50) by one of type (51). Denoting (50) by A and (51) by B, the matrix A of any displacement can be written
£0 ft ft ftV
If A' = B'A' is the matrix of a second displacement, and B1 and A' are of the types (50) and (51) respectively,
(27) A'. A =B'A'BA = B'B- A' A,
because any displacement leaving all lines of R* invariant is com- mutative with any displacement leaving all lines of J?* invariant.
378 NON-EIVLIDKAN GEOMETRIES [CHAP, vm
Thus any displacement
x = alox0 + anXl + auxt + alaxt,
is given parametrically in terms of two sets of homogeneous parameters «o, a,, at, as and £0, J3V £2, fts by means of the formulas obtained by equating «0- to the corresponding element of the last matrix in Equation (26).
The formulas for the parameters of the product of two displace- ments are determined by (27), for if A" = B"A" = A' A, then B"=B'B and A" = A' A, and hence
a" = «X + a|a0 4- aX - « -
The formulas for the a's are, by § 127, the same as for the multi- plication of quaternions, and the formulas for the yS's are given by the following quaternion formula:
Now let \t and X2 be two symbols defined by the multiplication table (31)
|
\ \ |
|
|
\ \ |
\ o o \2 |
and the conditions \^ = q\lt \zq = q\z, where q is any quaternion. If we write
(32) [\X + «,'i + *'J + a'Jc
the a'"s and /S'^s are given in terms of the a's, /S's, a"s, and /3"s by tlm equations (29) and (30).
§14.-,]
ELLIPTIC <;I-:<».MKTI:Y
379
The number system whose elements are \<y,+ \?2» where q^ and q are quaternions, is one of the systems of hiquaternions referred to in the footnote of § 130. It is often given a form which may be derived as follows :
(33)
Then e^ and e3 obey the multiplication table
(34) ~e
and we have 2(\1?1
>~ft*~fty~
+ (°^A}i+(a^^}J +
pp)
ao "o _j_ / i """ "i \ » _L I a "* "a \ „• ^ I t
*\ 2
Let us write
(35)
7! =
222
«5_aa + & »_<
o 2 J 2 2 2 2
The rule for multiplying biquaternions,
gives the following equations : 7i'= 7^7o- 7(7, - 7&-
^= 7^72- 7(7.+ 7^70+
' = 7& - 7,'*. + 7.;
NMN Kl ( LIDKAN GEOMETKIKS [CHAP.VIII
Tin1 7~s and 8's given by (36) may be regarded as a new set of I'aruiiH'U-rs for the elliptic displacements. Since the a's and y3's are separate sets of homogeneous variables, they may be subjected to the relation
(37) < + «« + a* + a* = ft + ft + ft + ft.
I'.y means of (35) the relation (37) becomes
(38) 7«A+7A+7A+73s3=o.
The formulas for the coefficients of a displacement (28) in terms of the new parameters are found by substituting
in the formulas for «0- in terms of the as and /S's. In other words, the matrix of the displacement corresponding to (70, 7^ y2, 78 ; 80, $lt S2, S3) is
'70 -7j -72 -7,
-78 72|.
r/T."T,
7l 7° IK 73
\78 -7,
S Vl 3 X2 o"5i/J
o o o o
72 7,' / o. 6. o. 8
10 1 2 _
1032 _82 g3 go_si
0
7o 73 72
7ry ry — ry
2 '3 '0 '1
^-78-72 7x 70/ and the formulas for the composition of two displacements are (36).
EXERCISE
The elliptic displacements are orthogonal transformations in four homo- geneous variables. Work out the parameter representation determined by
the formula „ /i cwi _L c\-i
K = (l— .S )(!+>)
of § 125.
146. Parameter representation of hyperbolic displacements. Let the equation of the absolute be taken in the form (39) < + /*2 (x* + xl + x-) = 0.
If /i is real, the corresponding geometry is elliptic ; and if ft is a pure imaginary, the corresponding geometry is hyperbolic. No gener- ality is lost by taking /u, = 1 (as in the section above) for the elliptic case and /* = V— 1 in the hyperbolic case. For the sake of the limit- ing process referred to at the end of the section, we shall, however, carry out the discussion for an arbitrary /i.
§ 14t5]
HYPERBOLIC GEOMETRY
By precisely the reasoning used in § 126 it is seen that any colline- ation leaving one regulus on the absolute invariant has the matrix
1
OL — «„ '
1 a.
A= -a a
a 2 8 °
1
x ^ a« a* ai
and any collineation leaving the other regulus invariant has the matrix
1
1
/i 2 3 1
fJL 8 2 '
Hence any displacement has a matrix .ZL4. In other words, if
(40)
the transformation (28) is a displacement.
882
NON-EUCLIDEAN GEOMETRIES
[CHAP.VIH
As we have already seen in the elliptic case, if A' and B' are matrices analogous to A and B,
B'A' >BA=B'B.A'A.
Hence the product of two displacements BA and B'A' is a displace- ment B"A" such that
and
A ' = A A B" = B'B.
On multiplying out the two matrix products A' A and B'B, it is evident that the elements of A" and B" are given by the formulas (29) and (30) found above for the elliptic case. These formulas are associated with the biquaternions determined by the table (31).
The remark must now be made that if n =V— 1, the parameter representation above does not give real values of afj for real values of the a's and /S's. Suppose, however, that we transform the biquater- nions \lql + \qn as follows :
\\ > _t*€i-€*
\)> \--~2jr
Then cl and e., obey the multiplication table
(41)
e
and we have 2 p (\qt + \q2) = (/*et 4- e2) q^ + (pe^ — e0) qn
or
^^-f- «,» + «
where
(42)
-I- 7j 4-
(So + 8^'
_
78—
§146] HYPERBOLIC GEOMETRY 383
The rule for multiplying biquaternions
= (X(7o + 7,'* + 7-L/ +7s*) + «,(8J + 8ft + *'J + W]
• [«i(70 + 7,t + 7,7 + 7/) + M*. + Si* + V + V) according to (41), gives the following equations:
7?= 7^70- 7(7,- 7^7,- 7^73 + /*'(8JS0- 81'81'- 8J8,- W,
? = 7^7,- 7173+ 7^7,+ 7{7i i'= 7^73 + 7(72- 7,'
7.'*,
For p = 0 these equations reduce to (64) and (65) of § 130, and for /i2=l they reduce to (36). For /x2= — 1 they give the standard formulas for combining hyperbolic displacements. Thus there are three essentially distinct systems of biquaternions, determined respec- tively by the conditions /*2= 1, /t*2= — 1, p = 0. The first corresponds to the elliptic, the second to the hyperbolic, and the third to the parabolic geometry. The geometry in each case is determined by an absolute whose equation in point coordinates is (39), and in plane coordinates,
(44) /*X + ui + < + < = 0.
Since the same geometry corresponds to any two real values of /*, there must be a simple isomorphism between any two systems of biquaternions corresponding to positive values of /*2 ; and a like state- ment holds with regard to the systems of biquaternions corresponding to negative values of /*2. The biquaternions for which /* = 0 may be regarded as a limiting case between those for which ft2 is positive and those for which ^ is negative, just as the parabolic geometry is regarded as a limiting case between the hyperbolic and elliptic (§ 144).
In these remarks it is understood that the coefficients yn, ylt 7,, 7^ 80» ^i> 82, 88 are always real. From the geometrical discussion above it is clear that if these coefficients were taken as complex, the
:5.M NON-EUCLIDEAN GEOMETRIES [CHAP.VIII
hiquaternions for which /r=l would be isomorphic with those for which /*" = —!.
The inultijilication table (41), in case i? = — 1, is satisfied if we take *,= 1 and es = V^T. Hence the biquaternions with real coefficients,
* i7.+ 7,* + 7a +
are equivalent, in case /*2= — 1, to the quaternions with ordinary complex coefficients,
The biquaternions for which /* = 0, when taken with complex coefficients, may be regarded as a number system of sixteen units with real coefficients. This is the number system (§ 130) which is needed to study the displacements in the complex Euclidean geometry, and it may be regarded as containing the other systems of real biquaternions.
CHAPTER IX
THEOREMS ON SENSE AND SEPARATION
147. Plan of the chapter. The theorems and definitions of Chapter II are for the most part special cases of more general concepts of Analysis Situs. The present chapter develops these ideas further, so that the two chapters together lay the founda- tion for the class of theorems which are particularly of use in the application of geometry to analysis, and vice versa.
In most of the chapter attention is confined to theorems which can be proved without the use of the continuity assumptions (C, R). Many of the theorems are proved on the basis of A, E, S alone and others on the basis of A, E, S, P.
In the first sections (§§ 148-153) of this chapter we prove some of the general theorems about convex regions. These are followed (§§ 154-157) by the definitions of some very general concepts, such as curve, region, continuous group, etc. It will not be necessary (or possible in the remaining pages) to develop the corresponding gen- eral theory to any considerable extent. Nevertheless, these general notions underlie and give unity to the rest of the chapter, which may in fact be regarded as a study of certain continuous families of figures by special methods.
In §§ 158-181 the theory of sense-classes is developed in consid- erable detail for the various cases considered in earlier chapters and for other cases, the principal idea involved being that of an ele- mentary transformation. Finally (§§ 182-199), we prove the funda- mental theorems on the regions determined in a plane by polygons and in space by polyhedra.
148. Convex regions. THEOREM 1. If I is a line coplanar with a triangular region R and containing a point of R, the points of R on I constitute a segment.
Proof. A line coplanar with a triangle and not containing more than one vertex meets the sides of the triangle in at least two and at most three points. These points, by § 22, are the ends of two or
au
386 THKoRKMS <>N SENSE AND SEPARATION [CHAP, ix
thive segments. By Theorem 20, Chap. II, the points of any one of these segments are in the same one of the triangular regions deter- mined by the triangle, and two points in different segments ;nc in different triangular regions.
COROLLARY. The points common to a tetrahedral region and a line containing one of its points constitute a linear segment.
Proof. A line not on one of the planes of a tetrahedron meets these, planes in at least two and at most four points. The p-.st of the argument is the same as for the theorem above, replacing Theorem 20, Chap. II, by Theorem 21 of the same chapter.
Convex regions on a line have been defined and studied in § 23.
DEFINITION. A set of points in a plane is said to be a two-dimen- sional (or planar) convex region if and only if it satisfies the follow- ing conditions: (1) Any two points of the set are joined by an interval consisting entirely of points of the set, (2) every point of the set is interior to a triangular region containing no point not in the set, and (3) there is at least one line coplanar with and not containing any point of the set.
A triangular region, a Euclidean plane, and the interior of a conic are examples of planar convex regions.
THEOREM 2. If I is a line coplanar with a two-dimensional convex region R and containing a point of R, the points of R on I constitute a linear convex region.
Proof. The definition of a linear convex region is given in § 23. That the points of R on I satisfy (1) of that definition follows directly from (1) of the definition of a planar convex region. To prove (2) that any point P of R on I is interior to a segment of points of R on /, we observe that by (2) of the definition of a planar convex region P is interior to a triangular region consisting entirely of points of R and that by Theorem 1 the points common to / and this triangular region are a linear segment. Condition (3) of the definition of a linear convex region is satisfied by the points of R on I because / contains one point of the line coplanar with R and not containing any point of R.
DEFINITION. A set of points in space is said to be a three-dimen- sional (or spatial) convex region if and only if it satisfies the following conditions: (1) Any two points of the set are joined by an interval
5148] CoNVKX REGIONS 3*7
consisting entirely of points of the set, (2) every point of the set is interior to a tetrahedral region containing no points not in the set, and (3) there is at least one plane containing no point of the set.
A tetrahedral region, a Euclidean space, and a hyperbolic space are examples of three-dimensional convex regions.
THEOREM 3. If a line I contains a point of a three-dimensional convex region R, the points of R on I constitute a linear convex region.
The proof of this theorem follows the same lines as that of Theo- rem 2, the corollary of Theorem 1 being used instead of Theorem 1 in showing that the points of / in R satisfy Condition (2) of the definition of linear convex region.
In consequence of Theorems 2 and 3 the definitions {between, precede, ray, sense, etc.) and theorems of § 23 are applicable to col- linear sets of points in two- and three-dimensional convex regions. In the rest of this chapter the segment AB where A and B are in a given convex region R always means the segment AB of points of R.
THEOREM 4. If ABC are three noncollinear points of a convex region R, I) a point of R in the order {BCD}, and E a point of R in the order {CEA}, there exists a point F of R in the orders {AFB} and {DEF}.
Proof. Let F be defined as the point of intersection of the lines DE and AB (fig. 77, p. 351). By (3) of the definition of a two- or three-dimensional convex region there is a line /« coplanar with A, B, and C and containing no point of R. Hence /. does not meet any of the segments AB, BC, CA. Hence (Theorem 19, Chap. II) the line DE which meets the segment CA and does not meet BC must meet AB. Hence {AFB}.
The line L does not meet any of the segments FB, BD, DF, and the line A C meets the segment BD and does not meet the segment BF. Hence AC meets the segment DF. Hence {DEF}.
THEOREM 5. A three-dimensional convex region R satisfies As- sumptions I-VIII of the set given for a Euclidean space in § 29.
Proof. Assumptions I, II, III, V, VIII are direct consequences of Theorem 3 and the theorems of § 23. Assumptions VI and VII are consequences of Condition (2) of the definition of a three-dimensional convex region. Assumption IV is a consequence of Theorem 4.
J5SS THEOREMS ON SENSE AND SEPARATION [CHAP. i\
The theory of order relations hi convex regions can be Kasc.l rn- tirely on Theorem 5. This amounts to developing the consequences of Assumptions I- VIII of § 29. Since both the Euclidean and the hyperbolic spaces satisfy these assumptions, this method of treating CUM vex regions is of considerable interest from the point of view i.f foundations of geometry (cf. references in § 29). The methods required to prove the theorems on this basis are but little different from those used in the next section.
COROLLARY. In a real protective space a convex region also satisfies Assumption XVII of § 29.
EXERCISES
1. The set of all points common to a set of convex regions which are all contained in a single convex region is, if existent, a convex region. (In other words, the logical product of a set of convex regions contained in a convex region is a convex region.)
2. Prove on the basis of Assumptions I-VIII of § 29 that for any set of points Pv 7*2, • • •, PH, finite in number, there is a line / such that Pv Pv ••-,/*„ are all on the same side of /.
*3. A set of points in a projective space such that any two points of the set are joined by one and only one segment consisting entirely of points of the set and such that every point of the set is interior to at least one tetrahedral region consisting entirely of points of the set, is a convex region.
*4. Study the set of assumptions for projective geometry consisting of A, E and the assumption that in the projective space there is a set of points satisfying the Assumptions I-VIII, XVII for a convex region.
149. Further theorems on convex regions. THEOREM 6. If A, B, C are three noncollinear points of a convex region R, they are tie vertices of one and only one triangular region consisting entirely of points of R. This triangular region consists of all points on the segments joining A to the points of the segment BC.
Proof. By Theorem 4 a line joining B to a point of the segment CA meets a segment joining A to any point Al of the segment BC\ and by the same theorem any point of the segment A Al is joined to B by a line meeting the segment CA. Hence the set of points [P] on the segments joining A to the points of the segment BC is identical with the set of points of intersection of lines joining A to points of the segment BC with lines joining B to points of the segment CA. By similar reasoning [P] is the set of points of inter- section of lines joining A to points of the segment BC with lines
§ws>] <'ONYK\ KI:<;H>NS 389
joining C to points of the segment All. The points [P] form a triangular region because they are all the points not .separated i'mm a particular P by any pair of the three liiu-s . | /;, HC, CA.
The other three triangular regions having .1. /•', C as vertices contain points of the line which by (3) of the definition of a convex region is coplanar with ABC and contains no point of R. Hence [P] is the only triangular region satisfying the conditions of the theorem.
In the rest of this section the triangular region determined by three noncollinear points A, It, C of a convex region R according to Theorem 6 shall be called the triangular region ABC. It is also called the interior of the triangle ABC.
COROLLARY. If ABCD are four noncoplanar points of a convex region R, they are the vertices of one and only one tetrahedral region consisting entirely of points of R. Tit is tetrahedral region consists of the segments of points of R joining A to points of the triangular region BCD.
Proof. Let [a] be the set of segments joining A to points of the triangular region BCD and [P] the set of all points on the segments [«]. Any P is also on a segment joining B to a point of the tri- angular region ACD, as is seen by applying the theorem above to the figure obtained by taking a section of the tetrahedron ABCD by the plane ABP. In like manner any P -is on a segment joining C to a point of the triangular region DAB, and on a segment join- ing D to a point of the triangular region A /if.
The same argument shows that any point of intersection of a line joining A to a point of the triangular region BCD with a line joining B to a point of the triangular region CAD is in the set [P] and that every P is a point of this description. From this it follows that [/'] contains all points not separated from a particular P by the faces of the tetrahedron ABCD. Hence by Theorem 21, Chap. II, [P] is a tetrahedral region.
Any tetrahedral region having ABCD as vertices and distinct from [P] contains points not in R, because- it either contains points on the segments complementary to [a] or on the lines joining A to the points of the triangular regions different from BCD in the plane BCD.
390 THKnliKMS (>N SKNSE AND SEPARATION [CHAI-. ix
THK«'|:K.M 7. If a plane tr contains a point of a three-dimensional convex region R, the points of R on TT cunxtiluh- « j>i ncex
r> 1 1 tun.
Proof. The points of R on TT satisfy Conditions (1) and (3) of the definition of a planar convex region because R satisfies Con- ditions (1) and (3) of the definition of a three-dimensional convex region. To prove that the points of R on TT satisfy (2) of the defini- tion of a planar convex region, let P be a point of R on TT and / a line on P and TT. By Theorem 3 there are two points A, Al of R on I such that the segment APAl is composed entirely of points of R. Let a be a line on Al and TT but distinct from /. By the same reasoning as before there are two points £, C of R on a such that the segment BAfi is composed entirely of points of R. By Theorem 6 the triangular region having A, B, C as vertices and containing P contains no points not in R. Hence the points of R on TT satisfy Condition (2) of the definition of a planar convex region.
THEOREM 8. If I is any line coplanar with and containing a point of a planar convex region R, the points of R not on I con- stitute two convex regions such that the segment joining any point of one to any point of the other meets tht linear convex region which I has in common with R.
Proof. By definition there is a line m coplanar with R and con- taining no point of R. By Theorem 18, Cor. 1, Chap. II, all points of the plane not on / or m fall into two classes [0] and [P] such that (1) two points 0, P of different classes are separated by I and m and (2) two points of the same class are not separated by / and m. The region R contains points of both of these classes. For let / be any point of R on I. By Theorem 2 any line through / coplanar with R and distinct from / contains a segment of points of R of which /is one point. If A and B are two points of this segment in the order {AIB}, A and B are separated by / and m and also are points of R. Hence there exist two mutually exclusive classes [O1] and [/»'], subsets of [0] and [P] respectively, which contain all points of R not on /.
Since any ()' and any P1 are separated by I and m and no segment 0?P' contains a point of m, every segment O'P' contains a point of I.
§14'.'] CONVEX REGIONS
Since two points of the same class ([(/] or [/'']) are not separated l>\ / and m, and since the segment joining them does not contain ;i point of in, it does not contain a point of /.
It remains to show that any point of either of the classes, say [0'], is interior to a triangular region consisting entirely of points of this class. Let p be any line on a point (X and coplanar with R. Let O[ and 0'2 be two points of R on p in the order {()[(?(%} and such that the segment 0[O!2 does not contain a point of I. Let q be any line distinct from p, coplanar with R and on 0'.2, and let O's, 0[ be two points of R on q in the order {0'Z0'.20(} and such that the segment O'a0'4 does not contain a point of I. By Theorem 6 there is a unique triangular region with 0[, 0'8, O't as vertices consisting only of points of R and containing all points of the segment 0' 0 . Since I does not meet any of the segments 0(0'2, 0!20'9, 0'20'4, it can- not meet any segments joining 0{ to a point of the segment 0'aO't (Theorem 4). Hence the triangular region 0[O'30't consists entirely of points of [0'].
COROLLARY 1. If TT is any plane containing a point of a three- dimensional convex region R, the points of R not on TT constitute two three-dimensional convex regions such that the segment joining any point of one to any point of the other meets the planar convex region which TT has in common with R.
Proof. The proof is a strict generalization of that of the theorem above to space, using the corollary of Theorem 6 instead of Theorem 6.
COROLLARY 2. For a given line I (or plane TT) and a given convex region R, there is only one pair of regions of the sort described in Theorem 8 (or Cor. 1).
Proof. If 0 is any point of R not on /, the class containing () must include all points joined to O by segments not meeting /. Hence it must be identical with one of the classes given by the theorem.
FIG. 80
392 THKOKKMS OX SKNSH AND SKI'A KATloN [CHAP.I*
DKFINITION. The two convex regions determine 1 aee.inling to Theorem 8 by a line in a planar convex region are called the ,s<VA-.s- of the line relative to the convex region. The two convex regions determined aeeonling to Cor. 1 by a plane in a convex region are called the two * /»/»-.•.• of the plane relative to the convex region.
DEFINITION. Two seta of points [/'], [Q] in a convex region or in a projective plane or space are said to be separated by a set [»s'] if ••MTV segment of the convex region or of the projective plane or space which joins a P to a Q contains an X
EXERCISE
Given two lines containing points <>f a convex region Imt intersecting in a point P outside the region. Construct the lint- joining I1 to a point ({ in tin- region by means of linear constructions involving only points and lines in the region. Cf. Ex. 4, § 20, Vol. I.
150. Boundary of a convex region. DEFINITION. A point />' is a boundary point of a set of points [P] if every tetrahedral region containing B contains a point P and a point not in [/•*]. The set of all boundary points of [P] is called the boundary of [/'].
THEOREM 9. All boundary points of a set of point* on, n I in<- I are on I. All boundary points of a set of points on a plane TT are on TT.
Proof. If Q is a point not on a line /, any tetrahedron one of whose faces contains / and none of whose faces contains Q will determine a tetrahedral region (§26) which contains Q and does not contain any point of /. Hence Q is not a boundary point of any set of points on L A like argument proves the second statement in the theorem.
COROLLARY 1. A boundary point B of a set of points [P] on " line I is any point such that any segment of I containing B contains a P and a point not in [P].
COROLLARY 2. A boundary point B of a set of points [P] on. a plane TT is any point such that any trio nijnla r mi Ion of TT contain- in^ B contains a P and a point not in [/'].
THEOREM 10. Let a- be the convex region common to a line I and a planar convex region R and let R( and R.( be the convex nyionx formed by the points of R which are not on <r. The boundaries of Rt and of RZ contain <r and all boundary point* of a. Kuril, boundary
§iso] coNVKX REGIONS :Jl«:i
f ml nt of R is a boundary point of Rl or of RJf and each lnxuidary point of RI or of R2 which is not on I is a boundary point of R.
Proof. If Q is any point of <r, and ni a line on Q coplanar with Rj and distinct from I, any segment of m containing Q contains points both of RI and of R2. Since any triangular region contain- ing Q contains a segment of m containing Q, it contains points both of Rl and of R0. Hence Q is a boundary point both of Rt and of R2. If /? is a boundary point of <r, any triangular region containing B contains a point Q of <r, and hence, by the argument just given, contains points both of RI and of R2. Hence B is a boundary point both of RI and of Rg.
Let A be a boundary point of R. Any triangular region T con- taining A contains at least one point not in Rt or R2, namely, J itself. Since A is a boundary point of R, T contains at least one point of R, which may be in Rl or in R2 or in tr. In the latter case T contains points of RI and R2 both, by the paragraph above. Hence in every case T contains points of RI or R2. If every trian- gular region containing A contains points of Rt and of R2, A is a boundary point of both RI and R2. If this does not happen, some triangular region Ty containing A contains points of one of Rt and R2 (say Rj) and not of the other. Any triangular region T contain- ing A then contains points of RI because by an easy construction we obtain a triangular region T' containing A and contained in both T and TQ; and since T' contains A, it contains points of R, which because they are in TQ must be points of Rr Hence A is a boundary point of R^
Let C be a boundary point of RI which is not on /. Any trian- gular region T containing C contains points of R, because it contains points of Rj. It also contains points not in Rt. One of these points is not in R unless T consists entirely of points of Rt, R2, and /. If the latter case should arise, since C is not on / a triangular region T' could be constructed containing C, interior to T, and not containing any point of I. T' then would contain jxunts of both RI and R2 and hence would contain a segment joining a point of RI to a point of R2; which segment, by Theorem 8, would contain a point of I, contrary to hypothesis. Hence T contains points not in R, and C is a boundary point of R.
894 THKOKKMS ON SENSE AND SEPARATION [CH.M-.IX
COROLLARY. Let <r be the convex region common to a plane TT n thi-cf-iliint-nsional convex region R, and let RI and- R0 be the conn c rtffiuitx far in I'd l>i/ the points of R n-hich are not on IT. The boundaries of R and R., contain <r and all boundary points of <r. Each boundary fK'iitt of R /.-»• " l>ou Hilary point of Rj or of R2, and each boundary point of Rj or of RS which is not on TT is a boundary point of R.
It is to be noted that we have not proved that a convex region always has a boundary. Cf. Ex. 7, below.
EXERCISES
1. If .-1 and B are two points of the boundary of a convex region R, one of the segments joining them consists entirely of points of R or entirely of points of the boundary of R.
2. A line has no points, one point, two points, or one interval in common with the boundary of a convex region.
3. If a segment consists of boundary points of a given set, its ends are also boundary points.
4. Using the notation of Theorem 10, no point of / not in a- or its bound- ary can be a boundary point of R. Hence if P is a point of a two- or three- dimensional convex region R, and B a boundary point of R, the points P and B are joined by a segment consisting entirely of points of R.
5. Using the notation of the corollary of Theorem 10, no point of ir not in <r or its boundary can be a boundary point of R.
6. Using the notation of Theorem 10, if R and its boundary are contained in another convex region R', then no point of the boundary of Rt not on <r or its boundary can be on the boundary of R2.
7. Give an example of a space containing a convex region which has no boundary.
8. A ray whose origin is in the interior of a triangle meets the boundary of this triangular region in one and only one point.
*9. Let O be an arbitrary point of a Euclidean plane, and R0 an arbitrary convex region containing 0 and having a boundary which is met in two points by every line which contains a point of R0. Let any set of points into which the boundary of R0 can be transformed by a homothetic transforma- tion (§ 47) be called a circle. Let the point to which O is transformed by the homothetic transformation which carries the boundary of R0 into any circle be called the center of this circle. Let two point-pairs AB and A'B' be said to l>e congruent if and only if there is a circle with .1 ;is center ami |>;is*in;j through B which can be carried by a translation into one with A' as center and passing through B'. The geometry based on these definitions is analo- gous to the Euclidean plane geometry. Develop its main theorems. Cf. the memoir of H. Minkowski by D. Hilbert, Mathematische Annalen, Vol. LX V 1 1 1 (1910), p. 445.
§§ i.-*>, 151] TRIANGULAR REdloNS 3!»">
151. Triangular regions. The theorems of the last sections can be used to complete the discussion of the regions determined by a triangle. We shall continue to use the notation of §26 and shall denote the sides AB, BC, CA by c, a, and b respectively. The points of the plane which are not on a form a convex region, of which a is the boundary. By Theorem 8 the points not on a or b fall into two convex regions, of each of which a and b together (by Theorem 10) constitute the boundary. The line c meets a and b in the points B and J respectively and hence has the segment 7 in common with one of the regions and 7 in common with the other. By Theorem 8 the region containing 7 is separated into two convex regions, each having 7 on its boundary, and the other into two, each having 7 on its boundary. Thus the three lines a, b, c determine four planar convex regions which are identical with the four triangular regions of Theorem 20, Chap. II. Since the lines enter symmetrically, each of the segments a, ft, 7, a, ft, 7 is on the boundary of two and only two of the triangular regions.
The three vertices A, B, C are on the boundaries of all four tri- angular regions, because every point of the plane can be joined to these three points by segments not meeting the lines a, b, c. No point not on a, b, or c can be a boundary point of any of the triangular regions, because such a point is an interior point of one of them.
Since any line m which meets one of the four planar convex regions meets it in a segment the ends of which are the only points of m on the boundary, the three segments which bound one of the four triangular regions cannot be met by the same line. The boundaries of the four regions therefore consist respectively (cf. fig. 16) of the vertices of the triangle, together with
a, ft, 7 for Region I, a, ft, 7 for Region II, a, ft, 7 for Region III,
_
a, ft, 7 for Region IV.
In addition to what has already been stated in Theorem 2, the discussion above gives us the following information :
THEOREM 11. A triangular region /.%• h»t/,n/>'tl In/ the three vertices of the triangle, together with three segments joining them which cannot all be met by a linn.
896
I l!K<>i;i:MS ON SENSE AND SEPARATION [CHAP.IX
li AOB and APC are two noncol linear segments they may be denoted by a and ft. The two segments whose ends are B and C may U- »U'imted by 7 and 7, 7 being the one met by the line OP. As we have just seen, a, ft, and 7, together with the vertices of the triangle, are the boundary of a convex region, and there is one and only one of the four convex regions of whose boundary a ami ft form part. Hence
THKOKEM 12. For any two noncollinear segments a, ft having a «> /union end tliere is a unique triangular region and a unique seg- n« nt 7 such that a, ft, and 7, together with the ends of a and ft, form tht boundary of the triangular region.
COROLLARY 1. On any point coplanar with but not in a given triangular region T, there is at least one line composed entirely of points not in T.
COROLLARY 2. The triangular region determined according to Theorem 12 by two noncollinear segments CB'A and CA'B consists of tJie points of intersection of the lines joining B to the points of the first segment with the lines joining A to the points of the second segment.
The complete set of relations among the points, segments, and tri- angular regions determined by three noncollinear parts A, B, C may be indicated by the following tables,
|
a |
<r |
ft |
ft |
7 |
7 |
|
|
A |
0 |
0 |
1 |
1 |
\ |
1 |
|
B |
1 |
1 |
0 |
0 |
i |
1 |
|
C |
1 |
1 |
1 |
1 |
0 |
0 |
H2:
|
I |
II |
III |
IV |
|
|
a |
0 |
1 |
II |
1 |
|
a |
1 |
0 1 |
1 |
0 |
|
ft |
0 |
1 |
0 |
|
|
ft |
1 |
0 |
0 |
1 |
|
7 |
0 |
0 |
1 |
1 |
|
7 |
1 |
1 |
0 |
0 |
where in the first table a "1" or a "0" is placed in the zth row ;md yth column according as the point whose name appears at the be- ginning of the iih row is or is not an end of the segment whose name appears at the top of the jih column ; and where in the second table a " 1" or a "0" is placed in the iih row andytb column according as the segment whose name appears at the beginning of the iih row is or is not a part of the boundary of the ' triangular region whose name appears at the top of the /th column.
§§i:.i,i62l TRIHEDRAL REGIONS 397
EXERCISES
1. The lines polar (§ 18, Vol. I) with respect to a triangle ABC to the points of one of the four triangular regions determined l>\ AJH' constitute one of the four sets of lines determined by ABC, according t<> tlir dual of Theorem 20, Chap. II. The points on these lines constitute the set of all joints coplanar with but not on the giwn triangular region or its boundary.
2. Divide the lines of the plane of a complete quadrangle into classes according as the point pairs in which they meet the pairs of opjKJsite sides separate one another or not. Apply the results to the problem : When can a real conic be drawn through four given points and tangent to a given line ? Dualize.
152. The tetrahedron. The discussion in § 151 generalizes at once to space. Let us use the notation of § 26. The points not on al con- stitute a convex region of which a1 is the boundary. By Theorem 8, Cor. 1, the points not on at and «2 constitute two convex regions, of each of which, by Theorem 10, at and a, form the boundary.
The plane aa has points in each of the three-dimensional convex regions bounded by «1 and «2 and hence by Theorem 7 has a planar convex region in common with each of them. By Theorem 8, Cor. 1, each of these planar convex regions separates the spatial convex re- gion in which it lies into two spatial convex regions, of each of which (Theorem 10, Cor.) it forms part of the boundary. Thus the points not on alt a,, aa form four spatial convex regions. Since any plane not on At meets alf a^ and «3 in a triangle, it meets each of these four spatial convex regions in a triangular region. Thus, since the planes al? «2, ag enter symmetrically, we have
THEOREM 13. DEFINITION. Three planes alt «2, aa meet by pairs in three lines, and each pair of these lines bounds two planar convex regions. The points not on a^ «2, and ag form four spatial convex regions (called trihedral regions) each bounded by the three lines and three of the planar convex regions. The relations among these regions are fully represented by the matrices of § 151 if the three lines are denoted by A, By C, the planar convex regions by a, ft, 7, a, ft, 7, and the three-dimensional regions by I, II, III, IV.
Each of the four spatial convex regions determined by a^, «a, at is met by «4 in a triangular region and separated by it into two convex regions each of which is partially bounded by the triangular region. Hence the points not on at, «s, at, at form eight convex
THEOREMS ON SENSE AND SEPARATION [CHAP.IX
spatial regions which must be identical with the tetrahedral regions of Theorem 21, Chap. II. Since the planes alt a_2, a3, at enter sym- metrically, there are sixteen triangular regions each of which is on the boundary of two and only two three-dimensional regions; :md, moreover, each tetrahedral region has one and only one triangular region from each of the four planes on its boundary.
Since any point not on a , «2, a3, at can be joined to any of the points Alt Az, A3, A4 by a segment not containing any point of
ai» as» as» or af ^ie P°mts -^i» A» *^8» A are on ^ne boundary of all eight tetrahedral regions ; and by similar reasoning each segment which bounds a triangular region also bounds each of the tetrahedral regions bounded by the triangular region.
THEOREM 14. The boundary of a tetrahedral region consists of its four vertices, together with four triangular regions and the six seg- ments bounding the four triangular regions and bounded by the four vertices.
COROLLARY. Three noncoplanar segments having a common end are on the boundary of one and only one of the tetrahedral regions having their ends as vertices.
The complete set of relations among the points, segments, triangular regions, and tetrahedral regions determined by Alt A2, Aa, At may be indicated by three matrices analogous to those employed in § 151. That the points At and Aj are ends of the segments a^ and <F0. is indicated in the first matrix, a "1" in the *th row and yth column signifying that the point whose name appears at the beginning of the t'th row is an end of the segment whose name appears at the top of the yth column, and a " 0 " signifying that it is not.
HI:
|
^12 |
*!« |
°"1S |
^8 |
Ou |
ff!4 |
^28 |
^28 |
<*u |
°v. |
0-84 |
"34 |
|
|
fl |
1 |
1 |
1 |
1 |
i |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
A, |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
|
-4, |
0 |
II |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
|
A* |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
The four triangular regions in the plane «,.(* = !, 2, 3, 4) determined by the lines in which the • other three planes meet a, may be denoted by riv riv r.3, riv Applying the results of
§ is-']
TETKAHEDKAL REGIONS
;5 1~>1 to each plane we have the following matrix, in which a "1" or a "0" appears in the tth row and yth column according as the segment whose name is at the beginning of the t'th row is or is not on the boundary of the triangular region whose name is at the top of the yth column.
|
Tll |
T12 |
T13 |
T14 |
T2l |
T,, |
•M |
T24 |
T81 |
TSJ |
T8« |
TS4 |
T41 |
r« |
'41 |
T44 |
|
|
(T12 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
*12 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
|
°-13 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
"13 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
|
°U |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
|
"14 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
|
"28 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
|
"28 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
<T84 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|
"24 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
"84 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
"34 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
H2:
Let us denote the eight tetrahedral regions by Tt, • • •, Tg and con- struct a matrix analogous to the preceding ones, in which a "1" or a " 0 " appears in the iih row and yth column according as the tri- angular region whose name is at the beginning of the t'th row is or is not on the boundary of the tetrahedral region whose name is at the top of the yth column. By definition there is a plane TT which meets all the six segments a-^ and none of the segments <TO.. There is one and only one tetrahedral region not met by TT. Let us assign the notation so that this region is called T,. As TT cannot meet the segments and triangular regions on the boundaries of Tt, these seg- ments must be the six segments cF^. and these triangular regions must be those bounded by <TO.. The latter can be found by means of the matrix H2. This determines the first column of the matrix to be constructed. The other columns are found by considering succes- sively the planes of the seven other classes of planes described in
400 THKOKKMS ON SENSE AND SEPARATION [CHAI-.IX
§ 26. Thus, for example, Ts is the region on whose boundary are the segments *u, cr», <r14, *„, <TM, <?„.
|
T, |
T3 |
T, |
T4 |
T6 |
T, |
T7 |
T8 |
|
|
Tll |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
|
fit |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
|
rl» |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
|
T14 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
|
T«l |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
|
fit |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
|
rit |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
|
T*4 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
|
TS1 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
Tsa |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
|
T88 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
|
T»4 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
T41 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
T4« |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
|
T48 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|
T44 |
0 |
. 0 |
0 |
0 |
1 |
1 |
0 |
0 |
EXERCISE
The planes polar (§ 18, Vol. I) with respect to a tetrahedron ABCD to the points of one of the tetrahedral regions determined by ABCD consti- tute one of the four sets of planes determined by ABCD according to § 26. The points on these planes constitute the set of all points not on the given tetrahedral region or its boundary.
* 153. Generalization to n dimensions. The generalization to n dimensions of the point pair, triangle, and tetrahedron is the (n + l)-point in n-space. This is any set of n + 1 points no n of which are in the same (n — l)-space, together with the lines, planes, 3-spaces, etc. which they determine by pairs, triads, tetrads, etc. By a direct generalization of § 26 orie proves that the points not on the n— 1 spaces of an (n + l)-point fall into 2W mutually
§§ i.™, 154] CURVES 401
exclusive sets Rt, • • •, R2. such that any two points of the same set are joined by a segment of joints of the set and that any segment joining two points not in the same set contains at least one point on an (n— l)-space of the (n + l)-point. Any one of the sets Rj, • • •, R... is called a simpler or n-iliinfiinio-nul wi/me-nt.
Thus the simplex is a generalization of the linear segment, trian- gular region, and tetrahedral region. By replacing triangular and tetrahedral regions by simplexes throughout §§ 148-152 we obtain i in mediately the theory of ?i-dimensional convex regions. A like proc- ess applied to §§ 154-157, below, gives the theory of ^^dimensional connected sets, regions, continuous families of sets of points, con- tinuous families of transformations, continuous groups, etc. We leave both series of generalizations to the reader.
154. Curves. DEFINITION. Let [T] be the set of all points on an interval TQT^ of a line I. A set of points [P] is called a con- tinuous curve or, more simply, a curve, if it is in such a correspond- ence T with [T] that
(1) for every T there is one and only one P such that P = F(T);
(2) for every P there is at least one T such that P= F(T);
(3) for eVery T, say T', and for every tetrahedral region R con- taining F(T'), there is a segment a- of / containing T' and such that for every T in <r, T(T) is in R.
A curve is said to be closed if F(T0) = T(T1). It is said to be simple if F can be chosen so as to satisfy (1), (2), (3) and so that if T'^T", F(T')*= F(T") unless the pair T'T" is identical with the pair T9Tf
The point F(T) is said to describe the curve as T varies. The curve is said to join the points F ( TQ) and F ( 7^).
In view of the definition of the geometric number system in Chap. VI, Vol. I, and the theorems in Chap. I, Vol. II, this definition could also be stated in the following form : Let (/) be the set of numbers such that 0 = / = 1. A set of points [P] is called a curve if it is in such a ciirresjxmdence T with [/] that (1) for every t there is one and only one P= T(t), (2) for every /' there is at l.-;ist one t such that P= T(t), and (3) for every t, say f, and for .-very tetra- hedral region R containing T(t') there is a number 8>0 such that if f-B<t<t'+ 8, T(0 is in R.
In the Euclidean or non-Euclidean spaces (3) may In- replaced by the con- dition: For every t' and every positive, number e there is ;i i>ositive number S such that if ( — §<Kt + 8, the distance between T (t) and T (f) is less than c.
402 THEOREMS ON SKNSE AND SEPARATION [CHA. ix
The most obvious examples of simple closed curves are the projective line and the point conic. The proof that these are simple closed curves will be given for the planar case, and may be extended at once to the three-dimensional case by substituting tetrahedral regions for triangular ones.
THEOREM 15. A projective line is a simple closed curve.
Proof. Let [P] be the set of points on a projective line and let P0, Pp Pj, Ps be four particular values .of [IJ] in the order {P^P^P^. Let T , T, T , T8, T4 be five collinear points in the order {T^T^T^T^T^, and let [T] be the set of all points of the interval TQ2\Tt. If T is on the interval T^T^T^ let F(T) be the point to which T is carried by a projective correspondence* which takes the points T0, T1? Ta into P0, Plt P respectively ; and if T is on the inter- val T^TsTt, let F(T) be the point to which T is carried by a projectivity which carries the points Tz, Tg, Tf into P,, Ps, P0 respectively.
The correspondence F is defined so that there is one and only one point P=F(T) for each T\ and also so that T(T')^T(T")t unless T' = T", or T' = TO and T"= Tt, or T' = Tf and T"= Tf Thus [P] satisfies conditions (1) and (2) of the definition of a curve and the condition that a curve be simple.
Let R be any triangular region containing a point P'= F(T'). By Theorem 1 there is a segment of the projective line [P] con- taining P' and contained in R; let P"= T(T") and P"'=Y(T'") be the ends of this segment. This segment is the image either of the points T between T" and T'" or of the points T not between T" and T'". Hence if o- be any segment of the line TQTl con- taining T' and not containing T" or T'", every point T on <r is such that F(T) is in R. Hence [P] satisfies Condition (3) of the defini- tion of a curve.
THEOREM 16. A point conic is a simple closed curve.
Proof. The proof is precisely the same as that of Theorem 15 except that [P] is the set of points on a conic, and the following lemma is used instead of Theorem 1.
•This does not use Assumption P, because it requires only the existence of a projectivity, and this may be set upas a series of perspectivities (cf . Chap. Ill, Vol I).
§
CURVES
I. KM MA. If a point P of a conic C* is in a triangular region R coplanar with C2, there -is a segment <r of 6'2 which contains P and is contained in R.
Proof. If C2 is entirely in R the conclusion of the theorem is obvious. If not, let Q be a point of C2 not in R. By § 75 the points of the line PQ interior to <72 constitute a segment having P and Q as ends. Let R be a point of this segment which is also on the segment con- taining P (Theorem 1), which the line PQ has in common with R (fig. 81).
Let T be the common point of the tangents at P and Q and let T' and T" be points of R in the order (TT'PT"}. Let S' and S" be the points in which QT' and QT" meet TR; so that {TS'RS"}- Let Sl and Sa be points interior to R, interior (Theorem 1). The lines QSt and QS^ meet TP in two points T( and T3 respectively in the order {TT'T^PT^T"}. Since these points are on the segment T'PT" they are in R. Since Q is oii the conic C* the lines QT^ and QTt meet C2 in two points Pl and P2 respectively.
Since ^ is interior and 1\ (a point of a tangent) exterior to C*t we have the order {QS^T^. But Sl and 7\ are-in R and Q is not in R. Hence by Theorem 1, P^ is in R. In like manner 7^ is in R.
The segment P^PP^ of the conic C2 is now easily seen to consist entirely of points of R. For if P is any point of this segment, and T and S the points in which QP meets PT and RT respectively,
Fir,. 81
in the order
Hence T is on the segment 1\PT3, and & is on the segment NjA'N,. Hence T and S are interior to R, and ti interior to C*. Since T is exterior to C2, it follows that £ and T separate P and Q. Therefore, as Q is not in R, P is in R.
404 THEOREMS ON SENSE AND SEPARATION [CHAP, ix
EXERCISE The boundary of a triangular region is a simple closed curve.
155. Connected sets, regions, etc. A set of points is said to be cotuwcted if and only if any two points of the set are joined by a curve consisting entirely of points of the set. A connected set is sometimes called a continuous family of points. In a space satis- fying Assumptions A, E, H, C (or A, E, K or A, E, J) a connected set is also called a continuum. A connected set in a plane such that every point of the set is in a triangular region containing no points not in the set is called a planar region. A connected set of points in space such that every point of the set is in a tetrahedral region containing no points not in the set is called a three-dimensional region.
A one-to-one transformation F carrying a set of points [X] into a set of points [F] is said to be continuous if and only if for every A', say X', and every tetrahedral region T containing F(A''), there is a tetrahedral region R containing X' and such that for every A" in R, F(A) is in T.
If a linear interval joining two points A, B is subjected to a con- tinuous one-to-one reciprocal transformation, it goes into a curve joining the transforms of A and B (§ 154). The set of points on the curve, excluding the transforms of A and B, is called a 1-cell.
If a triangular region, and its boundary are subjected to a con- tinuous one-to-one reciprocal transformation, the set of points into which the triangular region goes is called a simply connected element of surface, or a 2-cell.
If a tetrahedral region and its boundary are subjected to a con- tinuous one-to-one reciprocal transformation, the set of points into which the boundary goes is called a simply connected surface, or simple surface, and the set of points into which the tetrahedral region goes is called a simply connected three-dimensional region, or a 3-cell.
EXERCISES
1. A region contains no point of its boundary.
2. If J and H are any two points of a planar region R, there exists a finite number of triangular regions tv f2, •••,<„ such that f{ has a point in common with /, + 1 (i = 1, . . ., n — 1) and fj contains A and <„ contains B. This property could lw taken as the definition of a region in a plane.
§§155,156] CONNECTED SETS 405
3. (liven any set of regions all contained in a convex region. The set of all jH)ints in triangular regions whose vertices art- in tin- given regions in a convex region. This region is contained in every convex region containing the given set of regions (J. W. Alexander).
4. The set of all {joints on segments joining pairs of points of an arbitrary region R contained in a convex region constitutes a convex region R'. The region R' is contained in every convex region containing R.
5. The boundary (§ 150) of a region in a plane (space) separates (§ 149) the set of all }>oints in the region from the set of all points of the plane (space) not in the region.
6. A continuous one-to-one reciprocal transformation of space transforms any region into a region.
156. Continuous families of sets of points. The notion of con- tinuous curve has the following direct generalization:
DEFINITION. Let [T] be the set of all points on an interval T0Tt of a line /. A set of sets of coplanar points [S] is called a con- tinuous one-parameter family of sets of points if it is in such a correspondence F with T that
(1) for every T there is one and only one set S such that S = F(r) ;
(2) for every set S there is at least one T such that S = F(T);
(3) for every T, say T', and for every triangular region R includ- ing a point of the set F(T'), there is a segment <r of I containing T' and such that if T is in a- at least one point of the set F(T) is in R.
The definition of a continuous one-parameter family of sets of points in space is obtained by replacing the triangular region R in the statements above by a tetrahedral region.
If the sets S are taken to be lines, planes, conies, quadrics, etc., this gives the definition of one-parameter continuous families of lines, planes, conies, quadrics, etc., respectively. Cf. Exs. 1~5, below.
DEFINITION. A connected set of sets of points or a continuous family of sets of points is a set of sets of points [S] such that any two sets Sj, S2 are members of a continuous one-parameter family of sets of [S].
For example, the discussions given below in terms of elementary transformations establish in each case that a sense-class is a con- nected set of sets of points. Cf. also Exs. 6-7, below.
The definition of a continuous family may be extended in an obvious way so as to include sets whose elements are points, sets of points, sets of sets of points, etc.
406 THKoKKMS (>N SK.NSK AND SKPARATION [CHAP, ix
EXERCISES
1. Defining an envelope of lines as the plane dual of a curve, prove that an envelope is a continuous one-parameter family of lines.
2. The space dual of a curve is a continuous one-parameter family of planes.
3. Pencils of lines and planes are continuous one-parameter families.
4. A line conic or a regulus is a continuous one-parameter family of lines.
5. A pencil of point conies is a continuous one-parameter family of curves.
6. The set of all lines in a plane or space or in a linear congruence or a linear complex is a connected set of sets of points.
7. The set of all planes in space or of all planes tangent to a quadric is a connected set of sets of points.
157. Continuous families of transformations. Let [T] be the set of all points on au interval T^Tl of a line 1. Let [II T] be a set of transformations of a set of points [P]. If (1) to every T there cor- responds one and only one transformation II T, and (2) for every point P the set of points [IT7.(JP)] is a curve for which the defining correspondence F (in the notation of § 154) may be taken to be the correspondence between T and II7,(P), then [Hr] is said to be a continuous one-parameter family of transformations. The curves [[Ir(P)] are called the path curves of [II r],
The term " continuous one- parameter family of transformations " may also be applied to a set of transformations [IIr] of a set S of points P and of sets of points S (e.g. S may be a set of figures as defined in § 13, Vol. I). In this case (1) and (2) must be satisfied, and also the following condition : (3) For every set of points S, [IIr(S)] is a one-parameter continuous family of sets of points for which the defining correspondence F (in the notation of § 156) may be taken to be the correspondence between T and IIr(S).
If the set of correspondences [II T] is both a group and a continuous one-parameter family of transformations, it is called a one-parameter continuous group.
A set of transformations [II] of a set of points and of sets of points, such that any two transformations of [II] are members of a con- tinuous one-parameter family of transformations of [II], is called a continuous family of transformations. If [II] is also a group, it is called a continuous group.
If [Hr] is a continuous one-parameter family of one-to-one recip- rocal transformations of a figure F, and if II Tn is the identity, then F is said to be moved, or deformed, to the figure HTi(F) through the set
§§ I-.:, 158] DEFORMATIONS 407
of intermediate positions [Hr(F)]. Any one of the transformations nr is called a deformation; if /' is a set of points and all the transformations of the family [II T] are continuous, the deformation is said to be a continuous deformation.
158. Affine theorems on sense. Let us recapitulate some of the main propositions about sense-classes in Euclidean spaces by enumerating the one-dimensional propositions of which they are generalizations.
The group of all projectivities x'=ax + b on a Euclidean line has a subgroup of direct projectivities for which a > 0. This sub- group is self-conjugate, because if a transformation of the group be denoted by 2, and any other transformation x' = ax + ft by T, then T2T-1 is
x>=a(aU-x-^} + b\ + fr
a transformation in which the coefficient of x is positive. From the fact that the subgroup is self -con jugate, it follows as in § 18 that the same subgroup is defined by the condition a > 0, no matter how the scale is chosen, so long as P*, is the point at infinity. These statements are generalized to the plane in § 30 and to spaces of any dimensionality in § 31. The generalization consists in replacing a by the determinant
for the two-dimensional case, and by the corresponding w-rowed determinant in the w-dimensional case.
A sense-class S(AB) is the set of all ordered pairs of points into which a pair of distinct points can be carried by direct pro- jectivities (§ 23). This proposition is generalized to the plane in § 30 and to n-space in § 31.
A particular arbitrarily chosen sense-class shall be called positive and the other sense-class shall be called negative. This statement reads the same for any number of dimensions. In the three-dimen- sional case the positive sense-class is also called ri<jJ<t-l«i iided and the negative sense-class left-handed (see the fine print in § 162).
In the one-dimensional case a nonhomogeneous coordinate sys- tem is called positive if S(PQP^) is positive. In the two-dimensional
408
THEOREMS ON SENSE AND SEPARATION [CHAP, ix
case a nonhoinogeiieous coordinate system is called positive if N "AT) is positive when O = (0, 0), AT = (1, 0), and Y= (0, 1). In the three-dimensional case a nonhomogeneous coordinate system is called positive or right-handed if S(OXYZ) is positive wlu-n 0= (0, 0, 0), A'= (1, 0, 0), 1'= (0, 1, 0), and Z=(Q, 0, 1).
Oa tlm Euclidean line two ordered pairs of points AB and A'/:' are in the same sense-class if and only if
1 a 1 b
and
have the same sign, a,b,a',b' being the nonhomogeneous coordinates of A, B, A', B' respectively. Hence, if the coordinate system is posi- tive, S(AB) is positive or negative according as (b — a) is positive or negative. Similar criteria for the plane and space are given in §§ 30, 31. It follows immediately that if the coordinate system in the plane is positive, S(ABC) is positive or negative according
as the determinant
l a <
1 '!
is positive or negative, where A = (alt a2), B = (blt J2), C = (ct, c2). If the coordinate system in space is positive, S(ABCD) is positive or negative according as the determinant
1 at a,
1 } } 1 dl dt
is positive or negative, where A=(alt «2, a^, B = (blt ba, b3), C =
In the one-dimensional case B is on one or the other of the rays having A as origin according as S(AB) is positive or negative. In a Euclidean plane C is on one side of the line AB or the other according as S (ABC) is positive or negative (§ 30). In a Euclidean space D is on one side or the other of the plane ABC according as S (A BCD) is positive or negative.
The projectivities a/ = ax + b of the Euclidean line are in one-to- one reciprocal correspondence with the points (a, b) of the Euclidean plane. The direct projectivities correspond to the points on one side of the line a = 0 and the opposite ones to those on the
§§i.w,i»] ELEMENTARY TRANSFORMATIONS 409
other side. From this it readily follows that the set of all direct projectivities forms a continuous group, whereas the set of all projec- tivities is a group which is not continuous.
In like manner the transformations
can be set in correspondence with the joints of a six-dimensional Euclidean space, the direct and opposite collineations respectively corresponding to points of two regions separated by the locus
Similarly, the direct and opposite collineations in a Euclidean space of three dimensions may be represented by points of two regions in a space of twelve dimensions. In all three cases the set of all direct collineations forms a con- tinuous group, but the set of all collineations does not.
Another way of coming at the same result is this : Let the ordered pairs of points A B of a Euclidean line be represented by the points («, b) of a Euclid- ean plane, a being the nonhomogeneous coordinate of A, and b that of B. Under this convention the points representing pairs of the positive sense-class are on one side of the line b — a = 0 and those representing pairs of the negative s<-iise-clas8 on the other side of this line. The one-dimensional affine projec- tivities are in one-to-one reciprocal correspondence with the ordered point pairs to which they carry a fixed ordered point pair PQ. The direct projec- tivities thus correspond to point pairs represented by points on one side of the line b — a = 0 and the opposite projectivities to point pairs represented by points on the other side.
159. Elementary transformations on a Euclidean line. DEFINITION. Given an ordered pair of points AB of a Euclidean line, the oper- ation of replacing one of the points by a second point not separated from it by the other point is called an elementary transformation of the pair AB.*
Thus AB may be transformed into AB' if {ABB1} or {AB'B}. In other words (cf. § 23) B can be transformed to any point B' such that S(AB} = S(AB'), and into no other. Hence it follows that if AB is transformable to A' B' by any sequence of elementary trans- formations, S(AB} = S(A'B').
Conversely, if S(AB) = S(A'B'), it is easy to see, as follows, that by a sequence of elementary transformations AB can be transformed
* The transformations which we have considered heretofore have usually been transformations of the line, plane, or space as a whole. Here we are considering a transformation of a single pair of point*.
410 THEOREMS ON SENSE AND SEPARATION [CHAP.IX
to A' It'. From the theorems on linear order in Chap. II it follows that there are two points A" and B" satisfying the order relations
{ABA"B"} and {A'B'A"B"}.
By elementary transformations AB goes to AB"\ AB" to A"B"; A"B" to A'B"', and A'B" to A'B'. Hence we have
THEOREM 17. On a Euclidean line the set of all ordered pairs of points into which an ordered pair of distinct points AB can be transformed by elementary transformations is the sense-class S(Att).
An elementary transformation may be regarded as a special type of con- tinuous deformation (§ 157). If .4 B is carried by an elementary transformation to. AB', the point B may be thought of as moved (§ 157) along the segment BB" from B to B', and since this segment does not contain A, the motion is such that the pair of distinct points never degenerates into a coincident pair. Thus we may say that a sense-class consists of all pairs obtainable from a fixed pair by defoliations in which no pair ever degenerates.
When the ordered point pairs are represented by points in a Euclidean plane, as explained at the end of the last section, an elementary transforma- tion corresponds to moving a point (a, ft) parallel to the a-axis or the &-axis in such a way as not to intersect the line a = b.
DEFINITION. An elementary transformation of a pair of points AB is said to be restricted with respect to a set of points [P] if and only if it carries one of the pair, say B, into a point B' such that the segment BB' does not contain any one of the points P. (Any one of these points may, however, be an end of the segment BB'.)
It is evident that any elementary transformation can be effected as a resultant of a sequence of elementary transformations which are restricted with respect to an arbitrary finite set of points. Hence Theorem 1 7 has the following corollary :
COROLLARY. Let Plt P.,, • • •, Pn be any finite set of points on a line 1. Two ordered pairs of points are in the same sense-class if and only if one can be carried into the other by a sequence of elementary transformations restricted with respect to P^ P2, • • •, 1\ .
The concept of a restricted elementary transformation is intimately con- nected with the idea of a "small motion." In the metric geometry the points /',. /'2, • • •, Pn can be chosen so as to be in the order [Pv Pz, ••-,/'„} and so that the segments P{P,+i are arbitrarily small.. Any elementary transforma- tion of a pair of points on the interval P,/'1 + 1 will be effected by a small motion of one of the points in the pair.
§§i.v.»,ieo] KLKMKNTAKY TRANSFORMATIONS 411
160. Elementary transformations in the Euclidean plane and space. DKFINITION. Given an ordered set of three noncolliuear points in a Kuclidean plane, an elementary transformation is the operation of replacing one of them by a point which is joined to it by a
segment not meeting the line on the other two.
• As in the one-diinensionul case, an elementary transformation may be
regarded as effected l>y a continuous deformation of a point triad. A path is specified along which a point may be moved without allowing the triad to degenerate into a collinear one.
Let A, B, C be three noncollinear points and let C' and B' be points of the segments AB and CA respectively. Then by elemen- tary transformations (cf. fig. 84, p. 423) ABC goes to C'BC; and this to C'BB'\ and this to C'CB'] and this to BCB'\ and this to BCA. In like manner it can be shown that ABC can be carried to CAB by a sequence of elementary transformations. Hence any even permutation of three noncollinear points can be effected by elementary transformations.
By Theorem 27, § 30, an elementary transformation leaves the sense of an ordered triad invariant. Hence, by Theorem 26, § 30, no odd permutation can be effected by elementary transformations.
If A', B', C' are any three noncollinear points, ABC can be carried into some permutation of A'B'C' by elementary transforma- tions. For since at most one side of the triangle A'B'C' is parallel to the line AB, this line meets two of the sides in points which we may denote by A" and B". By one-dimensional elementary trans- formations on the line AB, the ordered pair AB can be carried either to A"B" or to B"A". These one-dimensional elementary transformations determine a sequence of two-dimensional elemen- tary transformations leaving C invariant and carrying ABC to A"B"C or to B"A"C. The point C can be carried by an obvious elementary transformation to a point C" such that A"C" is not parallel to any side of A'B'C', and then A"C" can be carried to two of the points, say A'"C'", in which the line A"C" meets the sides of the triangle A'B'C'. The points A'"B"C"' are on the sides of the triangle A'B'C', and the one-dimensional elementary transformations on the sides which carry them into the vertices determine two-dimensional elementary transformations which carry A'"B"C'" to some permutation of A'B'C'.
412 TIII-:OI;KM> ON >KNSK AND SEPARATION [CHAP. i\
Since ABC cannot be carried into A'B'C' if S(ABC) * S(A'B'< "), and since all even permutations of A'B'C' can be effected by ek-inrn- tary transformations, it follows that ABC can be carried into A'B'C' by a sequence of elementary transformations if S(ABC) = S (A'B'C'). Hence we have
THEOREM 18. In a Euclidean plane S (ABC) = S (A'B'C') if and only if there exists a finite set of elementary transformations carrying tht noncollinear points A, B, C into the points A', B', C' respectively.
DEFINITION. Given an ordered set of four noncoplanar points, an elementary transformation is the operation of replacing one of them by another point which is joined to it by a segment containing no point of the plane on the other three.
Let ABCD be four noncoplanar points. Holding D fixed, ABC may be subjected to precisely the sequence of elementary transformations given above in the planar case for carrying ABC into BCA. This effects the permutation /ABC D\
\B C A D/'
the symbol for each point being written above that for the point into which it is transformed. In like manner we obtain the permutations
(A B C D\ (ABC D\ (A B C D\ \B D C A) \C B D A/ \A C D Bj'
and it is easily verifiable that any even permutation of ABCD is a product of these permutations'. Hence any even permutation of a set of four points may be effected by elementary transformations.
By Theorem 23, § 27, an elementary transformation of four points K. «2. «,)» (*>!> *V £3). (<v <v c3), (dlt d2, da) leaves the sign of
«. a. 1
invariant, and hence leaves their sense-class invariant. Hence (§ 31) no odd permutations of four noucoplanar points can be effected by elementary transformations.
An ordered tetrad ABCD of noncoplanar points can be carried into some permutation of an ordered tetrad A'B'C' D' of noncoplanar points. For the line AB is not parallel to "more than two planes of the tetrahedron, and hence by the one-dimensional case AB can be
J§ 160, 161] ELEMENTARY TRANSFORMATIONS 413
carried into two points A"B"ot the planes of the tetrahedron A'B'C'D'. By repeating this argument it is easily proved that C and D can also be carried to points C"D" on these planes. By the two-dimensional case it follows that the ordered tetrad A"B"C"D" of points on the planes of the tetrahedron A'B'C'D' can be carried into some permu- tation of its vertices. Since ABCD cannot be carried into A'B'C'D* \ if S(ABCD)* S (A'B'C'D') it follows by the last paragraph but one that it can be carried into A'B'C'D' if S (ABCD) = S (A'B'C'D'}. Thus we have
THEOREM 19. Ina Euclidean space S (ABC D)=S (A'B'C'D') if and only if there exists a finite set of elementary transformations carrying the noncoplanar points A, B, C, D into the points A1, B', C', D1 respectively.
The theorems and definitions of the last two sections can be re- garded as based on any one of the sets of assumptions A, E, H, C, R or A, E, K or A, E, P, S. Assumption P is used wherever coordinates are employed, but it is possible to make the argument without the aid of coordinates and thus to base it on A, E, S alone (cf. Ex. 2, § 161).
161. Sense in a convex region. DEFINITION. Given a set of three noncollinear points of a planar convex region R, the operation of replacing any one of them by any other point of R on the same side of the line joining the other two is called an elementary transformation. The set of all ordered triads obtainable by finite sequences of elementary transformations from one noncollinear ordered triad of points ABC is called a sense-class and is denoted by S(ABC).
This definition is in agreement with the propositions about sense given for the special case of a Euclidean plane. Moreover, if R is any convex region, and I* is any line coplanar with R but containing no point of R, two triads of points of R are in the same sense with respect to R if and only if they are in the same sense with respect to the Euclidean plane containing R and having /. as singular line. Hence the theorems of § ICO may be taken over at once to convex regions in general. This result may be stated as follows:
THEOREM 20. In a planar convex region there are two and only two senses. Sense is preserved by even and altered by odd permu- tations of three noncollinear points. Two points C and. D are on opposite sides of a line AB if and only if S(ABC)=£ S(A/W).
414 THKORKMS <»N SKNSH AND SEPARATION [CHAP. IX
DEFINITION, (liven a set of four noncoplanar points of a three- dimensional convex region R, the operation of replacing any one of tlu-iii by any point of R on the same side of the plane of the other three is called an elementary transformation. The set of all ordered tetrads obtainable by finite sequences of elementary transformations from one noncoplanar ordered tetrad of points ABCD is called a sense-class and is denoted by 8 (ABCD).
The theories of sense in a three-dimensional convex region and in a three-dimensional Euclidean space are related in just the same way as the corresponding planar theories. Hence we have
THEOREM 21. In a three-dimensional convex region there are two and only two senses. Sense is preserved by even and altered by odd permutations of four points. Two points I) and E are on opposite sides of a plane ABC if and only if S(ABCD) =£ S(ABCE).
EXERCISES
1. The whole theory of order relations can be developed by defining sense- class on a line by means of elementary transformations instead of as in Chap. II.
*2. Develop the theory of order in two- and three-dimensional convex re- gions, defining sense-class in terms of elementary transformations and using Assumptions A, E, S or Assumptions I- VIII of § 29 (cf. Theorem 5, § 148) as basis.
3. An elementary transformation of a triad of points ABC in said to be restricted with respect to a set of points Pv P2, •••, Pn if it carries a point of the triad, say C, into a point C" such that the segment CC' does not contain any point collinear with two of the points Pv P2, •••, Pn. Two ordered triads of points are in the same sense-class if and only if there is a sequence of restricted elementary transformations carrying the one triad into the other.
4. Generalize the notion of restricted elementary transformation to space.
162. Euclidean theorems on sense. The involutions which leave the point at infinity of a Euclidean line invariant may be called point reflections. The product of two point reflections is a parabolic projectivity leaving the point at infinity invariant, and may be called a translation. A point reflection has an equation of the form
(1) a/ «-» + &, and a translation has one of the form
(2) x' = x + b.
§ itti] SENSE IN A EUCLIDEAN SPACE 415
The point reflections interchange the two sense-classes of the Euclid- ean line, and the translations leave them invariant.
In generalizing these propositions to the plane, the point reflec- tions may be replaced by the orthogonal line reflections (Chap. IV) or, indeed, by the set of all symmetries, and the one-dimensional translations by the set of all displacements in the plane. Since an orthogonal line reflection in the plane interchanges the two sense- classes, any symmetry interchanges them, but any displacement leaves each of them invariant. The generalization to three-dimensions is similar.
The equations of a displacement in two or three (or any number of) dimensions are a direct generalization of the one-dimensional equations, namely,
(3) *i = 2«<rO+6f (< = 1, 2,-..,n)
j=»
where the matrix («0-) is orthogonal and the determinant jatf| is -f 1. The equations of a symmetry satisfy the same condition except that the determinant a(>. is — 1 instead of -f 1.
It is worthy of comment that the distinction between displace- ments and symmetries holds in the complex space just as well as in the real, whereas the distinction between direct and opposite collineations holds only in the real space. Algebraically, this is because the distinction of sense depends merely on the sign of the determinant |«0.|, whereas the distinction between displacements and symmetries is between collineations satisfying the condition |atf| = + l and | «0. = — 1. In the representative spaces of six and twelve dimen- sions referred to in § 158, |«0-| = 1 and |atf = — 1 are the equations of nonintersecting loci.
From the point of view of Euclidean geometry, as has been said above, the two sense-classes are indistinguishable.* In the applications of geometry, however, a number of extra-geometrical elements enter which make the two
* This does not contradict the existence of a geometry in which one sense-class is specified absolutely in the assumptions. The group of such a geometry is unlike the Euclidean group in that it does not include symmetries though it does include displacements. Its relation to the Euclidean geometry is similar to that of the geometries mentioned in the fine print in § 110. Those geometries, however, corre- spond to groups which are not self -con jugate under the Euclidean group, when M- tin's one corresponds to a self-conjugate subgroup. ( >n the foundations of geometry in terms of sense-relations taken either absolutely or relatively, see the article by Schweitzer referred to in § 16.
4 Hi THEOREMS ON SENSE AND SEPARATION [CHA. ix
FIG. 82
aenae-cla8M>8 play essentially different roles. Thus any normal human b< -ins who identities the al>*tni<-t Kuelideaii space with the space in which he virus himself and other material objects may single out one of the sense-classes as follows: Let him hold his right hand in such a way that the index tinker is in line with his arm, his middle linger at right angles to his index finger, and his thumb at right angles to the two fingers (fig. 82). Let a jwint in his palm be denoted by O, and the tips of his thumb, index finger, and middle finger I \ A. F, Z respectively. The sense-class S (OXYZ) shall be called riyht-handed or positire, and the other left-handed or negative. This designation is unique be- cause of the mechanical structure of the body.
A nonhomogeneous coordinate system is called right-handed or positive if and only if S (OXYZ) is positive when O = (0, 0, 0), A' =(1, 0, 0) , Y= (0, 1, 0),
and Z = (0, 0, 1). The reader will find it convenient whenever an arbitrary sense-class is called positive to identify it with the intuitively rightrhauded sense-class.*
163. Positive and negative displacements. On a Euclidean line, if a translation carries one point A to a point B such that S(AB) is positive, it carries any point A to a point B such that S(AB) is positive. Such a translation is called positive. Any other translation is called negative and has the property that if it carries C to D, S(CD) is negative. Any translation carries positive translations into positive translations; Le. if T' is a positive translation and T any translation, TT'T~l is a positive translation. A translation a/= x + b is positive or negative according as b is positive or negative, provided that the scale is such that S(PQP1) is positive. The inverse of a positive translation is negative.
The distinction between positive and negative translations is quite distinct from that between direct and opposite projectivities, for all translations are direct.
• An interesting aox;ount of the way in which this choice is made in various branches of mathematics and other sciences is to be found in an article by E. Study, Archiv der Mathematik und Physik, 3d series, Vol. XXI (1913), p. 193.
§nw] POSITIVE AND NEGATIVE DISPLACEMENTS 417
\ A like subdivision of the Euclidean displacements of a plane
which are neither translations nor point reflections nor the identity may be made as follows : A rotation leaving a point 0 fixed and carrying a point A to a point B not collinear with O and A is said to be positive if S(OAB) is positive and to be negative if S(OAB) is negative. It is easily proved that if S(OAB) is positive for one value of A it is positive for all values of A. The inverse of a positive rotation is negative. Any displacement transforms a positive rotation into a positive rotation.
A rotation is a product of two orthogonal line reflections {/-£•} and {mJf,c} such that the lines I and m intersect in 0. Hence the ordered pairs of lines which intersect and are not perpendicular fall into two classes, which we shall call positive and negative respec- tively, according as the rotations which they determine are positive or negative.
In a three-dimensional Euclidean space let A be a point not on the axis of a given twist which is not a half-twist, let 0 be the foot of a perpendicular from A on the axis of the twist, and let A' and <y be the points to which A and 0 respectively are carried by the twist. The twist is said to be positive or right-handed if S(OAO'A') is posi- tive or right-handed and to be negative or left-handed if S(OA(yA') is negative.
It is easily seen that S(OAO'A') is the same for all choices of A, so that the definition just made is independent of the choice of A. The inverse of the twist carrying 0 and A to (X and Af carries CX and A' to 0 and A, and thus is positive if and only if S(O'A'OA) is positive. Since S(O'A'OA) = S(OAO'A'), the inverse of a positive twist is posi- tive. Any direct similarity transformation carries a positive twist into a positive twist.
With the choice of the right-handed' sense-class described in the fine print in § 162, the definition here given is such that a right-handed twist is the displacement suffered hy a commercial right-handed screw driven a short distance into a piece of wood.
Since a twist is a product of two orthogona1 line reflections, {II*} • {mm*}, it follows that the pairs of ordinary lines Im which are not parallel, intersecting, or perpendicular fall into two classes, according as the twist {mm*} • {II*} is positive or negative. We shall call the line pairs of these two classes positive and right-handed
418 TIIKOUKMS ON SKNSK AN1> SK1»A RATION [CHAI-. IX
,,r /« .UK! Ifft-ltiuuled respectively. Since the inverse of a
positive twist is positive, the ordered pair ml is positive if Im is posi- tive. Hence a pair of lines is right-handed or left-handed without regard to the order of its memhers. Any direct similarity transfor- mation carries a right-handed pair of lines into a right-handed pair and a left-handed pair into a left-handed pair.
EXERCISES
1. The collineations which are commutative with a positive displacement (<>r with a negative displacement) are all direct.
2. Hy the definition in § 69, 0<&AOB<ir or -n-<4. A O6<2ir according as S(U.IB) is positive or negative, provided that the points O, P0, P\ are so chosen that S(OP0P^) is positive.
3. By the definition in § 72, 0<m (IJZ) < - or — <m (/!/2) <ir according as the ordered line pair Itl2 is positive or negative.
4. Let us define an elementary transformation of an ordered line pair Zj/2 in a plane as being either the operation of replacing /t or /2 by a line parallel to itself, or the operation of replacing ll or /2, say lv by a line through the point /j/2 which is not separated from /t by /2 and the line through /j/2 perpendicu- lar to /2. Two ordered pairs of nonparallel and nonperpendicular lines are equivalent under elementary transformations if and only if they are both in the positive or both in the negative class.
5. Let us define an elementary transformation of a pair of nonparallel and nonperpendicular lines IJZ in space as the operation of replacing one of the lines, say lv by a line intersecting Zx and not separated from /t by the plane through the j>oint of intersection perpendicular to lv and the plane through this point and /2. The pair l^z can be transformed into a pair of lines mj7n2 by a sequence of elementary transformations if and only if both pairs are ri),') it-handed or both pairs are left-handed.
164. Sense-classes in projective spaces. It has been seen in Chap. II (cf. §§ 18 and 32) that the distinction between direct ami opposite collineations can be drawn in any projective space of an odd number of dimensions which is real or, more generally, which satisfies A, E, S. This depends (§ 32) on the fact that the sign of a determinant |ay| (i,j= 0, 1, • • •, ri) cannot be changed by multi- plying every element by the same factor if n is odd, and can be changed by multiplying every element by — 1 if n is even.
In a real projective space of odd dimensionality the direct collin- eations form a self-conjugate subgroup of the projective group ami thus give rise to the definitions of sense-class in §§19 and 32. The same remarks are made about the independence of this definition
§§1(4,165] SENSE IN PROJECTIVE SPACKS 419
<>f the frame of reference as in the Euclidean cases, and the criteria for sense in terms of products of determinants are given in §§24 and 32. If one forms the analogous determinant products for the projective spaces of even dimensionality, it is found that the sign of the product may be changed by multiplying the coordinates of one point by — 1, which verifies in a second way that there is only one sense-class in a projective space of an even number of dimensions. The projectivity
_
~
may be represented by means of a point (an, au, a21, aw) in a pro- jective space of three dimensions. The points representing direct projectivities are on one side of the ruled quadric
= 0,
and those representing opposite projectivities on the other side. This representation of projectivities by points is in fact identical with that considered in § 129. It can be generalized to any num- ber of dimensions just as are the analogous representations in § 158.
It readily follows that the group of all projective collineations in a real space of n dimensions is continuous if n is even, and not continuous if n is odd. If n is odd the group of direct collineations is continuous.
In the following sections (§§ 165-167) we shall discuss the sense-classes of projective spaces by means of elementary transfor- mations, the latter term being used as before to designate a particular type of continuous deformation. After this (§§ 169-181) similar considerations will be applied to other figures.
165. Elementary transformations on a projective line. DEFINITION. Given a set of three collinear points A, B, C, an elementary transfor- mation is the operation of replacing any one of them, say A, by another point A' such that there is a segment A A' not containing B or C.
THEOREM 22. Two ordered triads of points on a real projective line have the same sense if and only if one is transformable into the other by a finite number of elementary transformation*.
Proof. Comparing the definitions of elementary transformation and of segment (§ 22), it is clear that a single elementary trans- formation cannot change the sense of a triad of points. Hence two
420 THKnKKMS ON SKNSE AND SEPARATION [CHAP, ix
triads of points have the same sense if one can be transformed into the other by a tinite number of elementary transformations. The mn verse statement, namely, that a triad A, B, C can be transformed by elementary transformations into any other triad A'B'C' in the same sense-class, follows at once if we establish (1) that ABC can be transformed by elementary transformations into BCA and CAB and (2) that any ordered triad of points A, B, C can be transformed by elementary transformations into one of the six ordered triads formed by any three points A', B', C'.
(1) Let D be a point in the order {ABCD}. Then by elementary transformations we can change ABC into ABD, then into ACD, then into BCD, and then into BCA. By repeating these steps once more ABC can be transformed into CAB.
(2) If A' does not coincide with one of the points A, B, Ct it is on one of the three mutually exclusive segments (§22) of which they are the ends; and by (1) the points ABC may be transformed so that the ends of this segment are B and C. Hence we have {ABA1 C}, and by elementary transformations A B C goes successively into AA'C, BA'C, BA'A, BA'C. If A' does coincide with one of the points A, B, C, the triad ABC may be transformed according to (1) so that A' = A. In like manner the three points A'BC can be transformed into A', B', C in some order, and then A'B'C into A'B'C' in some order.
The proof given for this theorem holds good without change on the basis of Assumptions A, E, S. Cf. § 15.
DEFINITION. An elementary transformation of a triad of points ABC of a line I is said to be restricted with respect to a set of points Pv P2, • • • , Pn if it carries on-s point of the triad, say C, into a point C' such that C and C' are not separated by any pair of the points Pv Py, . . ., Pn. (C or Cf may coincide with any of the points
«••';.£)
It is obvious that any elementary transformation whatever is the
resultant of a finite number of restricted elementary transformations. Hence Theorem 22 has the following immediate corollary :
COROLLARY. Let Plt P3, • • •, Pn be any finite set of points on a line I. Two ordered triads of points of I have the same sense if and only if one is transformable into the other by a finite number of elementary transformation* restricted with respect to Pv P2, • • •, Pn.
6] ELEMENTARY TRANSFORMATIONS 421
The concept of "restricted elementary transit, rniaticm " .-onnects with the intuitive idea of "small motions." Let a line be set into projective corre- spondence with a conic, say a circle. For any n there is a set of points /'r />2>---, J\ on the circle such that the intervals /',/*,, etc. are equal. By increasing n these intervals can be made arbitrarily small, and thus the elementary transformations restricted with respect to Pv /*,,•••, /*„ can be made arbitrarily small.
166. Elementary transformations in a projective plane. I )KFIXITION. Given a set of four points in a projective plane, no three being col- linear, an elementary transformation is the operation of replacing one of them by a point of the same plane joined to the point replaced by a segment not meeting any side of the triangle of the other three points.
THEOREM 23. If ABCD and A'B'C'D' are any two complete quad- rangles in the same projective plane, there exists a finite set of elemen- tary transformations changing the points A, B,C, D into A', B', C1, D1 respectively.
Proof. It can be shown by means of the result for the one-dimen- sional case, just as in the proof of Theorem 18, first that the ordered tetrad ABCD can be carried by elementary transformations into an
ordered tetrad A"B"C"D" of points on the sides of the quadrangle A'B'C'D1 and then that A"B"C"D" can be carried by elementary transformations into some permutation of A'B'C' !>'.
\-2-2 THKOKKMS (>X SENSE AND SEPARATION [CHAV. ix
To complete the proof it is necessary to show that any permuta- tion of the vertices of a complete quadrangle can be effected by elementary transformations.
(liven a complete quadrangle A^A^A^A^ let B^ be the point of intersection of the lines AyAz and A^At, and let Cl and Dl be. two points in the order {A^B^C^D^^}. Let A6 be the point of intersec- tion of A^Cl with A^ and let Cz, 7>2, B3, C8, C4, Dt, Bt be the points defined by the following perspectivities (fig. 83) :
By Theorem 7, Chap. II, it follows that no two of the pairs of points A^A^ A^A^ AfA6, A^AZ, and A3A& are separated by the lines joining the other three of the points Alt Az, As, A4, A&. Hence there exist elementary transformations changing each of the following sets of four points into the one written below it:
A
A^ A^ A^ A^ Hence the permutation
n,=
can be effected by elementary transformations. By changing the notation in H1 it is clear that
n2 =
can be effected by elementary transformations. Hence the product UJlf (i.e. the resultant of Hl applied twice and followed by II2), which is
can also be effected by elementary transformations. Hence any two vertices of the quadrangle can be interchanged by a sequence
§§uw,i67] KKKMKNTAKY TKANSKOKMATIONS 428
of elementary transformations, and IKMKT any permutation of the vertices can be effected by means of elementary iransi'onnatinn.v
167. Elementary transformations in a protective space. I M.KIMTION. Given a set of five points in a protective space, no four of the points being coplauar, an elementary transformation is the operation of replacing any one of them by a point joined to it by a segment not meeting any plane on three of the other four.
It follows from § 27 that the determinant product (25) of § 32 is unaltered in sign by any sequence of elementary transformations of the points whose coordinates are the columns of (21) in § 32. Hence a sequence of elementary transformations cannot carry an ordered pentad of points from one sense-class into the other.
Hence the odd permutations of the vertices of a complete five- point cannot be effected by elementary transformations. That the even permutations can be thus effected may be seen as follows: Let the vertices be denoted by A, B, C, D, E and let the line DE meet the plane ABC in a point F. This point is not on a side of the triangle ABC. Let A' be the point of intersection of the lines FA and BC, B' that of FB and AU - B
CA, and C' that of FC and AB. Let FJG 84
Av be a point in the order
(fig. 84) and Bl the point in which the line FAl meets AC, so that
{AB^B'C}. Let J?2 be a point in the order {AB^BJfC}.
We now can transform ABCDE by elementary transforma- tions successively into AA^DE, AA&DE, BA^DE, BCB^DE, BCADE. Thus the even permutation
(ABCDE\ \BCADEJ
can be effected by elementary transformations. It is easily verifiable that any even permutation is a product of even permutations of this type.
It can be proved by the same methods as in Theorems 18 and 19 that any five points no four of which are coplanar can be carried into some permutation of any other such set of five points. The
4-J4 THKOKK.MS ON SENSE AND SKI'A K A'l'lo.N
of this proof are left as an exercise to the reader. When this is combined with the paragraph above, we obtain
TIIK< -KK.M 24. In a real protective space, S(ABCDE)=S(A' B'C'D' !<:') if and only if there exists a sequence of elementary transformation^ carrying the points A, B, C, D, E into A', B', C', D', E' respectively.
The proof just outlined for this theorem holds good on the basis of Assumptions A, E, S, P. Assumption P comes in because of the use of a coordinate system. This, however, can be avoided ; and the construction of a proof on the basis of A, E, S alone is recommended to the reader as an interesting exercise.
*168. Sense in overlapping convex regions. The discussion of sense in convex regions by means of elementary transformations (as made in §§159—161) is essentially the same for any number of dimensions. Now if two regions of the same dimensionality have a point in common, they have at least one convex region of that dimen- sionality in common. Assigning a positive sense in this region deter- mines a positive sense in each of the given regions. Thus if we have a set of convex regions including all points of a space, we should have, on assigning a positive sense to a tetrad of points in one region, a positive sense determined for any tetrad of points in any of the regions. Since, however, it is in general possible to pass from one region to another by means of different sets of intermediate regions, the possibility arises that this determination of sense may not be unique. In other words, it is logically possible that a given tetrad in a given region might, according to this definition, have both posi- tive and negative senses.
The determination of sense by this method is unique in projective spaces of odd dimensionality and is not unique in projective spaces of even dimensionality. We shall prove this for the two- and three- dimensional cases, but since it reduces merely to a question of even and odd permutations the generalization is obvious.
THEOREM 25. There exists a unique determination of sense for all three-dimensional convex regions in a real projective three-sj>'n •> . but not for all tivo-dimensional convex regions in a real projective plane.
Proof. Consider first the plane and in it a triangle ABC decom- posing it into four triangular regions, which we shall denote by the
§§168,169] OVERLAPPING CONVKX REGIONS 425
notation of § 26, Chap. II. Any one of these regions, say Region I. is contained in a convex region, say 1' (e.g., a Kuclideaii plane with line at infinity not meeting Region I), which contains the boundary of the triangular region. So the determination of sense for Region I c\u 'nds to all the points of its boundary and also to a portion of Region It.
Let the sense of ABC with respect to Region I be positive. The segment 7, one of the segments AJi (fig. 16), is common to the bound- aries of I and II and hence is contained in Region I'. If C' is any point common to I' and II, C and C" are on opposite sides of the line AB in Region I'. Hence, according to § 29, S(BAC') is positive in Region II. Hence S(BAC) is positive with respect to Region II.
Regions II and IV have in common a segment JiC, and thus by a repetition of this argument S(CAB) is positive with respect to Region IV. The latter region has a segment AC in common with Region I, and hence S(ACB) is positive with respect to Region L But by hypothesis S(ABC} is positive with respect to Region I. Hence there is not a unique determination of sense in a real projective plane.
To show that there is a unique determination of sense for a real projective three-space, let a given sense-class S(ABCDK) (cf. §164) be designated as right-handed, and in any convex region let a sense- class S(A'B'C'D'} be right-handed if S(OA'B'C'D'} is positive, where O is interior to the tetrahedron A'B'C'D'. This convention satisfies the requirements laid down above for overlapping convex regions and, by § 167, is unique for the projective three-space.
Any two-dimensional region whatever is, by definition (§155), the set of all points in an infinite set of triangular regions, i.e. in an infinite set of convex regions. In like manner, any three-dimensional region is the set of all points in a set of three-dimensional regions. The method given above may be applied to determine the positive sense-class in all convex regions in a given region R, and R may be said to be two-sided or one-sided according as this determination is or is not unique. Another, slightly different, method of treating this quastion is given in § 173.
*169. Oriented points in a plane. By the principles of duality the lines of a flat pencil or the planes of an axial pencil satisfy the same theorems on order as the points of a projective line.
426 THEOREMS ON SENSE AND SEPARATION H'HAP . ix
This proposition is valid whether the pencils are considered in a projective or in a Euclidean space.*
DKHMTION. In a plane any point associated with one of the sense- classes among the lines on this point is called an oriented point, and a line associated with one of the sense-classes among its points is called an oriented line. Two oriented points are said to be similarly oriented icith respect to a line I if their sense-classes are perspective with the same sense-class in the points of I. By Ex. 1, § 26, if two oriented points are similarly oriented with respect to a line /, they are similarly oriented with respect to a line m if and only if I and m do not separate the two points.
By § 30 a direct collineation of a Euclidean plane transforms any oriented point into one which is similarly oriented with respect to the line at infinity. Hence the oriented points fall into two classes such that any two oriented points of the same class are equivalent under direct collineations and that the two classes are interchanged by any nondirect collineation.
No such statement as this can be made about the oriented lines in a Euclidean plane, because any oriented line can be carried by a direct collineation to any other oriented line. This is obvious be- cause (1) an affine collineation exists carrying an arbitrary Hue to any other line and (2) the two sense-classes on any line are inter- changed by a harmonic homology whose center is the point at infinity of the line.
It is a corollary of the last paragraph that any oriented line of a projective plane can be carried into any other oriented line of the projective plane by a direct collineation. By duality the same propo- sition holds for oriented points in a projective plane.
The oriented points determined by associating the points of a seg- ment 7 with sense-classes in the flat pencils of which they are centers fall into two sets, all points of either set being similarly oriented with respect to any line not meeting 7. These two sets shall be called segments of oriented points and may be denoted by 7<+) and 7*~). If A and B are the ends of 7, the two oriented points deter- mined by A and B and oriented similarly to 7(+) with respect to a
•In general, the geometry of a Euclidean space or, indeed, of any space of n dimensions involves the study of the projective geometry of n — 1 dimensions, in order to describe the relations among the lines, planes, etc. on a fixed point.
ORIENTED POINTS 427
line / not meeting 7 or either of its ends are called the ends of 7(+) and may be denoted by A(" and B(+\ The other two oriented points determined by A and B are the ends of 7(~> and may be denoted by ^(-> and B(~\
In terms of these definitions it is clear that each of the two classes of similarly oriented points determined by a Euclidean plane satisfies a set of order relations such that it may be regarded as a Euclidean plane.
The situation in the projective plane is entirely different. Let us first consider a projective line, and let 7 and 5 be two complementary segments whose ends are A and B. Let A(Jf\ B(+\ A(~\ B(~\ y(+\ 7<-) be defined as above, and let 8(+) and B<~) be the two segments of oriented points determined by S and oriented similarly to -4(+) and A(~y respectively with respect to a line m not meeting 8 or either of its ends. Since A{+) and -B(+) are similarly oriented with respect to I, and A and B are separated by I and m,Aw and B^ are similarly oriented
with respect to m (cf. Ex. 1, § 26, and fig. 85). Hence the ends of Sw are A(+) and &~\ and the ends of S(-> are A^ and B<+\ Hence the oriented points and segments are arranged as follows :
the symbols for segments and their ends being written adjacent.
Let AI} BI} An> Bz be four points in the order {Afl^BJ on a pro- jective line or on a conic. They separate the line (§ 21) into four mutually exclusive segments 7^ Slt 72, S2 arranged as follows:
the symbols for segments and their ends being written adjacent. Letting A^ correspond to Aw, 7, to y+), etc., it is obvious that there is a one-to-one reciprocal correspondence preserving order between the points of a real projective line or conic and the oriented points of a real projective line.
428 THKoKEMS ON SENSE AND SEPARATION [CHAP, ix
Thus, if an oriented {>oint be moved along a projective lim- in such a way that all oriented ]«>int.s of any segment described an- similarly i>rient4'd with respect to a line not meeting the segment, the oriented p«'int must dcsrriU- the line twice before returning to its first position. A motion of this sort will obviously carry any oriented point of the projective plane into any other oriented point. Thus the oriented points either of a ]>n>- jective line or of a projective plane constitute a continuous family in the sense of § 156.
Let TT denote the projective plane under consideration here and let us suppose it contained in a projective space S, and let S' be a Euclidean space obtained by removing from S a plane different from TT which contains the line AB. Let S2 be a sphere of S' tangent to TT at a point Plt let O be the center of £2, and let Pz be the other point in which the line OPl meets the sphere. Let P<+) and P(~) be the two oriented points of TT determined by P^
A correspondence F between the points of the sphere S* and the oriented points of the projective plane TT may now be set up by the following rule : Let PI correspond to JF*+), and PZ to P*-^ ; if X is any point of TT not on the line at infinity, denote by A"t and A'2 the points in which the line OX meets the sphere, assigning the notation so that each of the angles &P1OX1 and 4.P2OX2 is less than a right angle (i.e. so that the points Xl are all on the same side as Pl of the plane through 0 parallel to TT, and the points X2 are on the other side of this plane) ; and denote by X(+y the oriented point of TT deter- mined by X and joined to P^+) by a segment of oriented points con- taining no point of the line at infinity AB, and by X^~} the other oriented point determined by X. Let Xl correspond to X*+\ and Xa to X(~\ If Y is any point of the line at infinity AB, and F(±> one of the oriented points determined by it, F(±) is an end of a segment <r(+) of points X(+) whose other end is P<+) and of a seg- ment <r(~) of points X(~} whose other end is f(~\ The line OF meets the sphere in two points one of which, Y{, is an end both of a seg- ment of points Xl corresponding to o-<+) and of a segment of points Xf corresponding to <r(~\ Let Y. correspond to F(±). This construc- tion evidently makes the oriented point other than lr(:fc> which is determined by Y correspond to the point other than 3^ in which O Y meets the sphere.
The correspondence T is one-to-one and 'reciprocal and makes each segment of oriented points of TT correspond to a segment of points
§§i«y,i70] ORIENTED 1'niNTS 429
on S2. In view of the correspondence between the sphere and the inversion plane, this result may be stated in the following form :
TIIKOREM 26. There is a one-to-one reciprocal correspondence pre- serving order-relations between tlie oriented points of a real protective plane and the points of a real inversion plane.
The treatment of oriented points in this section does not generalize directly to three dimensions, because there is only one sense-class in a projective plane and, therefore, also only one in a bundle of lines. The discussion of sense in terms of the set of all lines through a ^.omt is therefore possible along these lines only in spaces of an even number of dimensions.
A discussion which is uniform for spaces of any number of dimensions can, however, be made in terms of rays. An outline of the theory of pencils and bundles of rays which may be used for this purpose is given in the next three sections,, and an outline of one way of generalizing the contents of the present section is given in § 173.
Another type of generalization of the theory of oriented points in the plane is the theory of doubly oriented lines in three dimensions which is given in § 180, below.
*170. Pencils of rays. The term "ray"* is defined in § 23 for a linear convex region and extended to any convex region in § 148. The definition of angle in § 28 will be carried over to any convex region.
DEFINITION. The set of all rays with a common origin in a planar convex region is called a pencil of rays. The common origin is called the center of the pencil.
The order relations in a pencil of rays are essentially the same as those among the points of a projective line. This can be shown by setting up a correspondence between the rays through the center of a circle and the points in which they meet the circle, as hi § 69. It can also be done on the basis of Assumptions A, E, P alone by proving Theorems 27~33, below. The proofs of the theorems are not given, because they are not very different from those of other theo- rems in this chapter. A third way of deriving these relations is indicated in Theorems 34, 35, and a fourth in Theorems 37-41.
THEOREM 27. If a, I, c are three rays of a pencil, and if any seg- ment joining a point of a to a point of c contains a point of b, then en /•// segment joining a point of a to a point of c contains a point of b.
* In some books the term " ray " is used as synonymous with " projective line," and "pencil of rays" with "pencil of lines."
430 THKOKKMS ON SENSE AND SEPARATION L« HAI-.IX
DEFINITION. If a, b, c are three rays of a pencil, I is said to be between a and c if and only if (1) a and c are not collinear and (2) any segment joining a point of a to a point of c contains a point of /;.
THEOREM 28. If b is any ray between two rays a and c, any otln-r ray between a and c is either between a and b or between b ami c. No ray is both, between a and b and between b and c. Any ray between a and b is between a and c.
THEOREM 29. There is a one-to-one reciprocal correspondence pre- xi- r ring all order relations between the points of a segment of a line and the rays between two rays of a pencil.
THEOREM 30. If three rays a, b, c of a pencil are such that no two of them are collinear and no one of them is between the other two, then any other ray of this pencil is between a and b or between b and c or between c and a.
DEFINITION. Given a set of three distinct rays a, b, c of a pencil, by an elementary transformation is meant the operation of replac- ing one of them, say c, by a ray c' not collinear with c and such that neither a nor b is between c and c'. The class consisting of all ordered triads into which abc is transformable by finite sequences of elementary transformations is called a sense-class and is denoted by S(dbe).
An elementary transformation of abc into abc' is said to be restricted with respect to a set of rays ar an, • • •, an of the pencil if none of the rays alt a0, • • ., an is between c and c'.
THEOREM 31. Let «t, «2, • • •, an be an arbitrary set 6f rays of a pencil. Two ordered triads of rays of the pencil are in the same sense-class if and only if one can be transformed into the other by a sequence of elementary transformations which are restricted with respect to alt aa, • • ., an.
THEOREM 32. Let ajy «2, as be three distinct rays of a pencil such that no one of the three is between the other two. There exists a one- to-one reciprocal correspondence F between the rays of the pencil and the points of a protective line such that to each elementary trans- formation of the rays which is restricted with respect to a^, an, ag there corresponds an elementary transformation on the protective line which is restricted with respect to ' the points corresponding to olf a2, av
§170] PENCILS OF HAYS 431
The correspondence F required in this theorem may be set up as follows : Let three arbitrary collinear jxjints Alt Af, At be the corre- spondents of ajf a2, a8 respectively ; let 1^ be a projectivity which carries the lines which contain the rays between al and aa to the 1« nuts of the segment complementary to AvAtAa and carries the line containing al to Al; for the rays between at and af let F be the correspondence in which each ray between a and a corresponds to the point to which the line containing it is carried by F^ let F2 be the projectivity which carries the lines which contain the rays between «2 and ag to the points of the segment complementary to A,tAlAa and carries the line containing «2 to At ; for the rays between aa and «8 let F be the correspondence in which each ray corre- sponds to the point to which the line containing it is carried by F2; let Fg be a projectivity which carries the lines which contain the rays between «3 and al to the points of the segment complemen- tary to A8A2Al and carries the line containing ag to At', for the rays between ag and «1 let F be the correspondence in which each ray corresponds to the point to which the line containing it is carried
byr,
COROLLARY. There is a one-to-one reciprocal correspondence between the points of a projective line and the rays of a pencil such that two ordered triads of rays of the pencil are in the same sense-class if and only if the corresponding triads of points are in the same sense- class on the line.
THEOREM 33. If a, b, c are three rays of a pencil and a', b', c' are the respectively opposite rays, S(abc) = S(a'b'c').
DEFINITION. If a and b are any two noncollinear rays of a pencil, by an elementary transformation of the ordered pair ab is meant the operation of replacing one of them, say b, by another ray, b', of the pencil, such that no ray of the line containing a is between b and b' or coincident with b'. The set of all ordered pair* (i.e. angles) into which an ordered pair of rays ab can be carried by sequences of elementary transformations is called a sense-class and is denoted by S (ab).
THEOREM 34. If 0 is the center of a pencil of rnyx ami A, B, C, D are points of rays a, b, c, d respectively of the pencil, then S(ab)=S(cd) if and only if S(OAB)= S (OCD).
THI:»>KI:MS <>N SKNSI-: AND SEPARATION [CHAP.IX
Tn KUK KM 35. If a and b are any two noncollinear rays of a pencil, S(ab)3=S(ba). Every ordered pair of noncollinear rays in the pencil is either in S (ab) or in S (ba). If a' is the ray opposite to a, S (ab) * S (a'b).
THKORKM 36. If a, b and a', b' are two ordered pairs of rays of a pencil and c and c' are the rays opposite to a and a' respectively, then S (ab) = S (a'b') if and only if S (abc) = S (a'b'c1). The same conclusion holds if c is any ray between a and b and c' any ray between a' and b'.
THEOREM 37. DEFINITION. The points not on the, sides or vertex of an angle £ ab fa^ into two classes having the sides and vertex as boundary and such that any segment joining a point of one class to a point of tlw other contains a point of the sides or the vertex. If the angle is a straight angle, both of these classes of points are convex regions. If not, one and only one of them is convex and is called the interior of the angle ; the other is called the exterior of the angle.
THEOREM 38. If A' is any point of the side OA of an angle 4 A OB, and B' is any point of the side OB, then S(OAB) = S(OA'B'). If C is any point interior to the angle, S(OAB) = S(OA C) = S(OCB), and any point C satisfying these conditions is interior to the angle.
THEOREM 39. Any ray having the vertex of an angle as origin, and not itself a side of the angle, is entirely in one or the other of the two classes of points described in Theorem 37. If it is in the interior it contains one and only one point on each segment joining a point of one side of the angle to a point on the other side.
DEFINITION. Two rays a, b of a pencil are said to be separated by two other rays h, k of the same pencil (or by the angle 4/'/') if and only if a is in one and b in the other of the classes of points determined according to Theorem 37 by 4.hk. A set of rays having a common origin are said to be in the order {a^a^a^a^ ' • - an} if no two of the rays are separated by any of the angles 4 «,«2' 4 «2a,, • • •, 4 «„_!«„, 4 <W
THEOREM 40. A set of rays in the order {tfj«2#8 • • • #„_!#,,} a^ also in the orders {a2«8 • • • «„«.} and {anan_ t • • • a,^}.
COROLLARY. Any two rays a, b having a common origin are in the orders {ab} and {ba}. Any three rays a, b, c having a common origin are in the orders {abc}, {bca}, {cab}, {acb}, {bac}, {cba}.
§§170,171] DIRECTIONS 4:1:5
Til in IK KM 41. To any finite number n S 2 of rays having a com- mon origin- nun/ be aligned a notation so that they are in the order
aH}.
*171. Pencils of segments and directions. The notion of a ray belongs essentially with that of a convex region, but the theorems of the last section may easily be gut into a form which is nut limited to convex regions. The proofs are all omitted for the same reasons as in the section above.
DEFINITION. A set of all segments having a common end and lying in the same plane is called a pencil of segments. The common end is called the center of the pencil. Two segments or intervals having a common end A are said to be similarly directed at A if either of them is entirely contained in the other. The set of all segments similarly directed at a given point with a given segment is called a direction-class or, more simply, a direction. The set of all directions of the segments of a pencil at its center is called a pencil of direc- tions. The directions of two collinear segments having a common end A and not similarly directed are said to be opposite, and the two segments are said to be oppositely directed at A.
Thus if A BCD are four collinear points in the order {ABCD} the segments ABC and ABD are similarly directed, while ABC and ADC are oppositely directed. At a given point on a given line there are obviously two and only two directions, and these are opposite to each other. Two noncolliuear segments with a common end are con- tained in one and only one pencil, namely, the one having the common end as center and lying in the plane of the two segments.
DEFINITION. A segment a- is said to be between two noncollinear segments <rlf cr2 if the three segments are in the same pencil and (T is similarly directed with a segment which is in the pencil and contained entirely in the triangular region determined by <rl and <rs (Theorem 12). A direction d is said to be between two noucollinear directions dlt d.2 if there exist three segments <r, o-^ tro in the direc- tions d, d^y d3 respectively such that a- is between <r^ and <rf
This extension of the notion of betweenness to directions is justified by the following theorem.
THEOREM 42. If a and ft are two noncollinear segments vritb a common end 0, and a' and ft' are similarly directed unth a and ft
434 TIIKollEMS ON SENSE AND SEPARATION [CHAP.IX
ns/xctifili/ at O, the segments between a and /9 are similarly directed with the segments between a' and (3'.
I)KFiMii"N. Let o-j, <r2, <r8 be three segments of a pencil no two of tin-in U'ini,' similarly directed. By an elementary transformation is meant the oj>e ration of replacing one of them, say <rg, by a seg- ment <r4, which is in the pencil and such that neither <r^ nor <r,2 is between trt and <r4 or similarly directed with cr4. A class consisting of all ordered triads into which 0\<72<r8 is transformable by finite sequences of elementary transformations is called a sense-class and is denoted by S(o-^r2<T^. If dlt d3, da are three directions of a pencil, and 0-j, <r2, <r8 three segments in the directions rft, d^, d8 respec- tively, the sense-class S(djdad^) is the class of all triads of directions which are the directions of triads of segments in the sense-class
8(*??±
THEOREM 43. If <rlt <rz, <r3 are three segments of a pencil, no two of them being similarly directed, and <r'a is similarly directed with
'* tftw^^K'X).
THEOREM 44. There is a one-to-one reciprocal correspondence be- tween the directions of a pencil and the points of a line such that two triads of directions are in the same sense if and only if the corresponding triads of points have the same sense.
We now take from §§ 21—23 of Chap. II the definitions of sepa- ration, order, etc., and on account of Theorem 44 we have at once
COROLLARY 1. The Theorems of §§ 21-23 remain valid when applied to the directions of a pencil instead of to the points of a line.
COROLLARY 2. Two pairs of opposite directions separate each other.
DEFINITION. Let <TI and <ra be two noncollinear segments of a pencil ; by an elementary transformation is meant the operation of replacing one of them, say <r2, by any segment <rg of the pencil such that no segment collinear with <TI is between <72 and <TS. The set of all ordered pairs of segments into which o^o^ is transformable by sequences of elementary transformations is called a sense-class and is denoted by 8(0-^^.
THEOREM 45. If a pair of segments v^ <r2 is transformable by elementary transformations into a pair 0"^r2, then 0^0^ is transform- able by elementary transformations into <r<trf
§§171,172] BUNDLES OF HAYS 435
THEOREM 46. If a, segment <r^ is similarly directed with a segment a'n and not collincar with a Mi/incut <r^ whit-It In at the same origin 01 o-... *(°Va) = *s'(°X)-
THEOREM 47. If <rlt <rs, <rs, <r4 are segments of a pencil and <rl is not collinear ivith <r2, nor <TS with <r4, then either S^^o-J = *S'(o-3o-4) or ^Va^i) = -^(""a0*)- ^(^i^z) ^ 's'(0Vr,)- If °"! w opposite to <T
DEFINITION. Let d^ and dz be two directions of a pencil and let <TI and <r2 be two segments in the directions dl and dt respectively. By the sense-class S(d]d^) is meant the class of all ordered pairs of directions which are the directions of ordered pairs of segments in the sense-class 18(0-^^.
It is evident that the last two theorems may be restated, without material change, in terms of directions instead of segments.
*172. Bundles of rays, segments, and directions. DEFINITION. The set of all rays in a three-dimensional convex region which have a common origin 0 is called a bundle of rays. The point 0 is called the center of the bundle.
Let a, b, c be three noncoplanar rays of a bundle. By an elemen- tary transformation is meant the operation of replacing one of the rays, say a, by a ray a' such that no ray of the plane containing b and c is between a and a'. The set of all ordered triads of rays into which abc can be carried by sequences of elementary trans- formations is called a sense-class and is denoted by S(abc).
THEOREM 48. If abc and a'b'c' are two ordered triads of non- coplanar rays having a common origin 0, and A, B, C, A', B', C1 are points of the rays a, b, c, a', b', c' respectively, then S(abc) = S (a'b'c') if and only if S(OABC) = S(0 A'B'C'}.
THEOREM 49. If a, I, c are three noncoplanar rays of a bundle, S(abc) = S(bca) ¥= S(acb). If a', b', c' are any other three noncoplanar rays of the bundle, either S (a'b'c') = S(abc) or S (a'b'c') = S(acb).
THEOREM 50. If a, b, c are three noncoplanar rays of a bundle and a' is the ray opposite to a, S(abc) =£ S(a'bc).
THEOREM 51. If abc are three noncoplanar rays of a bundle, the set [x] of rays of the bundle which satisfy the relation S(xab) = S(xbc) = S(xca) are in such a one-to-one reciprocal correspondence T with the points of a triangular region ///"/ //' r>iys x^, x%, *,,
THKOKKMS <>N SKNSK AND SKPAKATION [CHAI-. i\
to points Xlt A'a> Xa, A\, A'6, A'6 respecting/, /,*,) = S(xtxfixt) if and only if X(A\A\A'J = S(A\A\A\). If A, B, C are point* of the rays a, b, c respectively, and the triangular region, is the interior of thi triangle ABC, F may be taken as tin' spondence in which each x corresponds to the point in which it meets the triangular region.
THEOKF.M .V_'. If a, b, c, d are four rays of a bundle such that any plane containing two of them contains a ray between tiie other two, mil/ othtr ray of the bundle is between two rays of the set a, b, c, d or in one of four sets [x], [y], [2], \w~\ such that [x\ satisfies the condition S(xbc)=S(xcd) = S(xdb), [y] satisfies S(yac) = S(ycd)= S(yda), [z] sat- isfies S(zab) = S(zbd) = S (zda), [iv] satisfies S(wab) — S(wbc) = S(wca).
COROLLARY. Under the conditions of the theorem if A, B, C, D are points of the rays a, b, c, d respectively, the center of the bundle is interior to the tetrahedron A BCD.
DEFINITION. A set of all segments having a common end is called a bundle of segments. The set of all directions of the segments of a bundle is called a bundle of directions.
DEFINITION. Let a-^ o-2, <rg be three segments of the same bundle, but not in the same pencil ; the operation of replacing any one of them, say <rs, by a segment <74 of the bundle such that no segment of the pencil containing <r^ and <r2 is between <7g and <r4 or coincident with <r4 is called an elementary transformation. A class consisting of all ordered triads of segments into which tr^^i cau be carried by finite sequences of elementary transformations is called a sense-class and is denoted by £(°"i°Vr3)-
The generalization of Theorems 48-52 to the corresponding theorems for a bundle of segments presents no difficulty.
* 173. One- and two-sided regions. A discussion of the order rela- tions in projective spaces which is closely analogous both to § l'-S and to § 169 may be made according to the following outline. The details are left as an exercise for the reader.
Let 0 be any point of a planar region R. Let A, B, C be the vertices of a triangular region T containing 0 and contained in R, and let a, $, 7 be the segments in R joining 0 to A, B, C respec- tively. Then S(a/3) = S(/3v) = S(ya).
If O' is any other point of T, and a', f-y the segments of R joining O* to A and B respectively, S(a/3) is said to be like S(a'/3'); and
§§17:5,174,175] SENSE-CLASSES ON A SI'IIKKK 487
if *("£) is like *(«'£'), and S(a'ft') like S(a"ft"), then S(a/3) is said to be like S(a"ft"). A region for which a given sense-class at one point is like the other sense-class at that point is said to be one- fiit/fil. Any other region is said to be two-sided.
A con vex region is two-sided. A projective plane is a one-sided region.
Let 0 be any point of a three-dimensional region R. Let A, B, C, D be the vertices of a tetrahedral region T containing O and contained in R, and let a, ft, y, 8 be the segments in R joining O to . [, B, C, D respectively. Then S(afty) = S(ftaS) = S(Byft) = S(y8a).
If 0' is any other point of T, and a', ft', y are the segments of R joining 0' to A, B, C respectively, S(afty) is said to be like S(a'ft'y'); if S(afty) is like S(a'ft'y'), and S(a'ft'y') is like S(a"ft"y"), then S(a/5y) is said to be like S(a"ft"y")-
One- and two-sided regions are defined as in the two-dimensional case.
Any region in a three-dimensional projective space is two-sided.
174. Sense-classes on a sphere. The theorems in § 172 can be regarded as defining the order relations among the points of a sphere if carried over to the sphere by letting each point of the sphere correspond to the ray joining it to the center of the sphere. Another way of treating the order relations on a sphere and one which connects directly with § 97 is as follows:
DEFINITION. Let A, B, C, D be four points of a sphere not all on the same circle. By an elementary transformation is meant the operation of replacing one of them, say A, by a point A' on the same side of the circle BCD. The set of all ordered tetrads into which ABCD is transformable by sequences of elementary transfor- mations is called a sense-class and is denoted by S(ABCD).
THEOREM 53. There are two and only two sense-classes on a sphere. S(ABCD}* S(ABDC}.
THEOREM 54. S(ABCD) = S(A'BfCfD>) if and only if bb'>O, where U(AB, CD) = a + bV^J, K(A'B', C'D')= a' + J'V^7, <tll,l a, a', b, b' are real.
175. Order relations on complex lines. In view of the isomor- phism between the geometry of the real sphere and the complex projective line (cf. §§ 91, 95, and 100) the theorems of the section above and of § 97 determine the order relations on any coinjilf\ line.
438 THKoiiKMs ON SKNSE AND 8EPAEATION [CHAP.IX
One very important difference between the situation as to order in the real and tlu» complex spaces is the following: In a real plane or space one sense-class on a line is carried by projectivities of a continuous group into both sense-classes on any other line. So that fixing a particular sense-class on one line as positive does not deter- mine a positive sense-class on all other lines. On a complex line, however, an ordered set of four points ABCD is in one sense-class or the other according as b is positive or negative, where a •+- b V— 1 = &(AB, CD) and a and b are real (Theorem 54). In conse- quence of the invariance of cross ratios under projection, a given sense-class on one line goes by projectivities into one and only one sense-class on any other line. Hence if one sense-class is called positive on one line, the positive sense-class can be determined on every other line as being that sense-class which is projective with the positive sense-class on the initial line.
This connects very closely with the convention for purposes of analytic geometry that by Vc is meant that one of the square roots of c which takes the form a -f- b V — 1, where a and b are real and b > 0, or if b = 0, a > 0. The symbol V— 1 is taken to represent that one of the square roots of — 1 for which S(<x> 0 1 V— l) is positive.
176. Direct and opposite collineations in space. From the algebraic definition of direct collineation in terms of the sign of a determinant, we obtain at once
THEOREM 55. Any collineation of a real three-dimensional pro- jective space which leaves a Euclidean space invariant is direct if and only if the collineation which it effects in the Euclidean space is direct.
In a Euclidean space a point D is on the same side of a plane ABCD with a point E if and only if S(ABCD) = S(ABCE}. Hence a homology whose center is at infinity is direct or opposite according as a point not on its plane of fixed points is transformed to a point on the same or the opposite side of this plane. Extending this result to the projective space by the aid of the theorem above we have
THEOREM 56. A homology which carries a point A to a point A', distinct from A, is opposite or direct according as A <and A' are separated or not separated by the center of the homology and the point in which its plane of fixed points is met by the line A A'.
§176] COLLINEATIONS IN SPACE l:l«»
COROLLARY 1. A harmonic hvmology is opposite. COROLLARY 2. The inverse of a direct homology is direct.
Since any collineation is expressible as a product of homologies (§ 29, Vol. I), it follows that
COROLLARY 3. The inverse of a direct collineation is direct.
Since an elation is a product of two harmonic homologies having the same plane of fixed points it follows from Cor. 1 that
COROLLARY 4. An elation is direct.
Since a line reflection (§ 101) is a product of two harmonic homologies,
COROLLARY 5. A line reflection is direct.
THEOREM 57. A collineation leaving three skew lines invariant is direct.
Proof. Denote the lines by llt 12, 13 and the collineation by F. The projectivity on ^ which is effected by F is a product of two or three hyperbolic involutions (§ 74). Each involution on ^ is effected by a line reflection whose directrices are the lines which pass through the double points of the involution and meet 13 and /g. The product II of these line reflections leaves l^ lz, la invariant and effects the same transformation on ^ as F. Hence II^F leaves /2, /8 and all points on ^ invariant. It also leaves invariant any line meeting llt lt, and /g, and hence leaves all points on /2 and /g in variant. Hence H^F is the identity, and hence F = H. Since the line reflections are all direct, F is direct.
COROLLARY 1. Any collineation leaving all points of two skew lines invariant is direct.
Proof. Such a colliueation leaves invariant three skew lines meeting the given pair of invariant linea
COROLLARY 2. Any collineation transforming a regulus into itself is direct.
Proof. Such a collineation is a product of a collineation leaving all lines of the given regulus invariant by one leaving all lines of the conjugate regulus invariant. Hence it is direct, by the theorem.
Corollary 2 is also a direct consequence of Cor. 5, above, and Theorem 34, Chap. VL
440 THKoKKMS ON SENSE AND SEPARATION [CHAP.IX
COROLLARY 3. Any collineation carrying a regulus into its con- jn'i'itt rcyulus is opposite.
Proof. The two reguli are interchanged by a harmonic homology whose center and axis are pole and polar with regard to the regulus. This harmonic homology is opposite by Cor. 1, Theorem 56, ami since its product by any collineation F interchanging the two reguli leaves them both invariant and hence is direct by Cor. 2, it follows that F is opposite.
DEFINITION. By a doubly oriented line is meant a line / associated with one sense-class among the points on I and one sense-class among the planes on I. The doubly oriented line is said to be on any point, line, or plane on I.
A doubly oriented line may be denoted by the symbol (ABC,a/3y) if A, B, C denote collinear points and a, /S, 7 planes on the line AB. For this symbol determines the line AB and the sense-classes S(ABC) and S(aj3y) uniquely. Since there are two sense-classes S(ABC) and S(ACB) among the points on a line AB and two sense-classes S(a/3y) and S(ayl3) among the planes on AB, there are four doubly oriented lines, (ABC a 6 )
(ACB, ay/3), (ABC, ayfy,
(ACB, a/3y), into which AB enters.
THEOREM 58. The collineations which transform a doubly oriented line into itself are all direct.
Proof. Let (ABC, afiy) be a doubly oriented line, F a collineation leaving it invariant, I any line not meeting AB, and /'= F(/). The line /' cannot meet AB, because AB is transformed into itself by F« If I' does not intersect /, let ra be the line harmonically separated from AB by I and I' in the regulus containing AB, I, and I'. If /' meets / let m be the line harmonically separated by I and I' from the point in which the plane IV is met by AB. In either case . I /•' does not intersect m, and if A is the line reflection whose directrices are AB and m, A(/') = I. Hence AF leaves both AB and m invariant. Since A and F preserve sense both in the pencil of points AB and in the pencil of planes a/3, AF preserves sense both on AB and on m. Hence by § 74, AF effects a projectivity on AB which is a
§§i76,m] DOUBLY ORIENTED LINE 441
product of two hyperbolic involutions, {/'4/'s} • {/J3/J,}, and it effects a projectivity on m which is a product of two hyperbolic involutions,
K^y!' ' l^./'V' Let 'i» 's» 's» ^ ^ ^ie ^ne8 A^i' ^»^a» •'>i^»» ^'4^4
respectively. The product
leaves all points on AB and on m invariant and is therefore direct by Cor. 1, Theorem 57. All the collineations in this product except F are direct by Cor. 5, Theorem 56. Hence F is direct.
COROLLARY 1. Any collineation which reverses both sense-classes of a doubly oriented line is direct.
Proof. Let T be a collineation reversing both sense-classes of a doubly oriented line (ABC, cc/3y). Let a and b be two lines meeting ; 1 /.' but not intersecting each other. The line reflection {ab} reverses both sense-classes of (ABC, a/3y) and is direct. Hence {ab} • T leaves them both invariant and is direct by the theorem. Hence F is direct.
COROLLARY 2. Any collineation which transforms each of two skew lines into itself and effects a direct projectivity on each is direct.
COROLLARY 3. Any collineation which transforms each of two skew lines into itself and effects an opposite projectivity on each is direct.
177. Right- and left-handed figures. The theorems of the last section can be used in showing that other figures than the ordered pentads of points may be classified as right-handed and left-handed. For this purpose the following theorem is fundamental.
THEOREM 59. If the collineations carrying a figure F0 into itself are all direct, the figures equivalent to FO under the group of all collineations fall into two classes such that any collineation carrying a figure of one class into a figure of the same class is direct and any wllineation carrying it into a figure of the other class is opposite.
Proof. Let [F] be the set of all figures into which F0 can be carried by direct collineations. There is no opposite collineation carrying F into an F; for suppose F were such an opposite collin- eation, let P be one of the direct collineations which by definition of [F] carry /' into F; then P-ir would be an opposite collineation carrying Fn into itself. In like manner it follows that any collinea- tion carrying any F into itself or any other F is direct.
442 TH KOI: K.MS ON SENSE AND SEPARATION [CHAP.IX
Let [Ft] be the set of all figures into which F0 is carried by oppo- site collineations. Aii argument like that above shows (1) that any cnllineation carrying FQ into an F' is opposite and (2) that any colliiu'utioii carrying an F1 into itself or another F' is direct. It follows at once that any collineation carrying an F into an F' or an F' into an F is opposite.
Since the direct colliueations form a continuous family of trans- formations, we have
COROLLARY. The figures conjugate to F^ under the group of direct collineations form, a continuous family.
The propositions about the sense-classes of ordered tetrads of non- collinear points are corollaries of this theorem because the only collineation carrying an ordered pentad of noucollinear points into itself is the identity.
By Theorems 57 and 59 all triads of noncollinear lines fall into two classes such that any collineatiou carrying a triad of one class into a triad of the same class is direct and any collineation carrying a triad of one class into a triad of the other class is opposite. It is to be noted particularly that the triads of lines here considered need not be ordered triads, since by Cor. 2, Theorem 57, the collineation effecting any permutation of a set of three noucollinear lines is direct.
Similar propositions hold with regard to doubly oriented lines, reguli, congruences, and complexes (cf. § 178).
Let us now suppose that a particular sense-class S(ABCD) in a Euclidean space has been designated as right-handed (cf. § 162). Any ordered tetrad of points in this sense-class is also called right- handed and any ordered tetrad in the other sense-class is called left-handed.
Let P be a point interior to the triangular region BCD, Q the point at infinity of the line AP, ft the plane APB, 7 the plane APC, and 8 the plane APD. All doubly oriented lines into which (APQ, fiy&) is carried by direct collineations shall be called right-handed and all others shall be called left-handed.
The set of points ABCDQ and the sense-class S(ABCDQ) in the projective space ABCD shall be called right-handed and all other ordered pentads of noncollinear points a'nd the other sense-class shall be called left-handed.
§§177,178] KIGHT- AND LEFT-HAM >K1> I N.I I; I :s 443
These conventions give the same determination of right-handed doubly oriented lines and ordered pentads of points no matter what point of the triangular region BCD is taken as P, because any col- liueation leaving A, B, C, D invariant and carrying one such P into another is direct. In like manner these conventions are independent of the choice of AH CD, so long as S(ABCD) is direct.
A triad of skew lines llt /a, /8 shall be said to be right-handed or left-handed according as the doubly oriented line (ABC, afiy) is right-handed or left-handed, provided that m is a line meeting lt, /.,, /3, and A, B, C are the points nillt ra/a, mlt respectively, and a, ,8, 7 are the planes mllt mla, ml3 respectively.
This convention is independent of the choice of m, by Theorems 57 and 58. By the same theorems any collineation carrying a right-handed triad of noncollinear lines into a right-handed triad of lines is direct, and any colliueation carrying a right-handed triad of lines into a left-handed triad is opposite.
The reader should verify that a pair of skew lines Im in a Euclid- ean space is right-handed or left-handed in the sense of § 163 according as lmlx is right-handed or left-handed, /x being the line at infinity which is the absolute polar of the point at infinity of /. If mx is the absolute polar of the point at infinity of m, lmmx is right-handed if and only if lmlx is right-handed.
Let A be a point of the axis of a twist F in a Euclidean space, let B = T(A), and let C be the point at infinity of the line AB ; let a be any plane on the line AB, ft = F(«), and 7 the plane on AB perpendicular to a. Then F is right-handed in the sense of § 163 if and only if the doubly oriented line (AJiC, afiy) is right-handed. This is easily verified.
178. Right- and left-handed reguli, congruences, and complexes. By Cor. 2, Theorem 57, every triad of lines in a regulus is right- handed or every triad is left-handed. In the first case the regulus shall be said to be right-handed and in the second case to be left-handed,
THEOREM 60. The collineations which leave an elliptic linear con- gruence invariant are all direct.
Proof. An elliptic congruence has a pair of conjugate imaginary lines as its directrices (§ 109), and there is one real line of the
444 THKt'KK.MS ON SENSE AND SEPARATION [CHAP.IX
congruence through each point of a directrix. Any collineation F which carries each directrix of the congruence into itself effects a projectivity on that directrix. This projectivity is a product of two involutions (§ 78, Vol. I). Each involution may be effected by a line reflection whose lines of fixed points are the (real) lines of the congruence through the (imaginary) double points of the involution ; since such a line reflection leaves both directrices invariant, it leaves the congruence invariant. Hence there exist two line reflections Aj, As, each transforming the congruence into itself, such that A AjF leaves all points on a directrix invariant. Hence A2AtF transforms each line of the congruence into itself. By Theorem 57, A.^AjF is direct, and by Theorem 56, Cor. 5, At and A2 are direct. Hence F is direct.
If / is any real line not in the congruence, the lines of the con- gruence meeting I form a regulus, and the directrices are double lines of an involution in the Hues of the conjugate regulus. If I' is the line conjugate to I in this involution, the line reflection {II'} must interchange the two directrices. Hence if F' is any colline- ation interchanging the directrices, {II'} • F' is a collineation which leaves each of them invariant. Hence by the paragraph above {//'} • F' is direct. Hence F' is direct. Hence any real collineation leaving an elliptic linear congruence invariant is direct.
COROLLARY 1. The triads of lines of an elliptic linear congruence are all right-handed or all left-handed.
For any triad can be carried into any other triad by a direct collineation.
COROLLARY 2. If four linearly independent lines are such that all sets of three of them are right-handed or such that all sets of three of them are left-handed, the linear congruence which contains them is elliptic.
An elliptic congruence shall be said to be right-handed if every triad of lines in it is right-handed ; otherwise it is said to be left-handed.
A pair of conjugate imaginary lines of the second kind (§ 109) is said to be right-handed or left-handed according as it is deter- mined by a right-handed or a left-handed congruence.
A pair of Clifford parallels (§ 142) is said to be right-handed or left-handed according as the congruence of Clifford parallels to which
§178] RIGHT- AND LEFT-HANDKD COMIM.KXI-.S 445
they belong is right-handed or left-handed. This distinction is in agreement with that introduced in § 142, because according to Ixith definitions a colliueation carrying a system of right-handed Clifford parallels into a system of right-handed ones is direct, and a colliuea- tioii carrying a system of right-handed Clifford parallels into a system of left-handed ones is opposite.
THEOREM 61. Tlie collineations which leave a nondegenerate linear complex invariant are all direct.
Proof. Let F be a collineation leaving a complex C invariant, and let I be any line of C and V = F (1). Let I" be any line of C not meeting / or I'. The lines of C which meet / and /" constitute a regulus, and three lines of this regulus together with I and /" con- stitute a set of five linearly independent lines (§ 106, Vol. I) upon which, therefore, all the lines of C are linearly dependent. Hence a collineation F' which leaves this regulus invariant and inter- changes I and I" leaves C invariant. Let F" be a collineation, simi- larly obtained, which interchanges I" and I' and leaves C invariant. The product F'F'T leaves C and I invariant, and F' and F" are direct.
Any collineation leaving C and I invariant leaves invariant the projectivity II between the points on / and the planes corresponding to them in the null system determined by C. The projectivity II transforms an arbitrary sense-class among the points on I into an arbitrary sense-class among the planes on /. These two sense-classes determine a doubly oriented line, I. The other sense-class of the points on / is carried by II into the other sense-class of planes on I, and these two sense-classes determine a doubly oriented line i. Since any collineation leaving C and I invariant leaves II invariant, it either transforms this doubly oriented line into itself or into the one obtained by reversing both its sense-classes. Hence any such colliueation is direct by Theorem 58 and its first corollary. In particular FT'T is direct, and since F' and F" are direct, it follows that F is direct.
By Theorem 61 all the doubly oriented lines analogous to 7 which are determined by C are all right-handed or all left-handed. In the first case C shall be called right-handed, and in the second case C shall be called left-handed.
44(1 THKDKK.MS <»N SKNSK AND SEPARATION [CHAP.IX
Tin- algebraic criteria hi the exercises below are taken from the article by E. Study referred to in $ 162. See also F. Klein, Auto- graphierte Vorlesungeii liber nicht-Euclidische Geometric, Vol. II, Chap. 1. Uottingen, 1890.
EXERCISES
1. Classify parabolic congruences (§107, Vol. I) as right-handed and left- handed.
2. For two lines p and p' let
(/>» X) = PuPu + PuPa + Pit Pn + P**P\* + PttPi* + PrnPu* where p^ are the Plucker coordinates (§ 109, Vol. I) of p, and p'{j those of //. Three lines /», />', f>" are right-handed or left-handed according as
(/>»/)•(/»'»/»")• (P">P)
is j>ositive or negative.
3. A pair of conjugate imaginary lines of the second kind whose Pliicker coordinates are p^ and p^ respectively are right-handed or left-handed ac-
PuPu + Pi* T'ia + PuPu + PMPat + Pta />42 + PnPn is positive or negative.
4. The linear line complex whose equation in Pliicker coordinates is (§110, Vol. I)
is right-handed or left-handed according as
a!2rt34 + °13rt42 + «14n28
is positive or negative.
5. A twist given by the parameters of §130 is right-handed if a:0/?0>0 and the coordinate system is right-handed.
6. The linear complex C determined by a twist according to Ex. 7, § 122, is right-handed or left-handed according as the twist is right-handed or left- handed.
*179. Elementary transformations of triads of lines. Let F0 be a figure such that all the collineations which transform it into itself are direct, and let [F] be the set of all figures equivalent to F under direct transformations. From the fact that the group of all direct collineations is continuous, it can be proved that [F] is a continuous family of figures.
This can also be put into evidence by generalizing the notion of elementary transformation to other figures. This is essentially what. has been done in §§ 169 and 173. For- triads of skew lines the following theorem is fundamental.
§§179,180] ELEMENTARY TRANSFORMATION 447
Til KOI; KM 62. If /j, lt, la are three skew lines, and 14 is a ////.• coplanar with /g and such that the points in which ^ and lf meet the plane l^ are not separated by the lines /8 and lt, then l£J,t can be carried to IJJ.^ by a direct collineation.
Proof. Let a be a line meeting /lf la, and lt (fig. 86) in points Alt A3, At respectively, which are all distinct. Let a be the plane n .11- taining la and the point B of intersection of lt and /4. If Al is in a, an elation with A1 as center, a as plane of fixed points, and carrying As to A4 will carry llf lt, la into llt lz, 14 respectively. -By Theorem 56, Cor. 4, this elation is direct.
If A^ is not in a, the points A9 and At are not separated by
A1 and the point A in which a Fn. go~
meets a ; for by hypothesis Av
and the point in which lz meets the line EA are not separated by the lines la and lf Hence the homology with Al as center and a as plane of fixed points which carries /3 to /4 is direct (Theorem 56). This homology carries l^ la, 13 into llt lz, 14 respectively.
An elementary transformation of a triad of skew lines IJJ^ ma)- be defined as the operation of replacing one of them, say Jg, by a line 14 which is coplanar with 19 and such that 13 and /4 do not sepa- rate the points in which their plane is met by ^ and ly.
By Theorem 62 an elementary transformation may be effected by a direct collineation. A sequence of elementary transformations therefore carries a right-handed triad into a right-handed triad and a left-handed triad into a left-handed triad.
Conversely, it can be proved that any right-handed triad can be carried into any right-handed triad by a sequence of elementary transformations and that the two classes of lines determined by a pair of skew lines ab according to Ex. 2, § 25, are the lines [x] such that abx is right-handed and the lines [y] such that aby is left-handed. These propositions are left to the reader.
* 180. Doubly oriented lines. The theory of sense-classes in three dimensions could be based entirely on that of doubly oriented lines (§ 176). We shall prove the earliest theorems of such a theory in
448
THKOUKMS ON SENSE AND SEPARATION [CHAP, ix
this si-t tion. The proofs are based on Assumptions A, E, S and d<- not make use of the preceding discussions of order in three-space.
DEFINITION. Two doubly oriented lines are said to be doubly perspective if they can be given the notation (ABC, a@y) and (A'Jf'C1, a'fi'y') respectively in such a way that A, B, C, a, /9, 7 are on «', /S', 7', A1, B', C' respectively. Two doubly oriented lines lt and / are said to be similarly oriented if and only if there exists a sequence of doubly oriented lines l^ lz, • • •, /„ such that 10 is doubly perspective with llt lv with /2, • • •, ln_^ with ln, and ln with /. Two doubly oriented lines which are not similarly oriented are said to be oppositely oriented.
From the form of this definition it follows immediately that
THEOREM 63. If a doubly oriented line l^ is similarly oriented with a doubly oriented line /„, and la with a doubly oriented line ly, /t is similarly oriented with lf
THEOREM 64. If three doubly oriented lines mQ, m^ m2, no two of which are coplanar, are such that raQ is doubly perspective with m^ and m1 with mn, then m2 is doubly perspective either with m0 or with the doubly oriented line obtained by changing both sense-classes on mtf
Proof. Let ABC be an ordered set of points of the sense-class of points of mQ and let /0, llt lz be the three lines on A, B, C respectively which meet ml and mf The sense-class of planes of raQ contains either the ordered triad of planes Iom0, ^m0, 12m0 or the ordered triad Zm,
case (fig. 87)16^=*,^ = A
I2m0 = 7. In the second case
FIG. 87
/,w0 = 7. In both cases let
At, Blf Cl be the points I0mlt
/,7/ij, I2m1 respectively, alt /3lf
7, the planes I0mlf ljn,lt I2ml
respectively, A^ BZ, C2 the points ZQw2,
a,» ^,, 7« the planes / ma, l,m , I m respectively.
J»*2 021222 *• *
In the first case (ABC, aj3y) is doubly perspective with (AJ^C^ "1^1) and tnis with (AZBZC<I> aAvt)- Since m0 = (ABC, a &i), and
respectively, and
§180]
DOUBLY ORIENTED LINKS
449
»/to is doubly perspective with mlt ml = (A1B1(\, a,^^); and since MJ is doubly perspective with mt, mt = (AaBtCt, «J3ty.J. But by construction (ABC ,
J \ 2 2 3*
«j3a72) is doubly perspec- tive with (ABC, afty), Le. 7/ia is doubly perspec- tive with w0.
In the second case (ABC, afty) is doubly per- spective with (A .C B ,
FIG. 88
afty), and
m1 is doubly perspective with mo, m^^C^,
afiifi)> and since 77^ is doubly perspective with raa, wia = (^.B^, «o7o/30). But by construction (A^B.f^ a^y^ft^ is doubly perspective with (ACB, ay ft)', i.e. w2 is doubly perspective with the doubly oriented line obtained by changing both sense-classes of TOO.
THEOREM 65. A doubly oriented line (ABC, afty) is similarly oriented with (ACB, ay ft) and oppositely oriented to (ABC, ay ft) and (ACB, afty}.
Proof. Let ?0, llf /2 be three lines distinct from AB and such that 1Q is on A and a, ^ on B and 7, 12 on (7 and # Let ml and ma be two lines distinct from AB, each of which meets 10, llt and /2. Let -4X, Blt Cl be the points l^m^ ^m^ lyml respectively and af, ftt, yt the planes Ijm.^ Ijn^ I2ml respectively ; let Ay, B^, Ct be the points ^m2> ^imaj ^W12 respectively and «2, ft0, 72 the planes /oma, ^m^, /O?HS respectively. Then by construction (tig. 88) and definition the oriented line (ABC, afty) is doubly perspective with (A^Cfl^ «,/3171), and this with (^2#2£v #.,7.,$,), and this with (ACB, ay ft). Hence (ABC, afty) is similarly oriented with (ACB, ay ft). By a change of notation it is evident that (ABC, ay ft) is similarly oriented with (ACB, afty). It remains, therefore, to prove that (ABC, afty) is not similarly oriented with (A CB, afty).
If these two oriented lines were similarly oriented, there would be a sequence of doubly oriented lines mo, m,, wn, • • -, m, such that
450 THKOKKMS ON SENSE AND SEPARATION [CHAP ix
m0 — (ABC, afty) and mm=(ACB, afty), and such that each oriented line of the sequence would be doubly perspective with the next one in the sequence. Let m be a doubly oriented line not coplanar with any of m0, mlt • • •, mn, and doubly perspective with mQ; let m be the doubly oriented line obtained by changing both sense-classes on m. By Theorem 64 ml is doubly perspective with m or m. By a second application of this theorem w2 is doubly perspective with m or m, and by repeating this process n times we h'nd that raB is doubly perspective with m or m. But this means that mn is (ABC, afty) or (ACB, a
THEOREM 66. There are two and only two classes of doubly oriented lines such that any two doubly oriented lines of the same class are simi- larly oriented and any two of different classes are oppositely oriented.
Proof. Let (ABC, afty) be an arbitrary fixed doubly oriented line and let K be the class of doubly oriented lines similarly oriented to it. This class contains (Theorem 65) (ACB, ay ft) but not (ACB, afty) or (ABC, ayft). If I is any line and m any line not meeting I or AB, (ABC, afty) is doubly perspective with one of the doubly oriented lines determined by m and this with one of those determined by /. Hence K contains two of the four doubly oriented lines determined by any line of space. Let K' be the class of doubly oriented lines similarly oriented with (A CB, afty). It also contains two of the four doubly oriented lines determined by any line of space. K and K' cannot have a doubly oriented line in common, because this would imply that (ABC, afty) and (ACB, afty) were similarly oriented. Hence every doubly oriented line is either in K or in K'.
There can be no other pair of classes of similarly oriented doubly oriented lines including all doubly oriented lines of space, because one class of such a pair would contain elements both of A" and of K', and this would imply, by Theorem 63, that (ABC, afty) was simi- larly oriented with (ACB, afty).
From the construction which determines whether two doubly oriented lines are similarly oriented or not, it is evident that any col- lineation carries any two doubly oriented lines which are similarly oriented into two which are similarly oriented. Hence, if a collinea- tion carries one doubly oriented line into-a similarly oriented one, it carries every doubly oriented line into a similarly oriented one ; and
§§180,1811 GENERAL THEORY 01 SENSE 451
if it carries one into an oppositely oriented one, it carries every doubly oriented line into an oppositely oriented one.
Any collineatiou which carries a doubly oriented line into a simi- larly oriented one is said to be direct, and any collineation which carries a doubly oriented line into an oppositely oriented one is said to be opposite. This definition of direct and opposite collineations is easily seen to be equivalent to that in § 32.
*181. More general theory of sense. The theory of sense-classes in the preceding pages can be extended to analytic transformations by means of simple limiting considerations. For example, consider a transformation of a part of a Euclidean plane
where both series are convergent for all points in a region including the point (0, 0). If the determinant
= J
, 0) 0/(0,0)
dx dy
dx dy
is not zero, it can be shown that there is a region including (0, 0) which is transformed into a region including (a^, bM) in such a way that all ordered point triads of a sense-class in the first region go into ordered point triads of one sense-class in the second region ; and if (x1, y') is in the same plane as (x, y), the two sense-classes will be the same if and only if J > 0.
By a similar limiting process the notions of right- and left-handed- ness can be extended to curves, ruled surfaces, and other figures having analytic equations. A discussion of some of the cases which arise will be found in the article by Study referred to in § 162.
This sort of theory of sense relations belongs essentially to differential geometry, although the domain to which it applies may be extended by methods of the type used in §§168 and 173.
The theory of sense may, however, be extended in a different way so as to apply to the geometry of all continuous transforma- tions instead of merely to the protective geometry or to the geometry of analytic transformations. The main theorems are as follows:
452 THKoKKMS ON SENSE AND SEPARATION [CHAI -. i\
Any one-to-one reciprocal continuous transformation of a curvi- into itself transforms each sense-class on the curve either into itself or into the other sense-class. A transformation of the first kind is called direct and one of the second kind opposite, A direct trans- formation is a deformation (§ 157), and an opposite transformation is not a deformation.
Any simple closed curve consisting of points in or on the boundary of a 2-cell R (§ 155) is the boundary of a unique 2-ceil which consists entirely of points of R.
A 2-cell can be deformed into itself in such a way as to trans- form an arbitrary simple closed curve of the cell into an arbitrary simple closed curve of the cell. Any one-to-one reciprocal contin- uous transformation of a 2-cell and its boundary into themselves is a deformation if and only if it effects a deformation on the boundary ; i.e. if and only if it transforms a sense-class on the boundary into itself.
If the sense-classes on one curve /t of a 2-cell and its boundary are designated as positive and negative respectively, any sense-class on any other curve j\ is called positive or negative according as it is the transform of the positive or of the negative sense-class on jl by a deformation of j\ into j through intermediate positions which are all simple closed curves on the 2-cell and its boundary. By the theorems above, this gives a unique determination of the positive and negative sense-classes on any curve of the given convex region. A curve associated with its positive sense-class is called a positin-l>/ oriented curve, and a curve associated with its negative sense-class is called a negatively oriented curve.
Any transformation of a 2-cell which is one-to-one, reciprocal, and continuous either transforms all positively oriented curves into positively oriented curves or transforms all positively oriented curves into negatively oriented curves. In the first case the trans- formation is said to be direct and in the second case to be opposite. The transformation is a deformation if and only if it is direct. A 2-cell associated with its positively oriented curves or with its negatively oriented curves is called an oriented 2-cell.
The oriented 2-cells of a simple surface fall into two classes such that any oriented 2-cell of one class can be carried by a continuous deformation of the surface into any other oriented 2-cell of the
§i'i] ORIKNTKD CKU.S 453
.same class, but not into aiiy oriented 2-cell of the other class. The two oriented 2-cells determined by a given 2-cell are in different classes. A simple surface associated with one of these classes of oriented 2-cells is said to be oriented.
A similar theorem does not hold for the oriented 2-cells of a projective plane. Instead we have the theorem that every contin- uous one-to-one reciprocal transformation of a projective plane is a deformation. Consequently any oriented 2-cell can be carried into any other oriented 2-cell by a deformation.
The oriented simple surfaces in a 3-cell and its boundary fall into two classes such that any member of either class can be deformed into any other member of the same class through a set of intermediate positions which are all oriented simple surfaces, but cannot be deformed in this way into any member of the other class. A continuous one-to-one reciprocal correspondence which carries a 3-cell and its boundary into themselves either interchanges the two classes of oriented simple surfaces or leaves them invariant. In the second case the transformation is a deformation and in the first case it is not. A 3-cell associated with one of its classes of oriented surfaces is called an oriented 3-cell.
The oriented 3-cells of a projective space fall into two classes such that any member of one class can be carried by a continuous deformation of the projective space into any member of the same class but not into any member of the other class. A continuous one-to-one reciprocal transformation of the projective space either transforms each class of oriented 3-cells into itself or into the other class. In the first case it is a deformation and in the second it is not. A projective space associated with one of its classes of oriented 3-cells is called an oriented projective space.
The one-dimensional theorems outlined above are easily proved on the basis of the discussion of the sense-classes on a line in §§ 159 and 165. The two-dimensional ones, though more difficult, are conse- quences of known theorems of analysis situs. They involve, however, such theorems as that of Jordan, that a simple closed curve separates a convex region into two regions ; and the theorems of this class do not belong (§§ 34, 39, 110) to projective geometry. The Jordan theorem in the special case of a simple closed polygon does, however, belong to projective geometry and is proved below (§ 187).
454 THKOKKMS ON SENSE AND SEPARATION [CHAP. IX
The tlmv-dinicMsional propositions outlined here have not all been proved as yet, but are (in form) direct generalizations of the one- and two-dimensional ones.
I. .•( us note that an ordered triad of points as treated in § 100 may be reM;irded as determining an oriented 2-cell. For the triangular region having the points as vertices is a 2-cell, and a sense-class is determined on its boundary by the order of the vertices. This sense-class determines a sense-class on every curve of the 2-cell and thus determines an oriented 2-cell.
In like manner an ordered tetrad of points as treated in § 160 determines an oriented 8-cell. For the tetrahedral region having points A BCD as vet* tices is a 8-cell. The triangular region BCD is a 2-cell which does not contain .1, and is oriented in view of the order of the points on its boundary. This oriented 2-cell determines an orientation of the boundary of the 3-cell, and thus of the 3-cell.
Likewise an ordered tetrad A BCD of a projective plane determines a 2-cell, i.e. that one of the triangular regions BCD which contains A ; and this 2-cell is oriented by the order of the points BCD. Similarly, an ordered pentad ABCDE of points in a projective space determines an oriented 3-cell, i.e. that one of the tetrahedral regions BCDE which contains A, oriented accord- ing to the order of the points BCDE.
182. Broken lines and polygons. DEFINITION. A set of n points Alt A2, • • •, An, together with a set of n segments joining AI to A2, A2 to At, ' • •, Att_l to An, is called a broken line joining Al to An. The points Alt • • ', An are called the vertices and the segments joining them the edges of the broken line. If the vertices are all distinct and no edge contains a vertex or a point on another edge, the broken line is said to be simple. If Al=An the broken line is said to be closed, otherwise it is said to be open. The set of all points on a closed broken line is called a polygon. If the vertices of a polygon are all distinct and no edge contains a vertex or a point on another edge, the polygon is said to be simple.
A broken line whose vertices are AI} A2, • • •, An, and whose edges are the segments joining Al to A2, A2 to A8, • • •, An_1 to An, is called the broken line A^A2 • • • Aa, and its edges are denoted by A^A^ A2A&, • ' -, An_lAn respectively. If A^ — An the corresponding polygon is denoted by A^2 • • • AH_lAl ; the vertex Al is sometimes denoted by An, At by An + l, etc.
The following theorem is an obvious consequence of the definition.
THEOREM 67. The polygon AtA9 - • • AnAl i* the same as A9.-iy • • • AnA^At and A^Am • - • A^Af If P is any point of the edge A^4a of
§ M2] POLYGONS 455
a simple polygon A^A^ • • • AnAlt this polygon is the same as a poly- gon AvPAt • - • A^A^ in which the edge AA\ the vertex P, and the edge PA9 constitute the same set of points as the edge A A . Jj « simple polygon AlA3Aa - - . AHAl is such that A^A^ are collinear and A^=t=- A^ this polygon is the same as the polygon A^t • • • AnA{ in which all the edges but A^A^ are the same as before and A A is the segment A^A^A^.
DEFINITION. If A, B, C are any three points on a simple polygon, an elementary transformation is the operation of replacing any one of these points, say C, by a point C' such that C and C' are joined by a segment consisting of points of the polygon and not containing either of the other two points. A class consisting of all ordered triads each of which is transformable by a finite number of elementary transformations into a fixed triad ABC is called the sense-class ABC and is denoted by S(ABC).
THEOREM. 68. There exists a one-to-one and reciprocal correspond- ence between the points of any simple polygon and the points of any line such that two triads of points on the polygon are in the same sense-class with respect to the polygon if and only if the correspond- ing triads of points on the line are in the same sense-class.
Proof. Let the vertices of the polygon be denoted by A^ AZ, • • •, AH_i and let Al be also denoted by AH. Let B^ B2, • • •, Bn_l be n arbitrary points of a line I in the order {Blt B^, • • •, Bn_l} and let Bn also denote B^ Let $,. denote that segment B{Bi + l, which con- tains none of the other points B. Let the edge joining Ai to . I correspond protectively to the segment y8(. in such a way that J, and Ai+l are homologous with B{ and Bi+1 respectively. (In general the projectivities by which two sides of the polygon correspond to two segments on the line will be different.) If we also let A{ correspond to B. (i= 1, . . ., n— 1), there is evidently determined a one-to-one and reciprocal correspondence F between the polygon and the line which is such that each side of the polygon with its two ends corresponds with preservation of order relations to a segment of the line and its two ends.
Let Pv Py, P3, P4 denote points of the polygon and Llt Lt, Lt, Lt the points of I to which they respectively correspond under F. The correspondence F is so defined that if P1P3PS goes into PJ\Pt by an
456 TH Koi: K.MS <>N SKNSK AM) SEPARATION [CHAP, ix
elementary transformation with respect to the polygon, thru LJ^L^ goes into LJ^L^ by an elementary transformation restricted with respect to /*,, #3, • • •, ^»(cf. § 165), and conversely. Hence the theorem follows at once from the corollary of Theorem 22.
COROLLARY 1. The theorem above remains true if the won I* "broken line with distinct ends" be substituted for polygon, mul " interval " for line.
The definitions of separation and order given in § 21 for the points on a line may now l>e applied word for word to the points on a simple polygon, and in view of the correspondence established in Theorem 68, the theorems about order relations on a line may be applied without change to polygons.
By comparison with the proof of Theorem 15 we obtain immediately
COROLLARY 2. A simple polygon is a simple closed curve. COROLLARY 3. A simple broken line joining two distinct point* AV, An is a simple curve joining Al and AH.
The order relations on a broken line which is not simple may be studied by the method given above with the aid of a simple device. Suppose we associate an integer with each point of a broken line A1A3''-AH as follows: With A^ and every point of the segment joining Av to Az the number 1 ; with A2 and every point of the seg- ment joining A0 to A3 the number 2 ; and so on, and, finally, with An the number n.
DEFINITION. The object formed by a point of the broken line and the number associated with it by the above process shall be called a numbered point ; and the numbered point is said to be on any seg- ment, line, plane, etc. which the point is on. If A, B, C are any three numbered points on a polygon, an elementary transformation is the operation of replacing any one of these numbered points, say C, l>y a point C' such that C and C' are joined by a segment of numbered points all having the same number. A class consisting of all ordered triads of numbered points each of which is transformable by a finite sequence of elementary transformations into a fixed triad ABC is called the sense-class ABC and is denoted by S(ABC).
By the proof given for Theorem 68 we now have
THI.MKKM f>9. There exists a one-to-one ~and reciprocal correspond- ence between flic numbered points of any broken line and the point*
§§i«M8»] POLYGONS 457
of any interval such tJuit tiro triads of numbered points t're in the sunie sense-class if and only if the corresponding triads of points on the interval are in the same sense-class.
We are therefore justified in applying the theorems and defini- tions about order relations on an interval to the numbered points of a broken line.
EXERCISE
*Any two points of a region can be joined by a broken line consisting entirely of ]K>ints of the region.
183. A theorem on simple polygons. In the last section a poly- gon was defined as the set of points contained in a sequence of points and linear segments. This is the most usual definition and doubtless the most natural. With a view to generalizing so as to obtain the theory of polyhedra in spaces of three and more dimen- sions, however, we shall find it more convenient to use the property of a simple polygon stated in the following theorem.*
THEOREM 70. A set of points [P] is a simple polygon if and only if the following conditions are satisfied : (1) [P] consists of a set of distinct points, catted vertices, and of distinct segments, called edges, such that the ends of each edge are vertices and each vertex is an end of an even number of edges; (2) if any points of [P] are omitted, the remaining subset of [P] does not have the property (1).
Proof. It is obvious that a simple polygon, as defined in § 182, satisfies Conditions (1) and (2), because no edge has a point in com- mon with any other edge or vertex and each vertex is an end of exactly two edges.
Let us now consider a set of points [P] satisfying (1) and (2). If two or more edges have a point in common, this point divides each edge into two segments. Hence the point may be regarded as a vertex at which an even number of edges meet. In like manner, if an edge contains a vertex the two segments into which the edge is divided by the vertex may be regarded as edges. Since there are originally given only a finite number of vertices and edges, this process determines a finite number of vertices and edges such that no edge contains a vertex or any point of another edgf.
*This form of the definition of a polygon and a corn-siMindinj; definition of a polyhedron arc due to N. J. Lennes, American Journal of Mathematics, Vol. XXXIII (11)11), p. 37.
458 THK»U:i:.MS OX SENSE AND SEPARATION [CHAI-. ix
Now let «, be any edge and Pl one of its ends. Since there are an even number of segments having Pl as an end, there exists tin- other distinct from ej let this be denoted by ef Let P2 be its other end, and let ea be a second segment having P2 as an end, and so on. By this process we obtain a sequence of points and segments
Since the number of vertices is finite, this process must lead by a Hnite number of steps to a point Pn which coincides with one of the jiiwious points, say P{. The set of points included in the points
and segments
P e P P e
•*<» ct+i» "*i+i' » »— V »
satisfies the definition of a simple polygon and has the property that each Pj(j=i, i-\-l • • • Pn_l) is an end of two and only two es. Hence it satisfies Condition (1). By Condition (2) it must include all points of the set [P].
COROLLARY. A set of points satisfying Condition (7) of Theorem 70 consists of a finite number of simple polygons no two of which have any point in common which is not a vertex.
Proof. In the proof of the second part of the theorem above, Condition (2) is not used before the last sentence. If Condition (2) l»e not satisfied, the set of points remaining when the segments *••+!»•••> en (and those of the points 7?, • • •, Pn-l which are not ends of the remaining segments) are removed continues to satisfy Condition (1). For on removing two segments from an even num- ber, an even number remains. Hence the process by which the simple polygon Pif ei+l, • • •, Pa_lt en was obtained may be repeated and another simple polygon removed. Since the total number of edges is finite, this step can be repeated only a finite number of times.
184. Polygons in a plane. In the next three sections we shall prove that the polygons in a projective plane are of two kinds, a polygon of the first kind being such that all points not on it con- stitute two regions, and a polygon of the second kind being such that all points not on it constitute a single region. The boundary of a triangular region is a polygon of the first kind, and a projective line a polygon of the second kind. In proving that the points not on a polygon constitute one or two regions, we shall need the following:
POLYGONS 4f>H
THEOREM 71. Any point coplanar with but not on a polygon p in a plane a is in a triangular ret/ion of a containing no point of }),
Proof. Let the polygon be denoted by A^ • • - AnAl and tin- point by P. By an obvious construction (the details of which are left to the reader; cf. § 149) a triangular region Tl may be found containing P and not containing Al or Aa or any point of the edge Jj.l... In like manner a triangular region T9 may be constructed which contains P, is contained in TI} and does not contain At or any point of the edge A^AZ. By repeating this construction we obtain a sequence of triangular regions Tlt Tt, - - •, Tn, each contained in all the preceding ones, containing P, and such that Tk does not con- tain any point of the broken line AtAt • • • At+l. Thus TH contains P and contains no point of the polygon A^A^ • • • AnAv
COROLLARY. Any point of space not on a polygon p is in a tetra- hedral region containing no point of p.
Let the set of lines containing the edges of a simple polygon in a plane be denoted by llt 12, • • •, /„. Since more than one edge may be on the same line, n is less than or equal to the number of edges. According to Theorem 67 we can first suppose that the notation is so assigned that no two edges having a common end are collinear except in the case of a polygon of two sides (which is a projective line), for two collinear edges and their common end can be regarded as a single edge. In the second place, according to the same theorem, we can introduce as a vertex any point in which an edge is met by one of the lines I, I. • • •, /„ which does not contain it.
1 -
Under these conventions the polygon may be denoted by A^A^ • • • AmAv where each point A((i = 1, 2, • • •, m) is a point of intersection of two of the lines llt /2, • • -, /„, and each edge is a segment join- ing two vertices and containing points of only one of the lines
t»i*~-:l>.
In like manner, when two or more simple polygons are under con- sideration, let us denote the set of lines containing all their edges by /j, la, - • ., ln. We may first arrange that no two edges of the same polygon which have an end in common are collinear, and then introduce new vertices at every point in which an edge is met by one of the lines /(, /„, • • -, /„ which is not on it. Thus in tins case also the polygons may be taken to have all their vertices at points
460 Til BUM-: MS ON SENSE AND SEPARATION [CHAP, ix
of intersection of the n lines lv /2, • • •, /, and to have no edge whii-h contains such a point of intersection.
We are thus led to study the points of intersection of a set of n coplanar lines and the segments of these lines which join the points of intersection.
185. Subdivision of a plane by lines. Consider a set of n lines /t, /2, . . ., ln all in the same plane TT. The number «0 of their points of intersection is subject to the condition
the two extreme cases being the case where all n lines are concur- rent and the case where no three are concurrent. According to § 22, Chap. II, and the definition of boundary (§ 150), the points of inter- section bound a number a^ of linear convex regions upon the lines. The number al is subject to the condition
n = «1 = n (n — 1),
the two extreme cases being the same as before.
THEOREM 72. The points of a plane which are not on any one of a finite set of lines l^ /2, • • •, ln fall into a number «2 of convex regions such that any segment joining two points of different regions contains at least one point of l^ lz, •••,/„. The number a* satisfies
the inequality 1X
n^a^^l + l.
Proof. The proof may be made by induction. If n = 1 the theorem follows directly from the definition of a convex region. We suppose that it is true for n = k, and prove it for n = k + I.
We are given k + \ lines lv, /2, • • •, lk+l. The lines l^ lv, • • •, lk
k(k—\)
determine a number Nt, not less than k and not more than v -4-1,
&
of convex regions. The line lt+l meets the remaining k lines in at least one point and not more than k points. The remaining points of lt+l therefore form at least one and at most k linear convex regions, each of which is the set of all points common to !t+1, and one of the planar convex regions (Theorem 3). By Theorem 8 each convex region which contains points of /i+-, is divided into two con- vex regions such that any segment joining two points of different
§185] sui!Di\ isio.N OF A PLAM: r,v LINES 461
regions meets /t + 1 if it does not meet one of the lines /t, / , • . ., l^ Hence the k -f 1 lines determine a number Jft+l of convex regions of the required kind such that Nt+ 1 S Nt + 1 S JVt+ k. Since
it follows that
2 9
COROLLARY 1. If n lines of a plane pass through a point, they determine n convex regions in the plane; if no three of them are
concurrent, they determine — — - — - -j- 1 con-vex regions.
A
Let us denote the «0 points of intersection of the lines ljf /2, •••,/„ by
or any one of them by a° ; the al linear convex regions which these points determine upon the lines by
a,1, a.,1, • • •, rtr.
1 > 2 > o,»
or any one of them by a1 ; and the «2 planar convex regions by
0 o *> O.*i a.,, •••,««,
1 " A ' * ™J
or any one of them by a2.
COROLLARY 2. If the lines llf /2, • • •, /„ are not concurrent, any line coplanar with and containing a point of an a2 has a segment of points in common with it. The ends of this segment are on the boundary of the a2, and no other point of tlie line is on this boundary.
Proof. The given line, which we shall call /, meets the lines llt /2, • • •, ln in at least two points, and, as seen in the proof of the theorem, one and only one of the mutually exclusive segments having these points as ends is composed entirely of points of the a*. Let a denote this segment. Its ends are boundary points of the aa by Theorem 10. Let Z,. and /,. be lines of the set l^ /.,,•••, 1H such that lt contains one end of cr and /,. the other. All points of the aa are separated from the points of the segment complementary to <r by the lines lt and /,. Hence any point of the complementary seg- ment is in a triangular region containing no point of the aa and is therefore not a boundary point of the a8.
\t\-2 THKOKKMS ON SKNSK AND SEPARATION [CHAF.IX
This argument carries with it the proof of
COROLLARY 3. Any interval joining a point of an a3 to a point not in the a* contains a point of the boundary of the a2.
THEOREM 73. If the lines /1? l^, • • •, ln are not all concurrent, the Ixx/ndary of each a9 is a simple polygon whose vertices are a°'s and whose edges are a1'*.
Proof. The theorem is a direct consequence of § 151 in case n='.\. Let us prove the general theorem by induction ; i.e. we assume it true for n = k and prove it for n = k + \.
Let the notation be so assigned that llt 12, 13 are not concurrent. Then any one of the convex regions, say R, determined by llt /„•••, lt is contained in a triangular region determined by llt lz, and lt, be-, cause no two points of R are separated by any two of the lines /t, 12, lf Let m be a line containing no point of this triangular region nor any of its vertices. The segments AtAj etc. referred to below do not contain any point of m.
If lt+1 contains a point of R, it contains, by Cor. 2, above, a seg- ment of points of R such that the ends of this segment are on the boundary of R. By Theorem 67, the ends of this segment may be taken as vertices of the polygon p which by hypothesis bounds R. Thus we may denote this polygon by A^A^ • • • Ai • • • AjAlt where Al and A{ are the points in which lt+1 meets the polygon.
There are just two simple polygons which are composed of the segment A^A{ and of sides and vertices of p. For any such polygon which contains A^A{ contains A^A^ or AvAj\ if it contains A^A.^ it must contain A^Aa and therefore A3At, • • •, A{_^Ait and since it con- tains A{AV it must be the polygon A1A2A3 • • • AtA^ ; if it contains A^Aj it must contain AJAJ_l and therefore AJ_1AJ_Z, • • •, Ai+lA{, and since it contains AiAl it must be AlAJAJ_l • • • A{AV
Neither of the lines lt+l and m meets any edge of the polygon A^A^AI • • • AfAv except A{Alt which is contained in ^.+1. Hence all points of this polygon except Ait A1 and those on the edge AiAl are in one of the two regions, which we shall call R' and R", bounded by lt+l and m. In like manner all points of the polygon A1AJAJ_l • • • AiAl except A{, Al and those on the edge AtA^ are in one of the two regions R' and R".
The points of R on any line coplanar with R and meeting the segment AiAl in one point form a segment <r (Cor. 2, above) which
§i>«] SUBDIVISION OF A PLANK UV LINKS
does not contain any point of w. Hence the cuds P, @ of <r are separated by lt+1 and m. But P and Q are boundary points of R by Cor. 2, above. Hence the boundary of R has points in both of tin- regions R' and R" bounded by lk+l and m. By the paragraph above, the points of the boundary of R in the one region, say R', must be the points, exclusive of the interval AiAl, of the polygon A^A^ • • • AtA^ and those in the other, R", must be the points, exclusive of the interval ./,./,, of the polygon A^ • • . J,./1.
Let RI and R2 be the two convex regions formed by the points of R not on lk + l. Since these two regions are separated by lk + l and m, we may assume that R: is in R' and R2 in R". Every boundary point of RI which is not a point of lt+1 is in R'. For if B is a point of the boundary of Rt it is not on m, by construction, and if it is not on lt + l it can be enclosed in a triangular region containing no point of /t+1 or m. Such a triangular region must contain points of Rj and hence can contain no point of R", since any segment joining a point of R' to a point of R" contains a point of /t + 1 or of m. Hence B is in R'. In like manner any boundary point of RS not on lt+l is in R". But by Theorem 10 every point!? of the boundary of R is on the boundary of RI or R2. Hence the boundary of RI contains all points of the boundary of R in R'; and by Theorem 10 it contains no other points not on lk+r Hence it is the polygon A^A^ • • • A(A^. In like manner the polygon A^ • • • AiAl is the boundary of R0.
Hence the boundaries of the two planar convex regions into which any one of the planar convex regions determined by llt /2, • • •, lt is separated by lk+l are simple polygons. The other planar convex regions determined by llt /„•••, lt+l are identical with regions determined by llt /,, • • •, lk.
COROLLAKY 1. Each a1 is on the boundaries of two and only two «**«.
COROLLARY 2. In case all the lines l^ lt,---, ln are concurrent, there is only one a°, the common point of the lines ; there are n al's, each consisting of all points except a° of one of the Unfit /. ; and there are n a*'s, each having a pair of the lines as its boundary.
THEOREM 74 The numbers «0, alt a3 satisfy the relation
Proof. We shall make the proof by mathematical induction. The theorem is obvious if n = 2, for in this case «0=1, a1 = 2,«.|=2.
4<i4 Til HOi; K.MS (»N SKNSK AND SKI'A KATloN [CHAP. IS
Let us now a>suine it to l>e true for n = k and prove that it follows for n = k+ 1.
The lines /t, /2, • • •, /t. determine a set of a'0 points, a[ linear convex regions, and a'., planar convex regions subject to the rela- tion a'0 — a[ -+- «2 = 1. The line /j^ meets a number, say r, of the planar convex regions and separates each of these into two planar convex regions. Hence a^ is increased to a2'+/\ The number of one-diineusional convex regions is increased by r for the number of convex regions on /t+1 and also by a number* s equal to the number of linear convex regions of the lines /t, /„, • • •, lt which are met by lt+l. The number of points of intersection of ^, 12, • • -, lt+1 also exceeds a'0 by s. Hence for ll} 1.2, • • -, lt+1 the numbers «0, alt a2 are a'0 + s, a[ + r + s, a!2 + r. Hence «0 — al + «2= (a'0 + s)
186. The modular equations and matrices. The relations among the points, linear convex regions, and planar convex regions may be described by means of two matrices of which those given in § 151 for the triangle are special cases. The first matrix, which we shall denote by H1? is an array of aQ rows and a1 columns, each row being associated with an a° and eacli column with an a1. The element of the ith row and /th column is 1 or 0 according as af is or is not an end of a\. The second matrix, H2, has al rows and «2 columns associated respectively with the a^s and a2's. The element of the tth row and jth column is 1 or 0 according as «/ is or is not on the boundary of af.
Since every segment a1 has two and only two ends, each column of Hj contains just two 1's; and since each a1 is on the boundary of two and only two «2's (Theorem 73, Cor. 1), each row of H2 con- tains just two 1 's.
For each of the al's let us introduce a variable which can take on only the values 0 and 1, these being regarded as marks of the field obtained by reducing modulo 2. We denote these variables by #j, xf, • - ', x0i respectively. There are 2<r' sets of values which can be given to the symbol t (xlf x2, • • • xaj.
•The number « is less than r if fo+i contains points of intersection of £,, /.,, • • -. lt.
Kxrluilini: the one in which all the variables are zero, these symbols constitute tin- ]><>ints of a finite project! ve space of a^ — 1 dimensions in which there are three points on every line (cf . S 72, Vol. I).
§18«] MODULAR EQUATIONS I-;;,
Every one of these symbols (xlt xtj, . . ., xttt) corresponds to a way of labeling each segment a1 of the original n lines with a 0 or a 1, the segment a/ being labeled with the value of xf We shall regard the symbol as the notation for the set of edges labeled with 1's. By the .sum of two symbols (xl xf, • . ., xfi) and (y , y , '",ya) we shall mean (a^+y^ #3+ys, • • •, xai+ ya), the addition being performed modulo 2. According to our convention the sum represents the set of al's which are in either of the sets represented by (xlt x2,"-, xaj and (ylt ya, • • •, y«,) but not in both. By a repe- tition of these considerations it follows that the sum of n symbols of the form (xlt xz, • • •, xtti) for sets of edges is the symbol for a set of edges each of which is in an odd number of the n sets of edges.
In the sequel we shall say that a polygon p is the sum, mod- ulo 2, of a set of polygons plt p2, • • •, pH if it is represented by a symbol (xlt xz, • • •, xai) which is the sum of the symbols for pjt pt, - • • , pn. Let us now inquire what is the condition on a symbol (xlt x^, • • ', xtti) that it shall represent a polygon ?
At every vertex of a polygon there meet two and only two edges. Hence, if we add all the x's that correspond to the a^s meeting in any point, this sum must be zero, modulo 2. This gives a0 equations, one for each a°, of the form
(4) Xf + JSq+...+ j:k = 0 (mod. 2)
(ap> aq> • • •> ak being the edges which meet at a given vertex), which must be satisfied by the symbol for any polygon. Obviously the matrix of the coefficients of these equations is Hj. For example, in the case of the triangle these equations are (cf. § 151)
#8+*4+*8+*,= 0,
(5) xl + xs + xs + *, = 0, (mod. 2) xt+ xf+ xt-\- xt= 0.
We shall denote the set of equations (4) by (H,). Since each column of H2 gives the notation for a polygon bounding an af, the columns of H2 are solutions of the equations (Ht). For example, the columns of the matrix H. in § 151 are solutions of (H().
Any solution whatever of these equations corresponds to a label- ing of the fl^'s with O's and 1's in such a way that there are an even number of 1's on the a*'s meeting at each a°. Hence, by the corollary of Theorem 70 the a's labeled with 1's must constitute
l»;r. THEOREMS ON SENSK AND SEPARATION [CHAP. IX
one «'i moiv simple polygons. Hence every solution of the equations (\\ ^ /v/'/v.sr///.s- (i si nt filt JKI///I/OH tir a set of simple polygons.
Since each column of the matrix Ht contains exactly twu 1's, any one of the equations is obtained by adding all the rest. Since the only marks of our field are 0 and 1, any linear combination of the fi [nations (Ht) would be merely the sum of a subset of these equa- tions. Consider such a subset and the points a° which correspond to the equations in the subset. Every a1 joining two points of the subset is represented in two equations, and the corresponding vari- able disappears in the sum. There remain in this sum the variables corresponding to the al's joining the points of the subset to the remaining points of the figure. These cannot all pass through the same point unless the subset consists of all points but one (since any two of the original n lines have a point in common). Hence while any one of the equations is linearly dependent * on all the rest, it is not linearly dependent on any smaller subset. Hence aQ — 1 of the equations (Ht) are linearly independent.
Since the number of variables is alt the number of solutions in a set of linearly independent solutions on which all other solutions are linearly dependent is o^— aQ+ 1. By Theorem 74 this number is a^ Thus the total number of polygons and sets of polygons is 2*1 — 1.
The simple polygons which bound the regions a2 are a set of solu- tions, namely, the columns of the matrix H2. Since each row of the matrix H2 contains just two 1's, it follows that if we add all the columns we obtain a solution of (H^ in which all the variables are 0. < )n the other hand, if we add any subset of the columns of Ho the sum will be a solution in which not all the variables are zero. For consider a segment joining an interior point A of the region a2 cor- responding to one of the columns in the subset to an interior point B of a region a2 corresponding to one of the columns not in the subset; this segment may be chosen so as not to pass through a point of intersection of two of the lines l^, 12,---, /„. Hence it contains a finite number of points on the polygons corresponding to the columns in the subset. The first one of these in the sense from Ji
•Since the only coefficients which can enter are 0 and 1, the statement that one solution is linearly dependent on a set of others is equivalent to saying that it is a sum of a number of them.
tin the modular space of or, — 1 dimensions this means that the a0— 1 inde- pendent (a, — 22)-Kpaces intersect in an («2 — l)-space.
i:ni ATIONS 467
to A is on au a.] which is on the boundary of a region in the sul»rt and a region not in the subset. The variable corresponding to tins interval therefore appears in only one of the a's in the subset and so does not drop out in the sum. Hence any «„— 1 of the boundaries of the a2 convex regions correspond to a set of lim-arli/ i/i</<j>f,i <lmt solutions of (4). In other words, 2"l~a°— 1, or one less than half of all the solutions of (Hj), are linearly dependent on the solutions corresponding to the columns of H2. The solutions of HI are thus divided into two classes, those linearly dependent on the columns of H2 and those not so dependent.
Since each of the lines 1J} /2, • • •, ln is a polygon, it corresponds to a solution of the equations (Hj), but it does not correspond to a solu- tion which is linearly dependent on the columns of the matrix H . This is a corollary of the argument used in showing that the sum of any subset of the columns of H2 is not a solution in which all the variables are zero. For in that argument we showed that a certain segment AB contains a point on the polygon represented by the sum of such a subset. The same argument applies to the complementary segment. Hence the line AB has two points, at least, in common with the polygon or polygons represented by the sum of the subset of columns. Hence this sum cannot represent a line.
Thus, if we take the solution of the equations (Hj) corresponding to any one of the lines l^ 12, • • •, /„, together with any ag — 1 of the columns of the matrix H2, we have a linearly independent set of solutions. But since this set contains «2 independent solutions, all solutions are linearly dependent on this set.
187. Regions determined by a polygon. If p is any polygon it can, by § 184, be regarded as one whose vertices are a°'s and whose edges are avs of a set of lines /1( /.,, • • •, 1H.
Two cases arise according as p is represented by a symbol which is or is not a sum of a subset of the columns of the matrix H2. In the first case p corresponds also to the sum of all the remain- ing columns, because the sum of all the columns is (0, 0, • • •, 0). It cannot correspond to a third set of columns, for the sum of the columns in the second and third sets, which is also the sum of the columns not common to these two sets, would be (0, 0, • • •, 0). Hence there would be a linear relation among a subset of the columns of H2 contrary to what has been proved abovi-.
.pis THKOKKMS ON SiftfSE AND SEPARATION [CHAP.IX
Let us denote the two sets of columns of H2, whose sums are the symbol for p, by clt c3, - • • , ct and ek + v • • •, caj respectively, and sup- pus.- tin- notation so assigned that they represent the boundaries of «if» a*» • • •» a* ana< a* + i« '•'» a«, respectively. Let the points of the plane in a*, a2, • • •, a%, together with such points of the bound- aries as are not points of p, be denoted by [P]. Let the set of the points analogously related to a*+1, • • •, a2s be denoted by [Q]. Clearly, the sets of points [P], [Q], and p are mutually exclusive and include all points of the plane.
Consider any point P0 of the convex region a2 corresponding to ct. It is connected by a segment consisting entirely of points P to every point P in or on the boundary of af. If k>l, cl has an edge in common with at least one of cot • • •, ck, and the notation may 1 it- assigned so that c1 has an edge in common with c2. Then P0 can be joined to any point Pl of the common edge by a segment of P-points, and Pr by another segment of P-points to every P-point of the region a2 and its boundary. If k>2 there is a solution which may be called c3, with an edge in common with cl or c2 ; for if not, the solution c, +c would be one in which all the 1's correspond to the
12 •*•
edges of p, and as no subset of the edges of p forms a polygon, cl + c0 would correspond to p itself. As before, every point of the region «32 and its boundary can be joined to PQ by a broken line of at most three edges. Since there is no subset of clt • • •, ct whose sum corresponds to -rr, this process can be continued till we have any point R of the convex regions a*, «22, • • •, «A2 and their boundaries joined by a broken line b to P0. If R is on TT the process of ron- structing b is such that all points of b except E are in [P], whereas if R is in [P] all points of b are in [P].
Hence any two points of [P] can be joined by a broken linef con- sisting only of such points; and, since every point of p is on the boundary of one of a2, a2, • • •, a%, any P can be joined to any point R of p by a broken line every point of which, except R, is in [P]. A precisely similar statement is true of [Q].
Consider now any broken line b' joining a point P to a point ^>. The points which are on this broken line and also on any a1 and its ends constitute a finite number of points and segments. Hence b' meets the lines lr 12, • . ., ln in a finite -number of points and seg- ments, each of the segments being contained entirely in an a1
§187] POLYGONS 469
These points and the ends of these segments we shall denote by AltAa, • • -,Ak taken in the sense on the broken line from P t< Since P and Al are within or on the boundary of the same convex region, A^ is either in [P] or on p. If Al is in [P] the same con- sideration shows that A2 is in [P] or on p. If none of the A'a were on p, this process would lead to the result that Ak is in [P], and In-nee Q would also be in [P], contrary to hypothesis. Hence one of the A's is on p, and hence any broken line joining a point P to a point Q contains a point on p.
It now follows that [P] and [Q] are both regions. For we have seen that any point P can be joined to any other P by a broken line consisting entirely of points of P. By Theorem 71 any point P is contained in a triangular region containing no points of p. This tri- angular region contains no Q, because if it did a segment joining it to P would, by the argument just made, contain a point of p. Hence [P] satisfies the definition of a two-dimensional region given in § 155. A similar argument applies to [Q]. Hence we have
THEOREM 75. Any simple polygon p which corresponds to a sym- bol (.Cj, xz, • ' ', xa^ which is the sum of a set of columns of HS is the boundary of two mutually exclusive regions which include all points of the plane not on p and are such that any two points of the same region can be joined by a broken line which is in the region. Any broken line joining a point of the one region to a point of the other region contains a point of the polygon.
COROLLARY 1. Any point R of p can be joined to any point not on p by a broken line containing no other point of p.
COROLLARY 2. If a segment ST meets p in a single point 0 which u not a vertex of p, S and T are in different regions with respect to p.
Proof. Let S' and T' be two points in the order {SS'OT'T} and such that the segment S'OT1 contains no point of /,, /„, •••,/„ ex- cept 0. By §185, S' and T1 are in two convex regions a* which have an edge in common. Since this edge is an edge of p, the columns of H2 corresponding to these two a*s must be one in the set cr c2, • • -, ck, and the other in the set'ct + 1, • • •, emf Hence, if S' and S are in [P], T' and T are in [Q], and vice versa.
THEOREM 76. Any simple polygon p which corresponds to a sym- bol (xlt z2, • • •, xai) which is not the sum of a set of columns of Ht
470 THEOREMS ON SENSE AND SEPARATION [CHAP, ix
is the boundnrtf of a region which includes all points of the plane not on p. Any two points not on p can be joined by a broi • n lint not meeting p.
Proof. By Theorem 71 any point not on p can be enclosed in a triangular region containing no point of p. Hence the theorem will be proved if we can show that any two points not on p are joined by a broken line consisting only of such points. If this were not so, we could let P0 be any point not on p and let [P] be the set of all points not on p which can be joined to PQ by broken lines not meeting p. As in the proof of Theorem 75, [P] would have to consist of a number of regions a2, together with those points of their boundaries which were not on p] and the boundary of [/*] could consist only of points of p. But the boundary of [/'] must consist of the polygon or polygons whose symbol is obtained by adding the columns of H2 corresponding to the «2's in [P]. By § 183 no subset of the points of p can be a simple polygon. Hence p would be the the boundary of [P] and be expressible linearly in terms of the boundaries of a2's, contrary to hypothesis.
Every polygon whose edges are on llt lz, •••,/„ corresponds to a symbol (xv, #2, • • -, xa^ which either is or is not expressible linearly in terms of the columns of H2. Hence the arbitrary simple polygon p with which this section starts and which determines the lines /j, /2, • • -, ln is described either in Theorem 75 or in Theorem 76. Hence we have
THEOREM 77. DEFINITION. The polygons of a plane a fall into two classes the individuals of which are called odd and even m-/«r- tively. A polygon of the first class is the boundary of a single region comprising all points of a not on the polygon. A polygon of the second class is the boundary of each of two regions which contain «U points of a not on the polygon, have no point in common, and are such that any broken line joining a point of one region to a point of the other contains a point of the polygon.
The odd polygons are also called unicursal, and the even polygons are also called bounding. A line is an example of an odd polygon, and the boundary of a triangular region is an example of an even one. The segments a, /S, 7 as defined in § 26 are the edges of an odd polygon.
§187] ODD AND EVEN POLYGON > 471
THKOKKM 78. Two polygons of which one is even ami which art M/r// that neither polygon hits n n rtc.r on the other have an even (or zero) number of points in common.
Proof. Let pl be an even polygon, let p% be any other polygon, and let the points of intersection of the polygons be A^, .... /. in the order {Rl R2 • . . Rn} with respect to pf If n = 0 the theorem is verified. If n were 1 the edge of p^ containing 7?, would have its ends in different regions with respect to irlt and hence the broken line composed of all pt except the side containing Rl would have to contain a point of plt contrary to hypothesis. If n > 1 the inter- val of pt which has ltl and R^ as ends and contains no other points R is a broken line which belongs (except for its ends) entirely to one of the two regions [P] and [Q] determined by pj and by Cor. 2, Theorem 75, the interval of pz similarly determined by R^ and Rt belongs entirely to the 'other of the two regions [P] and [ Q]. Thus, if Slt Ss, • • -, SH are a set of points of p2 in the order {'•',*,'«'., •*?,#, • • • SH_lRnSn}i and ^ is in [P], all the S's with odd subscripts are in [P] and all the S's with even subscripts are in [Q]. But by Cor. 2, Theorem 75, SH is in [Q] since Sl is in [P]. Hence n is even.
COROLLARY. A line coplanar with and containing no vertex of an even polygon meets it in an even (or zero) number of points.
THEOREM 79. Two odd polygons suck that neither has a vertex on the other meet in an odd number of points.
Proof. Let the polygons be pl and pt, let the lines containing the sides of pl be ll} • • •, ln_lt and let /„ be a line containing no vertex of either polygon. According to the results stated at the end of the last section, pl is expressible by addition, modulo 2, as the sum of ln and a number of boundaries of aa's. The latter combine into a number of even polygons, the edges of which are either edges of pl or of ln. Hence these even polygons have no vertices on pz and contain no vertices of pf Hence by Theorem 78 they have an even (or zero) number of points in common with jia. Thus our theorem will follow if we can show that /„ has an odd number of points in common with pf
By the argument just used pt can be expressed as the sum, modulo 2, of a line m and a number of even polygons which have no vertices on I . The latter meet /„ in an even (or zero) number
47l> THK<»Ki:.M> ON SKNSE AND SEPARATION [CHAP. IX
of points, and m meets /rt in one point. Hence p3 meets lit in an mid mimU-r (»f points.
COROLLARY 1. Two odd polygons always have at least one point. in common.
COROLLARY 2. If p is a simple polygon and there exists an odd polygon p meeting p in an even (or zero) number of points and such tli nt neither polygon has a vertex on the other, then p is even.
Since the plane of a convex region always contains at least one line not having a point in common with the region, the last result has the following special case, which, on account of its importance, we shall list as a theorem.
THEOREM 80. Any simple polygon lying entirely in a convex region is even.
To complete the theory of the subdivision of the plane by a polygon, there are needed a number of other theorems which can be handled by methods analogous to those already developed. They are stated below as exercises.
EXERCISES
1. If a simple polygon p lies entirely in a convex region R, the points of R not on p fall into two regions such that any broken line joining a point of one region to a point of the other has a point on p. One of these regions, called the interior of the polygon, has the property that any ray (with respect to R) whose origin is a point of this region meets p in an odd number of points, provided it contains no vertex of p. The other region, called the exterior of the polygon with respect to R, has the property that any ray whose origin is one of the points of this region meets p in an even (or zero) number of points, provided it contains no vertex of p.
2. If p is any even polygon in a plane a, one of the two regions determined by/), according to Theorem 77, contains no odd polygon and is called tin- interior of p. The other contains an infinity of odd polygons and is called the. exterior of p.
3. If one line coplanar with and not containing a vertex of a simple jxily- gon meets it in an odd (even or zero) number of points, every line not <-<>n- taining a vertex and coplanar with it meets it in an odd (even or zero) number of points.
4. If the boundary of a convex region consists of a finite number of linear segments, together with their ends, it is a simple polygon.
5. A simple polygon which is met by every line not containing a vertex in two or no points is the boundary of a convex region.
§§187,188] POLYGONAL REGIONS 473
6. For any simple polygon .l,.-l, • • • .(„.!,, there exists a set of n - 1J tri- angular regions such that (1) every i><>int of the interior of the polygon is in or on the boundary of one of the triangular regions, (2) every vertex of one of the triangular regions is a vertex of the polygon, and (3) no two of the triangular regions have a point in common.
*7. By xise of convex regions and matrices analogous to H1 and H,, prove Theorem 77 for any curve made up of analytic pieces (i.e. 1-cells which satisfy analytic equations).
188. Polygonal regions and polyhedra. I>KFIXITIOX. A planar polygonal region is a two-dimensional region R for which there exists a finite number of points arid linear regions such that any interval joining a point of R to a point not in R, but coplanar with it, meets one of these points or linear regions. A (three-dimensional) polyhedral region is a three-dimensional region R for which there exists a finite number of points, linear regions, and planar polygonal regions such that any interval joining a point of R to a point not in R meets one of these points, linear regions, or planar polygonal regions.
Let R be a planar polygonal region and let llt 12, • • •, ln be a set of lines coplanar with R and containing all the points and linear regions such that any interval joining a point in R to a point not in R meets one of these points or one of these linear regions. Let us adopt the notation of § 185.
If a point P of one of the two-dimensioual convex regions a* is in R, all points of the a2 are in R, for all such points are joined to P by intervals not meeting llf /2, • • •, /„.
Since any point not on l^ l^, • • •, /„ is interior to a triangular region containing no points of l^ /.,, • • •, /B, no such point can be a boundary point of R.
Let a?, • • -, a£ be the a2's which have points in R. As we have seen, all points of these a3's are in R. All points of their bound- aries are either in R or on its boundary ; for every point of the boundary of an a.* (r = l, • • •, k) may be joined to a point of «£, that is, to a point of R, by a segment of points of R, and hence is either a point of R or of its boundary.
Any point B of the boundary of R is on the boundary of one of "*,•••, af- For any triangular region T containing B contains points of R and hence contains a triangular region T' of points of R. The region T' must have points in common with at least one a*. If T be chosen so as to contain no points of any u~ which does
474 THKOKKMS ON SENSE AND SEPARATION [CHAP. IX
not have B on its boundary, any a2 having a point in common with T' is one of «*,•••, «£• Hence every boundary point of R is on the boundary of one of a] - • •, «'£. Hence the set of points of R and its boundary is identical with the set of all points of «?,•••, a^ and their boundaries. In other words,
THKOHK.M 81. For any planar polygonal region R there is a finite set of convex polygonal regions Rt, • • •, Rn such that the set of oil ]>(>i,itx of R , . . ., RB and their boundaries is identical with R and its boiDulary.
As a consequence, any set of points which consists of planar polygonal regions and their boundaries can be described as a set of points in a set of convex polygonal regions and their boundaries. Therefore no generality is lost in the following definition of a polyhedron by stating it in terms of convex polygonal regions.
DEFINITION. A set of points [P] is called a polyhedron if it sat- isfies the following conditions and contains no subset which satisfies them: [P] consists of a set of distinct points a°, a°, • • ., a°o, seg- ments a\, a\, • • ., a\ , and convex planar polygonal regions a'\, a\, • • •, aza such that each a1 is bounded by two a°'s and each a" by a simple polygon whose vertices are a° 's and whose edges are a1 's ; no a1 or a2 contains an a° and no two of the al's or a2's have a point in common; each a1 is on the boundary of an even number of a2's. The points a° are called the vertices, the segments a1 the edges, and the planar regions a2 the faces of the polyhedron.
Just as any point of a polygon can be regarded as a vertex, so any point of an edge of a polyhedron can be regarded as a vertex, and any segment contained in a face and joining two of its vertices can be regarded as an edge.
The relations among the vertices, edges, and faces of a polyhedron can be described by means of matrices Ht and H2 analogous to those of § 186. In the first matrix,
the element rj\ is 0 or 1 according as a? is not or is an end of aj. In the second matrix, TT _ / 2\
the element 77* is 0 or 1 according as a/ is not or is on the bound- ary of a2. The theory of the polyhedron can be derived from a
§§1SK,189] POLYIIKIH; A 47.')
discussion of these matrices just as that of the projective plane (a special polyhedron) has been derived in the sections above.
Thus the polygons which can be formed from the vertices and edges of the polyhedron are denoted by symbols of the form (xi> x# • • •> x<*^ as in § 186. They are all expressible as sums, modulo 2, of the boundaries of the faces together with P — 1 other polygons. The number P is called the connectivity of the polyhedron and is the same no matter how the polyhedron is subdivided into faces, edges, and vertices. It is determined by the following relation :
EXERCISES
1. Any polygonal region can be regarded as composed of a finite set of triangular regions together with portions of their boundaries, no two of the triangular regions having a point in common.
2. If R is a polygonal region, every broken line joining a point of R to a point not in R has a point on the boundary.
3. For any three-dimensional polyhedral region R there is a finite set of polyhedral regions Rv R2, • • •, RB such that the set of all points of Rp R,, • • •, Rn and their boundaries is identical with R and its boundary. R1? R2, • • •, RB may be so chosen as all to be tetrahedral regions.
4. If a polyhedron is the boundary of a convex region, each edge of tin- polyhedron is on the boundaries of two and only two of its faces.
189. Subdivision of space by planes. The theorems of § 185 generalize at once into the following. The proofs (with one excep- tion) are left to the reader.
THEOREM 82. The points of space which are not upon any one of a finite set of planes TT lt trn,— •,irnfall into a finite number a3 of convex ret/ions such that any segment joining two points of different regions contains at least one point of TT,, 7T2, • • •, irn. The number af satisfies
w (»-!)(» -2) the ^nequakty n^a^ — - -^ - - + n.
As in § 185, we indicate the «0 points of intersection of n planes IT,, *•,,•••,"•„ by «,...,<,
or any one of them by «°; the al linear convex regions determined by these points upon the lines of intersection, by
476 THKUIIKMS UN SENSE AND SEPARATION [CHAI-.IX
or anv <>iu- uf tli«-in by «'; the «2 planar convex regions determined by the lines of intersection upon the planes, by
ff,
or any one of them by aa; and the az spatial convex regions deter- mined by the planes, by « „* ns
tti» u->> ' ' *i **«,»
<>r any one of them by a*.
THEOREM 83. If the planes ir^ TTO, • • -, TT, are TW>£ all coaxial, (lie lo a ml < i r if of each a* is composed of a finite number of az's and of those al's and a°'s which bound the a*'s in question. Each a2 is vjiou the boundary of two and only two as's.
COROLLARY 1. If the planes TT^ 7T2, • • ., TTB are coaxial, aQ = 0, 0^=0, and the boundary of each a8 is composed of two a*'s together with the common line of the planes.
COROLLARY 2. If the planes are not all concurrent, any line through a point I of one of the regions a8 meets the boundary in two points P, Q. The segment PIQ consists entirely of points of the a3, and the complementary segment entirely of points not in the a*.
THEOREM 84. If an a1 is on the boundary of an a8, it is on the boundaries of two and only two a*'s of the boundary of the a8. Any plane section of an a* is a two-dimensional convex region bounded l>i/ a simple polygon which is a plane section of the boundary of the a8.
COROLLARY. The boundary of each a8 is a polyhedron..
THEOREM 85. The numbers ao, alt «2, «3 are subject to the relation
aO-ai + aa-«3=°-
Proof. The proof is made by induction. In the case of two planes, a0 = 0, al = 0, af = 2, ag = 2. Assuming that the theorem is true for n planes, let us see what is the effect of introducing a plane Trn+v This plane is divided by the other planes into a number of convex two-dimensional regions equal to the number of a8's in which it has points ; but it divides each of these a8's into two «8's. Hence the adjunction of these new «2>s and a*s increases «2 and «3 by equal amounts. The plane TTB+I, according to Theorem 8, Cor. 1, divides in two each a2 which it meets ; but it has a new a1 in com- mon with each such region. Here, therefore, aa and a1 are increased by equal amounts. The plane 7rn+, divides in two each a1 which it meets ; but it has a point in common with each such region. Hence,
§§is;>, 190] SUBDIVISION OF SPACK I'.V IM.ANKS 477
in this case, o^ and a0 are increased by equal amounts. Hence, if the formula is true for n planes, it is true for ?t + l.
COROLLARY. The number of a^s for ir^ ?r2, • • • , TTH which are not on lines of intersection of pairs of the planes TT^, rr2, • • •, irn_l is tie i tinnier I if which al — n^for the planes TT^ 7T2, • • •, irm exceeds al — aQ for the planes TT^ 7r2, • • •, irH_v
Proof. New a^s are produced by the introduction of TT, in two ways : (1) TTB may meet an a1 of TT^ 7r8, • • ., TT,., in a point ; if so, tins a1 is separated into two a1(s and a new «° is introduced ; (2) TT, may meet an a2 of TT^ 7r2, • • •, TTJI_I in a new a1. The only new a°'s pro- duced by the introduction of irn are accounted for under (1). Hence (2) accounts for the increase of «1 — #o, as stated above.
190. The matrices Hlf Ha, and H$. The relations among the convex regions determined by n planes which are not coaxial may be described by means of three matrices, which we shall call Hlf HS,
and H. In the first matrix.
H^W,
i = 1, 2, • • •, a0 ; j = 1, 2, • • •, a^ and rj^ = 1 or 0 according as a? is or is not an end of aj. In the second matrix,
H3=(^)>
t = l, 2, • • -, a^ j = 1, 2, • • •, «2 ; and ?;?. = 1 or 0 according as aj is or is not on the boundary of aj. In the third matrix,
H8 = «.),
i = 1, 2, • • •, a2 ; j = 1, 2, • • •, «8 ; and 17^. = 1 or 0 according as a\ is or is not on the boundary of aj. Examples of these three matrices are those given in § 152 to describe the tetrahedron. It will be noted that Hl has two 1's in each column, and Hg two 1's in each row.
Corresponding to the matrix Ht, there .is a set of aQ linear equations (modulo 2) ff)
Let the symbol (x0, xlf • • •, xa^, where the #t's are 0 or 1, be taken to represent a set of a^s containing a^1 if xk = l and not containing it if xk= 0. Just as in § 186, this set of o^'s will be the edges of a polygon or set of polygons if and only if (xtf xlt • • • , x^ is a solution of (Hj).
Just as in § 186, the sum of two sets of ]>olygon8 (modulo 2) will l>o
478 TIIF.oKKMS ON SENSE AND SEPARATION [CHAP.IX
taken to be the set of polygons represented by the sum of the symbols
•'#•••» x«) f°r tue two sets of Polyg°u8- Tne sum» modulo 2, of two sets of polygons pl and pt is therefore the set of polygons whose f.l;_;t>s appear either in pl or in p^ but not in both pl and p2.
By the reasoning in § 186, «0 — 1 of the equations (Ht) are linearly independent, and the other one is linearly dependent on these. The columns of H2 are the symbols (x0, x^ — -, xa) for the boundaries of the a2's and hence are solutions of (Ht).
Corresponding to the matrix H2, there is a set of al linear equations (modulo 2)
(H,)
Let the symbol (xlt x^ • • ., xai), where the #t's are 0 or 1, be taken to represent a set of «2's containing al if xt = l and not containing it if xt = 0. If this symbol is a solution of (H2), it represents a set of a2's such that each a1 is on the boundaries of an even number (or zero) of them ; i.e. it represents the faces of a polyhedron or a set of polyhedra.
The columns of Hg represent the boundaries of the «8's. By Theorem 84 any a1 of the boundary of an a8 is on the boundaries of two and only two a2's of this boundary. Hence the columns of H0 are solutions of (HJ.
8 \ 2'
Corresponding to the matrix Hg, there is a set of «2 linear equations (modulo 2)
Let the symbol (xlt x^ • • •, xa^, where the a^'s are 0 or 1, be taken to represent a set of «3's containing a*k if xk = 1 and not containing it if xt = 0. If this symbol is a solution of (H8), it represents a set of a*'s of which there is an even number on each a2. It is easily seen that the only such set of a8's is the set of all «8's in space. Hence the only solutions of (H3) are (0, 0, • • •, 0) and (1, !,•••, 1). Hence there are ag — 1 linearly independent equations in (Hg) on which all the rest are linearly dependent.
Let the ranks* of the matrices Ht, H2, Hg be r,, rt, r3 respectively.
•The rank of a matrix is the number of rows (or columns) in a set of linearly independent rows (or columns) on which all the other rows (or columns) are linearly dependent.
§§ 190, 191] THE RANK OF H, 479
By what has been seen above
ri=ao-l> V^.-1- The discussion in the next section will establish that
r2=ai-«o-
191. The rank of H2. Let us now suppose that wlf TT,, • • •, TTB are not all on the same point and that the notation is so assigned that •TJ, 7r2, 7rg, 7r4 are the faces of a tetrahedron. By inspection of the matrices given in § 152, it is clear that for the case n = 4, «0 = 4, al==12, «2 = 16, aa=8, and r2 = 8 (a set of linearly inde- pendent columns of H2 upon which the rest depend linearly is the set of columns corresponding to TU, TW, TW, TSI, ru, TM, TM, and TW). The number of solutions of (Ht) in a linearly independent set upon which all the other solutions depend is «l — «0 + 1 = 9. Hence one solution which does not depend linearly upon the columns of H3, together with a set of eight linearly independent columns of H2, constitute a set of linearly independent solutions of (Hj) upon which all the others depend linearly. Any solution representing a projective line, e.g. (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), will serve this purpose.
In case n>4, the columns of H,, fall into three classes: (1) those representing the boundaries of a2's in irn ; (2) those representing the boundaries of «2's which are not in Trn but have an a1 in TT, ; and (3) those representing the boundaries of a2's which have no a1 in 7rn.
Any column of Class (1) is expressible as a sum of columns of Classes (2) and (3). For the a2 whose boundary it represents is on the boundary of an a8 whose boundary has no other a2 in common with ?rn (cf. § 150). Since each a1 on the boundary of an a* is on the boundary of two and only two a2's of the boundary of the a8 (Theorem 84), it follows that the given column is the sum of the columns which represent the boundaries of the other a^s on the boundary of the a8. These columns are all of Classes (2) or (3).
Each a1 which is not on a line of intersection of two of the planes •nV •"•„,•••, 7rn_l is the linear segment in which one of the a^s determined by TT^ 7r2, • • •, irn_^ is met by TT.. Hence the row of H2 corresponding to this a1 contains just two 1's in columns of Class (2), and the sum of these two columns of Class (2) is the symbol for
480 THKOUKMS <>N SKNSK AND SEPARATION [CHAP. IX
the boundary of one of the aa's determined by 7rt, TT^ • - -, irn_r Moreover, the columns of Ha of Class (2) form a set of pairs of tliis sort, since every a1 of irn is either on a line of intersection of two of the planes TT^ TT,, • • -, TTH_I or is an edge of two and ouly two a*'s not in 7rM.
No one of such a pair of columns of H2 can enter into a linear relation among a set of columns of Classes (2) and (3) unless the other does. For this column would be the only column of the set containing a 1 in the row corresponding to the a1 common to th^ boundaries of the a2's represented by the two columns, and hence the sum of columns could not reduce to (0, 0, • • -, 0).
Let Hj be the matrix consisting of the columns of Class (3) of HS and the sums of the pairs of columns of Class (2) discussed in the last two paragraphs. According to the last paragraph the rank of H2 is less than the rank of H2 by the number of these pairs of columns ; and by the corollary of Theorem 85 this number is the difference between the values of «1— ao for TT^ 7r2, • • •, ?rn and for
The columns of H2 are the symbols in terms of the a!'s deter- mined by TT^ 7r2, • • •, ?rn for the boundaries of the a2's determined by TTj, 7r2, • • ., rrrn_l. Hence any two rows of this matrix which cor- respond to a pair of a^s into which an a1 determined by TT^ 7r2, •"•••, 7rB_1 is separated by TTH must be identical; and if one of each such pair of rows is omitted, H2 reduces to the H2 for TT^ TTO, • • •, irn_v Hence H2 has the same rank as the H2 for TT^ 7r2, • • •, irn_r
Since the difference in the ranks of H2 for -TT^ 7r2, • • -, irn and of Hj is the same as the difference between the values of al — aQ for TTj, 7r2, . . . ?TB and for TT^ 7T2, • • •, irn_v it follows that the introduc- tion of TT, increases the rank of H2 by the same amount that it increases at — aQ. Since a^ — #0 = r2 for n = 4, the same relation holds for all values of n. Hence we have
THEOREM 86. For a set of planes 7rjt 7r2, • • •, TTH which are not all concurrent,
By Theorem 85 this relation is equivalent to
192. Polygons in space. THEOREM 87. The symbol (xlt #2, • • •, xaj for a line is not linearly dependent on the columns of H2.
§192] POLYGONS IN SPACE 481
Proof. Let TT be any plane not containing any of the points a°. The boundary of any aa is an even polygon in the sense of § 187 and is met by TT in two points or none, the two points being on different edges, if existent. The sum, modulo 2, of two sets of poly- gons pv p^ each of which is met by TT in an even number (regard- ing zero as even) of points is a set of polygons p met by ir in an even number of points ; for if TT meets pl in 2 kl points and pn in 2 /,-._, points, and if ka of these points are on edges common to p and p2, TT must meet p in 2^+2 &a— 2 &g points. Hence any polygon which is a sum of the boundaries of the aa's is met by TT in an even number of points; i.e. any polygon represented by a symbol 0*v #2' * ' *' ^i) lmeai>ly dependent on the columns of H2 is met by TT in an even number of points. Since no line is met by ir in an even number of points, the symbol representing it cannot be a sum of any number of columns of H2.
THEOREM 88. All solutions of (HJ are linearly dependent on a set of r2(i.e. al — ao) linearly independent columns of H2 and the symbol (xt, #2, . . ., xai) for one line.
Proof. It has been shown that the rank of Ht is a0— 1. The number of variables in the equations (H^ is fly The number of linearly independent solutions in a set on which all the rest are linearly dependent is therefore o^— <ro+l. Since the rank of H2 is a\ ~ ao> anc^ ^ne col11111118 of H2 are solutions of (H^, there are al — «o linearly independent columns of H2 which are solutions of (H^ ; and since the solution of (H^ which represents a line is not linearly dependent on these, the statement in the theorem follows.
In the proof of Theorem 87 it appeared that any polygon which is a sum, modulo 2, of a set of polygons bounding a^s is met by a plane which contains none of its vertices in an even number of points. Since a line is met by a plane not containing it in one point, an argument of the same type shows that any polygon which is a sum, modulo 2, of a line and a number of polygons bounding a^s is met by a plane containing none of its vertices in an odd number of points. Thus we have, taking Theorem 88 into account :
THEOREM 89. DEFINITION. A polygon which is the sum, modulo 2, of a number of polygons which bound convex planar regions is met by any plane not containing a vertex in an even number of points
THKOREMS ON SENSE AND SEPARATION [CHAP.IX
is called an, even polygon. A polygon which is the sum, modulo 2, of a line and a number of polygons which bound convex planar regions is met by any plane not containing a vertex in an odd number of f KI i /its and is called an odd polygon. Any polygon is either odd or even.
Suppose a polygon p is the sum, modulo 2, of the boundaries of a set of convex regions a*, •••,«*. The set of points [P] in a*, a* or on their boundaries is easily seen (by an argument analogous to that given in the proof of Theorem 75) to be a connected set. By an extension of the definition in § 150 p may be said to be the boundary of [P]. From this point of view an even polygon is a bounding polygon and an odd polygon is not.
193. Odd and even polyhedra. It has been seen in § 190 that the solutions of (H2) represent polyhedra or sets of polyhedra. The converse is also true, as is obvious on reference to the definition of a polyhedron. The sum of two symbols (xlt x3, • • •, x^) which repre- sent sets of polyhedra is a symbol representing a set of polyhedra. This is obvious either geometrically or from the algebraic considera- tion that the sum of two solutions of (H2) is a solution of (H2).
The set of polyhedra p represented by the symbol which is the sum of the symbols for two sets of polyhedra pl and p2 is called the sum, modulo 2, of pl and pz. As in the analogous case of polygons, p is a set of polyhedra whose faces are in pl or in p2 but not in both P1 and pt.
The number of variables hi (H2) is «2 and the rank of H2 is «2 — ag by Theorems 86 and 85. Hence the solutions of (H2) are linearly dependent on a set of «3 linearly independent solutions. Since any «8 — 1 of the columns of Hs are linearly independent, such a set of columns, together with one other solution linearly independent of them, will furnish a set of linearly independent solutions of (H2).
The symbol for any plane is a solution of (H2) linearly inde- pendent of the columns of H3. For let I be any line meeting no a° or a1. Any column of H.t represents the polyhedron bounding an a8, and such a polyhedron is met by / in two points or none. By reasoning analogous to that used in the proof of Theorem 87, it fol- lows that I meets the sum, modulo 2, of the boundaries of any number of a*'s in an even number of points or none. Since / meets each plane TTj in one point, the symbol for IT. is not 'linearly dependent on the columns of Hg. By the last paragraph we now have
§;!<:;, im] ODD AND EVEN POLYHEDRA
THEOREM 90. Any solution of (Hs) is linearly dependent on at — 1 columns of Hg and the symbol for any one of the planes ir^ irt, • • •, irn.
COROLLARY 1. Any polyhedron is the sum, modulo 2, of a subset of a set of polyhedra consisting of one plane and all polyhedra which bound convex regions.
Proof. Let TT^ Tr2, • • •, irn be a set of planes containing all vertices, edges, and faces of a given polyhedron and such that TT^ 7T2, 7rg, irt are not concurrent. By the theorem the given polyhedron is either expressible as a sum of the boundaries of some of the a"s deter- mined by TJ-J, 7r2, • • -, 7r)4 or as a sum of one of these planes and some of the «8's.
In the course of the argument above it was shown that any poly- hedron expressible in terms of the boundaries of the a8's was met in an even number of points by any line not meeting an a° or an a\ One of the planes TT^ 7r2, • • -, trn is met by such a line in one point. Hence any polyhedron which is the sum of such a plane and a number of the boundaries of «8's is met by this line in an odd number of points. Hence
COROLLARY 2. DEFINITION. A polyhedron which is the sum, modulo 2, of a number of boundaries of convex three-dimensional regions is met in an even number of points by any line not meeting a vertex or an edge. Such a polyhedron is said to be even. A poly- hedron which is the sum, modulo 2, of a plane and a number of boundaries of convex three-dimensional regions is met in an odd number of points by any line not meeting a vertex or an edge. Such a polygon is said to be odd.
EXERCISE
Let p be a polygon and TT a polyhedron such that IT contains no vertex of p and p contains no vertex or edge of IT. If p and it are both odd they have an odd number of points in common. If one of them is even they have an even number (or zero) of points in common.
194. Regions bounded by a polyhedron. An even polyhedron p is the sum of the boundaries of a set of convex three-dimensional polyhedral regions, and we may assign the notation so that these regions are denoted by af, a.f, - - >, a*.
The polyhedron p is also the sum of the boundaries of
a*+i» at+2> * * *' a«,»
484 THKOKKMS ON SKNSE AND SEPARATION [CHAP.IX
because the sum of all the columns of Hg is (0, 0, • • -, 0). There is no other linear expression for p in terms of the boundaries of the a"s, because there is only one linear relation among the columns Of H8.
This is all a direct generalization of what is said at the beginning of £ 187. As in § 187, it is easily seen that the points of a*, a*, • • • together with those points of their boundaries which are not on j>, constitute a region bounded by p ; and that the points of at8, «*. + 1, • • •, a*, together with those points of their boundaries which are not on p, constitute a second region bounded by p. With a few addi- tional details (which are generalizations of those given in the proof of Theorem 75) this constitutes the proof of the following theorem :
THEOKKM 91. Any even polyhedron is the boundary of each of tiro and only two regions which contain all points of space not on the polyhedron. These regions are such that any broken line joinim/ « point of one region to a point of the other contains a point of the polyhedron. Any two points of the same region can be joined by a broken line consisting entirely of points of the region.
By a similar generalization of Theorem 76, we obtain
THEOREM 92. Any odd polyhedron is the boundary of a single
region containing all points not on the polyhedron. Any two point*
of this region can be joined by a broken line not containing any pi I nt
of the polyhedron.
COROLLARY. Any point P on a polyhedron can be joined to «n>/
point not on it by a broken line containing no point of the polyhedron
except P.
195. The matrices E1 and E2 for the projective plane. DEFINITION. A segment, interval, broken line, polygon, two-dimensional convex region, or three-dimensional convex region associated with a sense- class among its points is called an oriented or directed segment, interval, broken line, polygon, two-dimensional convex region, or three-dimensional convex region.
DKKIMTION. Let a1 be any segment which, with its ends A and B, is contained in a segment s, and let sl denote the oriented segment obtained by associating a1 with one of its sense-classes. The sense- class of a1 is contained -in a sense-class of-s which is either S (AO) or S (OA) if 0 is any point of a\ In the first case A is said to be
§195]
MATRICES OF ORIENTATION
485
related to s1 and in the second case A is said to be /
related to s1.
To aid the intuition, we may think of an oriented segment a« marked with an arrow, the head of which is at the end which is jHjsitively related to thr oriented segment.
Obviously, if one end of an oriented segment is positively related to it, the other end is negatively related to it, and vice versa.
J )KFIXITION. The sense-class S(A1AVA^ of a polygon^4^43 • • • AnAv and the sense-class S (AB) on the edge A^A% are said to agree in case of the order {A^ABA^ and to disagree in case of the order
Returning to the notation of § 185, the segments a,1, . . ., a] may each be associated with two senses. They thus give rise to 2 al directed segments. Assigning an arbitrary one of the two senses to each a\ we have al oriented segments to which we may assign the notation s,1, • • >,s^. We shall denote the oriented segment obtained by changing the sense-class of */ by — s.1 and call it the negative of s\
The relations of the 5lJs to the points a°, • • -, a°o may be indicated by means of a matrix which we shall call Er In the matrix Ej the element of the t'th row and yt-h column shall be 1, — 1, or 0, according as the point a? is positively related to, negatively related to, or not an end of, the oriented segment s?.
It is clear that the signs 1 and — 1 are interchanged in the /th column of this matrix if the sense-class of sj is changed. Since the sense-class of each segment is arbitrary, a matrix equivalent to Et can be obtained from the matrix Hj, § 186, by arbitrarily changing one and only one 1 in each column to — 3.
In the case of the triangle, by letting the segments a, a, ft, /9, 7, 7 give rise to sl} *2, • • • , *8 respectively, we derive the following matrix from H of 151:
|
*i |
•"2 |
":l |
»4 |
•« |
«e |
|
|
A |
0 |
o |
1 |
1 |
1 |
1 |
|
B |
1 |
1 |
0 |
0 |
-1 |
-1 |
|
C |
-1 |
-1 |
-1 |
-1 |
0 |
0 |
The elements of the matrix Ej may be regarded as the coefficients of a set of linear equations analogous to the equations (Ht) of § 186,
THKuKKMS ON SKNSK AND SKPARAT1ON [CHAP.IX
where, ln>\ve\ei. tin- variables and coefficients are not reduced with t to any modulus. These equations arise as follows:
Let (.r,, u'a, • • •, .<-0i) be a symbol in which the x's can take on any integral values, positive, negative, or zero, and let this symbol rep- resent a set of oriented segments comprising «/ counted xi times if I jxisitive, — ,s-/ counted — xf times if x{ is negative, and neither */ nor — */ if x. is zero, i taking on the values 1, 2, • • •, ar
The sense-class of an oriented polygon agrees with a definite sense- class of each of its sides and thus determines a set of oriented seg- ments. The symbol (xlt x2, • • •, xai) for this set of oriented segments may also be regarded as a symbol for the oriented polygon. Each vertex of the polygon is positively related to one of the oriented segments represented by (a^, «2, • • •, xai) and negatively related to another. Thus if s/ and s? meet at a certain vertex to which they are both positively related according to the matrix E^ we have that 3^=1 and #,. = — 1 or that xt=—\ and #,•=! in the symbol (xlt - • •, xai) for any directed polygon containing the sides a,, and a.. The x's corresponding to the segments not in the polygon must of course be zero. Hence the symbol (x^ • • •, #ttj) must satisfy the linear equation whose coefficients are given by the row of Et cor- responding to the vertex in question. If sf and s- are oppositely related to a vertex according to the matrix EJ} we must have «,.= ! and x.= \ or xi = —\ and x.— — \ in the symbol for any directed polygon containing the sides a. and a.. Hence in this case also the linear equation given by the corresponding row of Et must be satis- fied. Finally, the equation given by a row of Et corresponding to a point which is not a vertex of the polygon is satisfied because all the x.'a corresponding to edges meeting at that point are zero. Hence the symbol for a directed polygon must be a solution of the linear equations whose coefficients are the elements of the rows of the matrix Er These equations shall be denoted by (E^. In the case of the triangle they are x +x +x +x =Q
8456
(6) . a^ + ajj— 'a;6 — a?e=0,
-^-^-^-^ = 0.
By reasoning entirely analogous to that of § 186, it follows that any solution of (Et) in integers represents one or more directed simple polygons. The situation here differs from that described in
§ii«J .MA TRICKS OF ORIENTATION
the modulo 2 case, in that the same side iiHiy euter into more than one polygon and the same polygon may be counted any number of times in a set of polygons.
Since each column of the matrix Ej contains just one 1 and one — 1, the sum of the left-hand members of the equations (E ) vanishes identically. There can be no other linear homogeneous relation among the equations (E^, because the matrix Ej and the equations (E^ when reduced modulo 2 are the same as Ht and (H^, and so any linear relation among the equations. (Et) would imply one among (Hj).* Hence the number of linearly independent equations of (E^ is «0 — 1. The number of variables being av the number of linearly independent solutions is a^— «0+1. In view of Theorem 74, this number is equal to aa.
It will be recalled that in the modulo 2 case one class of solu- tions of Equations (H^ is given by the columns of the matrix H2. These columns are the notation for the polygons bounding the con- vex regions a*, - • •, a^. If each of these polygons be replaced by one of the two corresponding directed polygons, a set of solutions is determined for the equations (Et). These solutions are obtained directly from the matrix H2 by introducing minus signs so that the columns become solutions of (Ej). This is possible in just two ways for each column, because each polygon bounding an a* has two and only two sense-classes. A matrix so obtained shall be denoted by E2. In the case of the triangle such a matrix is
/ 0 -1 0 1\
1 0-1 0
0110 -1 0 0-1
0 0 1-1 ,1-1 0 0,
It is evident on inspection that the rank of this matrix is equal to the number of columns. That is to say, unlike those of HS, the
» The coefficients of any linear homogeneous relation among the rows of K, in;.y be taken as integers having no common factor. Hence on reducing modulo '2. it would yield a linear relation among the rows of II,. But as the only linear relation aiming the rows of H, is that the sum of all the rows is /..TO, there is no linear rela- tion among the rows of E, not involving all the rows. Tin-re rould not be two such relations among all the rows of E,, because by combining them we could drrivr a relation involving a subset of the rows.
488 THEOREMS <>N SENSE AND SEPARATION [CHAP.IX
columns of E are linearly independent. The same proposition holds good for the matrix Es in the general case. This can be pi"\cd us follows :
liv the reasoning used above for the rows of Et and Hj, it fol- lows that any linear relation among the columns of E2 implies one among the columns of H2. Since the only such relation among the columns of H2 involves all the columns, we need only investigate linear homogeneous relations among the columns of E2 in which all the coefficients are different from zero. If such a relation existed, two columns of E2 corresponding to regions having an edge in common would have numerically equal multipliers in the relation, else the elements corresponding to the common edge would not cancel. But since any two of the convex regions a2 can be joined by a broken line consisting only of points of these regions and of the edges of their bounding polygons, it follows that all the cot-tti- cients in the relation would be numerically equal, i.e. they could all be taken as 4- 1 or — 1.
Now the n lines llt • • ', ln containing all the points and segments of our figure are not all concurrent ; three of them, say llt /2, /3, form a triangle. Let us add together all the terms of the supposed rela- tion corresponding to regions a2 in one of the four triangular regions determined by l^ 12, 13. The elements corresponding to edges interior to this triangular region must all cancel, because they cannot cancel against terms corresponding to regions a2 exterior to the triangular region. The sum must represent an oriented polygon of which the edges are all on the boundary of the triangular region. This oriented polygon, by § 183, must be identical with the boundary of the tri- angular regions associated with one of its two sense-classes. If we operate similarly with the other three triangular regions determined by /j, /2, lg, we obtain three other oriented polygons. But since the linear combination of the columns of E2 is supposed to vanish, each edge of the four triangular regions should appear once with one sense and once with the opposite sense, and this would imply that in the case of a triangle there would exist a linear homogeneous relation among the columns of E2, contrary to the observation above. Hence in every case the «2 columns of E2 are linearly independent.
Since there are only aa linearly independent solutions of the equa- tions (Ej), it follows that all the solutions of (Ej) are linearly
§§u«,W8] MATRICES OF ORIENTATION 489
(h- [if nth- nt <nt the columns of Ea. This is in sharp contrast with the property of the equations (Hj) stated at the end of § 186.
196. Odd and even polygons in the projective plane. Ix-t us apply the results of the section above to the theory of odd and even poly- gons. Since any polygon is expressible in terms of the columns of E2, an odd polygon must be so expressible. Let us write this expres- sion in the form .-.
where s*, •••,«£ represent the columns of Es, p is the symbol for the given oriented polygon, and p and y^-'^y^ are integers which may be taken so as not to have a common factor.
Since the coefficients do not have 2 as a common factor, (7) does not vanish entirely when reduced modulo 2. But since an odd polygon is not expressible in terms of the columns of H2, p must contain the factor 2, and (7) must reduce, modulo 2, to an identity among the columns of H2. The only such identity is the one involving all the columns of H2. Hence the yt's are all odd. But in order that the edges not on the odd polygon p shall vanish, the y's correspond- ing to s'2's having an edge in common must be equal. Since any two points not on p can be joined by a broken line not meeting p (Theorem 76), it follows that all the y's are equal If they are all taken equal to ± k, it is obvious that p = 2 k. Hence we have the theorem:
THEOREM 93. The symbol p for any odd polygon is expressible in the form „.
(8) 2p=2>A»
1=1 where each e{ is + 1 or — 1.
This theorem may be verified in a special case by adding the columns of the matrix E2 given above for a triangle. The sum is (0, 0, 2, - 'J, <>, <>), whi.-h represents a line counted twice. The number 2 is called the <-t,,-jficirnt <>f torsion of the two-sided polygon (cf. Poincare', Proceedings of tin- London Mat lu-rnat i.-:tl Society, Vol. XXXII (1900), p. 277. The systematic use of th«- matri. •• -s E,. E.,, etc. is due to Poincare').
Another form of statement for Theorem 93 is the following : If tlie region bounded by an odd polygon p be decomposed into convex reyitmx each bounded by an even polygon, each edge of p is on the boundary of two of these convex regions.
490 IIIKOKKMS OX SENSE AND SEPARATION [CHAP. ix
Au even j>olygon p is also expressible in the form (7). Aside from a common factor of all the coefficients, there is only one ex- 1'ivssion for p of the form (7), for if not, by eliminating p we could obtain a linear homogeneous relation among the columns of E2.
Let R be one of the two regions bounded (Theorem 75) by p, which contains one of the convex regions «2 for which the corre- sponding //,- in (7) is not zero. Any two s2's corresponding to a?'s having an edge in common must be multiplied by numerically equal y's in (7) in order that the . symbol for the common edge shall not appear in p. Since any two points of R can be joined by a broken line consisting entirely of points of R, this implies that the coefficients y{ corresponding to the «2's in R are all numerically equal to an integer k. From this it follows that the sum of the terms in the right-hand member of (7) which correspond to «2's in R is equal to p, because each edge of p is an edge of one and only one of the «2's in R. Since the equality just found is of the form (7), and (7) is unique, we have that p and y1? • • •, yai are all numeri- cally equal to k. Obviously the factor k can be divided out of (7). Hence we have
THEOREM 94. The symbol p for an even polygon is expressible in the form
(9) F'
where et is 0 or + 1 or — 1. The af's such that the e-s with the same subscripts are not zero are the af's in one of the regions R referred to in Theorem 75.
DEFINITION. By the interior (or inside) of an even polygon is meant that one of the two regions determined according to Theorem 94 which contains the «f s having the same subscripts as the non-zero «/s in (9). The other region is called the exterior of the polygon.
EXERCISE
Identify the interior of a two-sided polygon as defined above with the int.-rior jis dffincd in § 187.
197. One- and two-sided polygonal regions. Let A^ Az, - • •, An be a polygon which is the boundary of a convex region R for which there is a convex region R' containing R and its boundary. If 0 and
§197] MATRICES OF ORIENTATION 491
O'are any two points of R, then X(".ll.lj=S((t'AlAj (<-f. £ 161) with respect to R' because O and O1 are on the same side of the line A A in R. Again, \
because Al and Aa are on opposite sides of the line OA9, At and At are on opposite sides of the line OAt, etc.
A sense-class in R, which we shall call positive, determines a positive sense-class in any convex region R' containing R, i.e. the sense-class containing the given sense-class of R. This, in view of the paragraph above, determines a unique sense-class on the poly- gon bounding R, by the rule that if S^A^AJ is positive, where O is in R, then S(A^AZA^ is positive on the boundary of R; and if S(OAZA^ is positive, then S(ASA1AH) is positive on the boundary of R. From § 161 it follows without difficulty that this determination is independent of the choice of the convex region R'.
Conversely, it is obvious that by this rule a sense-class on the boundary of R determines a definite sense-class in R.
DEFINITION. Let a2 be any planar convex region which, with its boundary, is contained in a convex planar region R, and let a1 be any segment on the boundary of a2. Let sl denote the oriented segment obtained by associating a1 with one of its sense-classes, and s2 denote the oriented region obtained by associating a2 with one of its sense-classes. The sense-class of s2 is contained in a certain sense-class of R which may be denoted by S(OAB), where 0 is in d2 and A and B are on a1. If S(AB) is the sense-class of s1, then 81 and s2 are said to be positively related ; and if S(AB) is not the sense-class of s1, they are said to be -negatively related.
As pointed out above, this definition is independent of the choice of R. Let Rj and R0 be two convex regions having no point in com- mon and bounded by two polygons A^A^At • • • Am and A^A^B^ •••!?„ respectively which have in common only the vertices A^ and Ay and the points of the edge A^f Suppose, also, that R,, R2 and their boundaries are contained in a convex region R. These conditions are satisfied if R4 and R., are a2's, and A^A,t is an a1, determined by a set of lines four of which are such that no three are concurrent.
The rule given above for determining positive sense on the boundaries of R, and R., requires that if S (OA^A.) is positive for 0 a point of Rt, then S(A^AA^ must be positive on the boundary
492 TH Koi: K.MS ON SKNSE AND SEPARATION ICHAP.IX
of R,, where A is a point of the edge A^A^ If (/ is any point of R2, it is on the opposite side of the line A^At from 0 in R. Hence S^AfAJ is positive, and hence S(ASAA1) must be the positive sense- class on the boundary of R.
Let R, and RS be two of the aa's determined by a set of lines '•'»•••>',> let tne boundary of R( associated with the positive sense-class as determined in the last paragraph be denoted 1>\ (./-,. xf> •••,««,) according to the notation of §195; and let the boundary of Ra associated with the jwsitive sense-class determined at the same time be (yv yz, • • •, yai). The notation may be assigned so that .t-j and y1 refer to the edge A^AZ common to the boundaries of R, and Rf In this case, if 2^ = 1, y1= — 1, and if xv= — 1, y^= + 1 ; for the positive sense for the boundary of RI is <S' (A^AA^ and for the boundary of R2 is S (A^AA^. Hence the sum of the two sym- bols (xlt xt, • • ., xaj) and (ylt yz, - - •, ya) is the symbol for the bound- ary of the region R' composed of Rlt R2 and the common edge J,./.,, this boundary being associated with a sense-class *S" which agrees with the positive sense-class on any edge of the boundary of Rt or R2 which is an edge of the boundary of R'.
By repeated use of these considerations it follows that if a set of a2's with their boundaries constitute a convex region R and its boundary, the symbol (a^, #2, • • -, xat) for the boundary of R asso- ciated with a sense-class which is designated as positive, is the sum of the symbols for the boundaries of the a8's, each associated with its positive sense-class. In other words, the symbol for the boundary of R associated with its positive sense-class is the sum of a set of columns of H2, each multiplied by + 1 or — 1 so that it shall be the symbol for the boundary of the corresponding a2 associated with the sense-class which is positive relatively to the positive sense-class of R. By comparison with Theorem 94, it follows (as is obvious from other considerations also) that any polygon which is the boundary of a convex region is even.
The argument in the paragraph above applies without essential modification to any region bounded by a polygon and having a unique determination of sense according to § 168. Hence any poly- gon bounding a two-sided region is even.
Moreover the steps of the argument may l>e reversed as follows : If the symbol for any oriented polygon p be expressible in terms
§§i'.>7, 198] MATRICES OF ORIENTATION l<»:{
of the columns of H2, in the form (7), where the non-zero coefficients are «,v eit, • • •, eit, p is the boundary of tlie region R consisting of <r, trt, • • •, (('a and those points of their boundaries which are not on p. If R' is a convex region contained in R, and ita positive sense-class be determined as agreeing with the positive sense-class of one of the regions a*, a^, • • •, a2t, it must agree with that of every a2 with which it has a point in common ; for otherwise the symbols for the common edges of two of the «2's would not cancel in (7). If R" is any other convex region contained in R, and its positive sense-class is also determined by this rule, the positive sense-classes of R' and R" must, by definition, agree in any region common to R' and R''. Hence R is two-sided according to § 168. Thus we have by comparison with § 196
THEOREM 95. The interior of an even polygon is a ttvo-sided region.
198. One- and two-sided polyhedra. Let the vertices of a poly- hedron be denoted by a", «2°, • • •, a«o, the edges by a*, «.], • • •, a^ and the faces by af, a2, • • -, a£. Assigning an arbitrary one of its sense-classes to each edge, there is determined a set of oriented segments s/, s.]> • • •> s«, and a matrix
in which i=l, 2, • • •, ao; j = 1, 2, • • • , ax ; and ej,. is + 1, — 1, or 0, according as a? is positively related to, negatively related to, or not an end of s*.
Assigning an arbitrary one of its sense-classes to each face, there is determined a set of oriented planar convex regions «*, *2S, • • •, «j?f
and a matrix ,-, , „.
^2 = W*
in which i = 1, 2, • . ., al ; j = 1, 2, • • •, aa ; and e* is + 1, — 1, or 0, according as «? is positively related to (cf. § 197), negatively related to, or not on the boundary of sf. By the last section each column of E2 is the symbol (xlt x2, • - •, xa), in the sense explained in § 195, for an oriented polygon obtained by associating the polygon bound- ing one of the a2's with one of its sense-classes. Changing the sense- class assigned to any a* to determine the corresponding sa amounts to multiplying all elements of the corresponding column of Ea by — 1. For simplicity let us at first restrict attention to polyhedra in which each edge is on the boundaries of two and only two faces.
494 Til Koi: K.MS ON SKNSK AND SEPARATION [CHAP. IX
In this case there are just two non-zero elements in each row of E2. Hence the sum of the columns of E2 will reduce to (0, 0, • • •, 0) if and only if the sense-classes have been assigned to the faces of the jH.lv lu-dron in such a way that one of these elements is + 1 and the other — 1 in each row. This means that each a1 is positively related to one of the s2's on whose boundary it is and negatively related to the other. Thus the faces are related as are the a"-'s which constitute a two-sided region bounded by an even polygon in the plane (§ 197).
DEFINITION. A polyhedron for which the sense-classes can be assigned to the edges and faces in such a way that each edge is positively related to one of the faces on whose boundary it is and negatively related to the other, is said to be two-sided, or bilateral; and one for which this assignment of sense-classes is not possible is said to be one-sided, or unilateral.
Changing the assignment of sense-classes on an edge amounts merely to multiplying the corresponding column of Et and row of E by — 1, and changing the assignment of sense-classes on a face amounts to the same operation on a column of E2. Consequently the polyhedron is two-sided if there is a linear relation whose coeffi- cients are 1's and — 1's among all the columns of E2, and it is one- sided if there is no such relation. It is also obvious from these considerations that if a polyhedron satisfies the definition of two- sidedness (or of one-sidedness) for one assignment of sense-classes to its edges, it does so for all assignments. We therefore infer at once :
THEOREM 96. A polyhedron is one- or two-sided according as the rank of E, is a or a — 1.
V £ £ £
By reference to § 195 we find
COROLLARY. Tlie projective plane is a one-sided polyhedron.
In the case of any polyhedron in which each edge is on the boundary of only two faces, it is seen that the only possible linear relation among the columns of E2 reduces to one in which each coefficient is + 1 or — 1, for any other relation would imply that a subset of the faces determines a i>olyhedron.
THEOREM 97. A polyhedron bounding a convex region R which is contained with its boundary in a convex region R', is two-sided.
Proof. Let sense-classes be assigned to the edges in an arbitrary way, but let sense-classes be assigned to the faces according to the
|19B] MATRICES OF ORIENTATION
following rule: Let a given sense-class S(PQRT) in R' be desig- nated as positive. Let 0 be any point of R and A, B, C, three nou- collinear points of a face of the polyhedron. The sense-class S(ABC) is assigned to this face if and only if S(OABC) is positive.
There is no difficulty in proving that if C and D are two points of an edge s.1 of the polyhedron bounding R, and E and E' points of the two faces having this edge on their boundaries, then E and E' are on opposite sides of the plane OCD. Hence
S(OCDE}^ S(OCDE').
Hence the sense-classes are assigned according to the rule above to the two faces having the edge s/ on their boundaries in such a way that Sf is positively related to one and negatively related to the other.
DEFINITION. By an oriented polyhedron is meant the set of oriented two-dimensional convex regions [s2] obtained by associating each face of a two-sided polyhedron with a sense-class in such a way that if sense-classes are assigned arbitrarily to the edges to determine directed segments, each of these directed segments is positively related to one of the oriented two-dimensional convex regions on whose boundary it is and negatively related to the other. The s2's are called the oriented faces of the oriented polyhedron, and the sl's its oriented edges.
COROLLARY. A given two-sided polyhedron determines tico and only two oriented polyhedra according to the definition above.
DEFINITION. Let a8 be a three-dimensional convex region which is contained with its boundary in a convex region R, and aa a two- dimensional convex region on the boundary of a8. Let 8* denote a* associated with one of its sense-classes, and let s2 denote a3 associated with one of its sense-classes. The sense-class of s8 is contained in one of the sense-classes, say S, of R. Let 0 be a point of a8, and A, B, C three points of a2, such that S(OABC) is S. Then if S(A 3d] is the sense-class associated with a2 to form s2, *2 and s* are said to be positively related. Otherwise they are said to be negatively related.
By § 161 this definition is independent of any particular choice of the convex region R containing ns and its boundary. From what has been proved above it follows that if each aa on the boundary of an a8 is associated with a sense-class in such a way as to be ]x»si- tively related to the oriented region drUTiiiiiu'd by «8 and one of its
496 THF.OKKMS ON SENSE AND SEPARATION [CHAP.IX
sense-classes, this set of oriented two-dimensional convex regions is an oriented polyhedron.
The definitions made in this section are extended to polyhedra in which each edge is on an even number of faces (instead of only two, as we have been supposing) as follows :
DEFINITION. A polyhedron is said to be two-sided if sense-classes can be assigned to the edges and faces in such a way that e;n h resulting oriented edge is positively related and negatively related to equal numbers of the resulting oriented faces.
EXERCISES
1. An odd polyhedron is one-sided and an even polyhedron is two-sided.
2. Make a discussion of one- and two-sided polyhedral regions in space analogous to the discussion for the two-dimensional case in § 197.
199. Orientation of space. The matrices of § 195 can be general- ized to the three-dimensional case. Let sf, s* , • • • , s^, be the oriented segments obtained by associating each of the segments at, al, - • •, ali with an arbitrary one of its sense-classes. In the first matrix,
i = 1, 2, • • •, «0; j— 1, 2, • • -, aj and e^. is + 1, — 1, or 0 according as «° is positively related to, negatively related to, or not an end of Sj. Ej can be formed from Hj by changing one 1 to a —1 in each column. The choice of the — 1 in the yth column amounts to the choice of the sense-class on a? which determines s?. As an exercise, the reader should form E1 from the Hx given for a tetra- hedron in § 152.
Sets of oriented segments sl are represented as in § 195 by sym- bols of the form (xlt xz> • • •, xai), where the x's are positive or negative integers. By the same argument as in § 195, if this symbol represents a set of oriented segments each of which is an edge of a polygon associated with that one of its sense-classes which agrees with a fixed sense-class of the polygon, it is a solution of the equations,
and, conversely, any solution of these equations is the symbol for one or more such sets of oriented segments. Thus any solution of may be regarded as representing one or more oriented polygons.
§199] MATRICES OF ORIENTATION 497
Let s2, s.;, • • •, s2t be the oriented two-dimensional convex regions obtained by associating each aa with an arbitrary one of its sense- classes. The oriented two-dimensional regions obtained by associating the /'s with the opposite- sense-classes may be denoted by — s'f, -*«>•••> — S4 respectively. In the second matrix,
i = 1, 2, • • ., a, ; / = 1, 2, . • •, «2 ; and e* is 1, — 1, or 0 according as «! is positively related to, negatively related to, or not on the boundary of sf. E2 can be formed from H3 by changing some of the 1's in each column of Ho to —1's in such a way that each column shall be a symbol (a^, #2, • • •, xai) for a set of sl's whose sense-classes all agree with that of the oriented polygon determined by associating the boundary of s2 with one of its sense-classes. This is possible by the argument at the beginning of § 197, since each column of H2 is the symbol for the boundary of one and only one a2. As an exercise, the reader should form E0 from the H2 given for a tetra- hedron in § 152.
A symbol of the form (xlt x2, • • •, x«t) in which each a; is a posi- tive or negative integer or zero may be taken to represent a set of oriented two-dimensional convex regions which includes sf counted Ff times if xf is positive, — sf counted — a\. times if a\. is negative, and does not include sf if xi is zero. If this symbol represents an oriented polyhedron (§ 197), it is a solution of the equations
For consider the ith of these equations:
If an oriented face of the oriented polyhedron is positively related to s], it contributes a term +1 to the left-hand member of this equation ; for if s£ is this oriented face, xt= 1 and €?*= 1 ; and if — 8% is this oriented face, xk— — \ and €?,.= — 1. An oriented face which i>i negatively related to s\ contributes a term — 1 to the left-hand member of this equation ; for if s% is this oriented face, xk = 1 and €2A = — 1 ; and if — s% is this oriented face, xk— — l and eit= 1. Hence there are as many terms equal to + 1 as there are oriented faces positively related to s}, and as many terms equal to — 1 as there are
198 THKOKKMS ON SENSE AND SEPARATION [CHAP. IX
oriented faces negatively related to a}. If neither sf nor — .s; is in tin- oriented polyhedron, or if s£ does not have s] on its boundary, tlu- /th term of this equation is zero, for in the first case ort= 0 and in the second case e?x.= 0. Hence by the definition of an oriented polyhedron, each of the equations (E2) is satisfied if (xlt #2, • • -, '.,.) represents an oriented polyhedron. In particular (Theorem (.)7) the symbol for either oriented polyhedron determined by the boundary of an a8 is a solution of (E2).
One-sided polyhedra do not give rise to solutions of (E0).
Let «* and — sf, s.* and — s* , • • • , «£, and — s's be the pairs of oriented three-dimensional convex regions determined by a*, a *, • • •, a *s respectively according to the definition in § 197. In the third matrix,
E3=(4)>
t = l, 2, • • •, «2; j = 1, 2, • • •, «8; and e^ is + 1, — 1, or 0 according as s? is positively related to, negatively related to, or not 011 the boundary of s?. The matrix Eg can be formed from Hg by changing 1's to — 1's in the columns of Hg in such a way that the resulting columns are the symbols for oriented polyhedra and therefore solu- tions of (E2). This is possible by Theorem 96. As an exercise, the reader should form Eg from the H8 given for a tetrahedron in § 152.
The sum of the columns of Et is (0, 0, • • -, 0) because each row of Et contains one + 1 and one — 1. There can be no other linear relation among the columns of Et, because this would imply, on reducing modulo 2, more than one linear relation among the columns of Hj. Hence the rank of Et is ao— 1, and the number of solu- tions of Ej in a linearly independent set on which all the solutions are linearly dependent is 0^—0^+!.
Since the rank of H2 is a^ — aQ, and since every homogeneous linear relation among the columns of E2 implies one among the columns of H2, the rank of E0 is at least al — aQ. It is, in fact, at least a, — a0 + 1 because, by Theorem 93, the symbols for a set of columns Cj, ca, • • •, ck which represent oriented polygons bounding all the s2's of a projective plane satisfy a relation of the form
(10) «1e1+«8e8+...+Vt = 2J,
where / is the symbol for a line in this plane and ex, e^, • • •, eH are + 1 or — 1. Reducing modulo 2, this gives rise to a homogeneous linear relation among the columns of H which is not one of those
§i«»] MATRICES OF ORIENTATION 499
obtained by reducing the homogeneous linear relations among tin- columns of Ea.
Thus there are at least al — «o+ 1 linearly independent columns of Effl. These are all solutions of (Et), and as there are not more than «t — aQ + 1 linearly independent solutions of (Ej), there are not more than al — aQ + 1 linearly independent columns of ES. Hence the rank of E2 is «t — «0 + 1, which by Theorem 85 is the same as «,-«. + !.
In consequence, the symbol (xl9 xi2, • • •, xai) for any oriented poly- gon is linearly expressible in terms of the symbols for oriented poly- gons which bound convex planar regions. It can easily be proved that in case of an odd polygon this expression takes the form (10) where, however, the polygons denoted by cx, C2, • • •, ck are not neces- sarily all in the same plane.
Since the number of variables in the equations (E2) is «2 and the rank of E2 is «2 — a.A + 1, the number of solutions in a linearly inde- pendent set on which all solutions are linearly dependent is «8 — 1. The columns of E, are all solutions of (E,). Hence the rank of E.
o > *£' >
cannot be greater than «b — 1. It cannot be less than ag — 1, because, on reducing modulo 2, this would imply that the rank of Hg was less than «0 — 1. Hence the rank of Ea is «„ — 1. Since the symbol
o o o »
for any oriented polyhedron whose oriented faces are s2's or — s2's is a solution of (E2), it follows that it is expressible linearly in terms of the symbols for oriented polyhedra which bound convex three- dimensional regions.
Since the rank of Eg is «8 — 1, the set of equations
must have one solution distinct from (0, 0, • • •, 0). When reduced modulo (2) this solution must satisfy (H8) and therefore, by § 190, reduce to (1, !,•••, 1). Since each equation in the set (E8) has only two coefficients different from zero, and these coefficients are ± 1, it follows that all the x's are numerically equal in a solution (*,» «2> • • ; *„,) of (E8). Since the equations are homogeneous, all the a^s may be taken to be +1 or — 1. The t'th of these equations is of the form
500 THKOKKMS ON SKNSK AND SKl'AK ATM )N [CHAP.IX
e^ being + 1 or — 1 according as sf is positively or negatively related to s8, and e^ being +1 or —1 according as sf is positively or negatively related to sft. Hence, if the set of regions represented by a solution in which the x's are ± 1 includes that one of sf and — s,* to which ,s,2 is positively related, it also includes that one of Sj* and — «*f to which sf is negatively related ; and if it includes that one of *? and — .sfj to which sf is negatively related, it also includes that one of sj* and — sft to which sf is positively related.
Hence the existence of a solution of (E^ other than (0, 0, • • «, 0) implies the existence of a set of s8's and — s8s such that each s2 is positively related to one of them and negatively related to another. Since the notation sf and — sf may be interchanged by multiplying theyth column of Eg by — 1, the notation may be so arranged that (1, !,•••, 1) is a solution of Eg. With the notation so arranged, each sa is positively related to one s8 and negatively related to another. We thus have
THEOREM 98. If each of the a^s determined by a set of planes T!» *"".,> •••>""„ *'"• a protective space is arbitrarily associated with one of its sense-classes to determine an oriented planar convex region s2, each of the a*'s can be associated with one of its sense-classes to deter- mine a three-dimensional convex region s8 in such a way that each s2 is positively related to one s8 and negatively related to another.
The set of s3's described in this theorem is a generalization of an oriented polyhedron as denned in § 198. If the definition* of uni- lateral and bilateral polyhedra be generalized to any number of dimensions, it is a consequence of this theorem that the three- dimensional space is a bilateral polyhedron. In general, it can easily be verified, by generalizing the matrices EI} E2, Eg etc., that projec- tive spaces of even dimensionality are unilateral polyhedra and projective spaces of odd dimensionality are bilateral polyhedra.
EXERCISE
An odd two-dimensional polyhedron iu a three-dimensional space is one-sided and an even one is two-sided.
IXDEX
A 1 tout. 159
Absolute conic, 850, 371
Absolute involutions. 1 in
Absolute polar systems, 2ii;-!. :\l'-l
Absolute quadric, 869, 37:i
Aildition of vectors, 84
Aftine classification of conies, 18(5
Atnne collineation, 72, 287
Afline geometry. 72. 147, 287
Affine irroups, 71. 72,287,305 ; subgroups
of the, 116
Agree (sense-classes), 485 Alexander, J.W., iii, 405 Algebra of matrices, 333 Algebraic cut, 15 Alignment, assumptions of, 2 Analysis, plane of, 208 Angle, 139. 231, 429, 432 Angles, equal. I'M; nuuibercd, 1">4 ; of rotation. :V2~>. :J27 ; sum of two, 154 Angular measure. 151, 153, 103, 105,
231, 311, 313, 302, 305 Anomaly, eccentric, 198 Antiprojectivities, 250, 251, 253 Apollonius, 235 Arc, differential of, 300 Area, 96, 149, 150, 157, 311, 312; of
ellipse, 150
Assumption, Archimedean, 146 Assumption A, 2 Assumption C, 16 Assumption E, 2 Assumption H, 11 A -sumption II,,. 2 Assumption H, 33 A--umption I, 30 Assumption J, 7 Assumption K, 3 Assumption P, 2; commutative law of
multiplication equivalent to, 3
A -sumption Q, 16
A-ssumptiou K, 23
imptioii K. 2!»
Assumption S. 32
Assumptions, of alignment. 2 ; cate- gorical ne>s ot'. 2:: ; consistency of, 23 ; iitinuity. 10; for Kuclii lean geom- etry, 59, 144, 302; of extension. 2; independence of. 23: of order, 32; of projectivity. •_'
Asymptotes of a conic, 78
Axis, of a circle, 854 ; of a conic, 191 ;
of a line retlertion. 2~>H ; of a parabola. 193; of a quadric, 310; radical, 1-V.» ; of a rotation. 299; of a translation. 317; of a twist, 320
Backward, 303
Barycentric calculus. 40. 104. 2'.'2. 2'.«:;
Barycentric coordinates, 100, 10* . ir.'_'
ISase i-in-le. L'.M
Base points of a pencil, 242
Beltrami, K., 361
Bennett, A. A., iii
Between, 15, 47, 48, 00. 350, 387, 430, 433
Bilateral polyhedron. 4'.t4
Bilinear curve. -jO'.t
Biquaternions. 347, 370, 882
Bisector, exterior, 179; interior, 179; perpendicular, li':;
Bocher, M., 256, 271
Bo-rer, K., 168
Bolyai, J., 301
Honola. K.. 59, 302, 3«53, 371, 376
Morel, K., 60
Boundary, 892, 474, 482
Bounding polygons. 470, 482
Broken lines. 4">4 ; directed, 4H4 ; ori- ented, 484
Bundle, of circles, 250 ; center of, 485 ; of directions, 430; of projectivities, :!4'J ; of rays. 485 ; of segments, 4:;r.
Burnside, W., 41
Calculus, barycentric, 40, 104, 292, 293
Carnot, L.N.M.. 90
Carslaw, H. S., 302
Cartan, E., 341
Casey, J., 108
Cateizoricalness of assumptions, 28
Cayley, A., 103, 335, 341, :;<M
Cells, 404 ; oriented, 452, 458
Center, of a bundle of rays. 485; of a circle, 131, 894; of a . onic. 78; of curvature, 201 ; of gravity, 94 ; of a pencil, 429, 433; of a rotation. 122: of similitude, 162, 163; of a sphere, 316
Center circle, 231
Centers, line of, 1 •">'.»
Ceva, 89
501
INDEX
Chain, 17, 21, 222. 229. 250; conjugate points witli respect to, 248; funda- mental theorem for, 22 ; n-dimen- sional, 250 ; three-, 284
Circle, 120, LSI. 142, 146, 148, 157, 269, 354, 894 ; axis ..1. 354 ; base, 254 ; bundle of. 25ti : .-filter. 2:51 ; center of, 181, 894 ; circumference of, 148 ; of curvature, 201 ; degenerate, 253, 260 ; director, 200 ; directrix of, 192 : Feuerl)acli. 169. 233; focus of. 192; fundamental. 2">4 : imaginary, 187, 22'.» ; at infinity. 29:5 ; intersectional properties of, 142; length of, 148; limiting points of pencils of, 150 ; linearly dependent, 256 ; nine-point, 160, 233 ; orthogonal, 161 ; pencils of, 157, 159, 242 ; power of a point with respect to, 162 ; sides of, 245
Circular cone, 317
Circular points, 120, 156
Circular transformations, 225 ; direct, 225, 452 ; types of direct, 246, 248
Clebsch, A., 366, 368, 360, 377
Clifford, W. K., 203, 347, 361, 374
Clifford parallel, 374, 375, 377, 444
Clockwise sense, 40
Closed curve, 401
Closed cut, 14
Coble, A. B., iii
Coefficient of torsion, 480
Cole, F. N., 222
Collinear vectors, 84 ; ratio of, 85
Collineations, affine, 72, 287; direct, 61, 64, 65, 107, 438, 451; direct, of a quadric, 260 ; equiaffine, 105 ; focal properties of, 201 ; involutoric, 257 ; opposite, 61, 438, 451 ; in real projec- tive space, 252
Commutative law of multiplication equivalent to Assumption 1\ 3
Complementary segmentsorintervals, 46
Complex elements, 166
Complex function plane. 268
Complex geometry, 6, 29
Complex inversion plane, 264, 265
Complex line, 8 ; order relations on, 437 ; and real Euclidean plane, correspond- ence between, 222
(''implex plane, 154 ; inversions in, 235
Complex point, 8, 156
( 'one, circular, 317
Con focal conies, 192
Confocal system of quadrics, 348
Congruence of lines. 275, 283 ; elliptic, 443 ; right-handed and left-handed elliptic, 444
Congruent figures. 79, 80, 94. 124, 134, 139, 144, 297, 303. 352. 369, 373. 875. 394
Conic, 82, 158, 199 ; absolute, 350, 371 ; asymptotes of. 7:» ; axisof. 191 : center of, 73; central, 73; confocal, 192:
diameter of, 73; directrix of, 191; eccentricity of, 196 ; eleven-point, 82 ; equation of, 202, 208 ; exterior of, 171, 174, 176; focus_ of^ 1»1 ; interior of, 171,174, 176~TTnvalTant.sof, 207 : latns rectum of, 198; metric properties of. 81; nine-point, 82; normal to, 173; ordinal and metric properties of , 170; outside of, 171; parameter of. I'.'*; projective, affine, and Euclidean clas- sification of, 186, 210, 212 ; a simple closed curve, 402 ; vertex of, 101 Conjugacy under a group, 80 Conjugate imaginary elements, 182 Conjugate imaginary lines. 281, 282, 444 Conjugate points with respect to a chain,
243
Connected set, 404 ; of sets of points, 405 Connectivity of a polyhedron, 475 Constructions, ruler and compass, 180 Continuity, assumptions of, 16 Continuous, 404 Continuous curve, 401 Continuous deformation, 406, 407, 410,
452
Continuous family of points, 404 Continuous family of sets of points. 4o.~, Continuous family of transformations,
406
Continuous group, 406 Continuum, 404
Convex regions, 385-394 ; linear, 47 ; sense in overlapping, 424; oriented or directed three-dimensional. 4s4 ; oriented or directed two-dimensional, 484
Coolidge, J. L., 229, 360, 362 Coordinate system, positive, 407, 408,
416 ; right-handed, 408, 416 Coordinates, barycentric, 106, 108, 292 : polar, 249; rectangular, 311; tetra- cyclic, 253, 254, 255 Correspondence, between the complex line and the real Euclidean plane, 222 ; between the real Euclidean plane and a complex pencil of lines, 238 ; between the rotations and the points of space, 328 ; perspective, 271 ; projective, 272 Cosines, direction, 314 Cremona, L., 168, 251, 348 Criteria, of sense, 49 ; of separation, 55 Crossings of pairs of lines, 276 Cross ratio, equianharmonic, 259 ; of
points in space, 56
Curvature, center of, 201 ; circle of. 201 Curve, 401 ; bilinear, 269; closed. 401 ; a conic a simple closed, 402; eqtii- distantial, 356; normal, 286; path, 249, 366, 406; positively or nega- tively oriented, 452; rational, 286; simple, 401
INDKX
Cut-point, 14, 21
Cuts, open and closed, 14 ; algebraic, 15
Cyclic projectivity. 258
Darboux, G., 261, 824
Dedekind, K.,60
Deformation, continuous, 406, 407, 410, 452
Dt-ucnerate circle, 253, 200
Degenerate sphere, 815
Dehn, M.. :.".'<»
De Paolis, R., 362
Describe, 401
Diagonals of a quadrangle, 72
Diameter, of a conic, 78 ; end of, 151 ; of a quadrilateral, 81
Dicks. .11. I.. E., 35, 339, 841
Differential of arc, 366
Dilution, 95, 848
Direct collineation, 61. 64, 65, 107, 438, 451 ; of a quadric, 260
Direct projectivities, 37, 38, 407
Direct similarity transformations, 135
Direct transformations, 225, 452
Directed, oppositely, 483 ; similarly, 433
Directed broken line, 484
Directed interval, 484
Directed polygon, 484
Directed segment, 484
Directed three-dimensional convex re- gion, 484
Directed two-dimensional convex re- gion, 4H4
Direction-class, 433
Direction cosines, 814
Directions, bundle of, 436 ; pencil of, 433
Director circle, 200
Directrices of a skew involution or line reflection, 258
Directrix, of a circle, 102; of a conic, 191 ; of a parabola, 193
Disagree (sen>c-classes), 485
Displacement, 123, 129, 138, 148, 297, 317, 825, 3r>2. :;»;;i. :;78 ; parameter rep- resentation of, 344; parameter repre- sentation of elliptic, 877 ; parameter representation of hyperbolic, 380 ; types of hyperbolic, 855
Distance, 147, 157, 311, 364, 373; air. - braic formulas for, 865 ; of transla- tion, :;•_'.'), 327; unit of, 147
Dnelileinann, K., 229, 230
I). ml>le elliptic plane, 875
Double elliptic plane geometry, 875
Double points of projectivities. '>. 1 14. 1 77
Doubly oriented line. 440, 44_'. 1 1.\ 117. 449
Doubly perspective, 448
Down," 303
!•'.• -centric anomaly, 198 Eccentricity of a conic, 196
Edges, of a broken line, 454 ; of a poly.
lieiln.ii, oriented, 4'.«'» Eisenliart, I.. IV. Elementary t ran>f. ,1111:1! j,.i,>. »<»9, 411-
414, 41*. 4 Hi. »•_'!. I. i;;}
487, 447. I.V.. 4:.i; I, 410,
414, »L'O. Klemeiit.s, eoniplex. }~tH ; imaginary, 7,
156, 182 ; ideal, 71, 287 ; improper, 71 Eleven-point conic, 82 Ellipse, 78, 140 ; area of, 150 ; foci of,
1W: imaginary, 187 Elliptic congruence. 443 Elliptic displacements, parameter rej»-
icseiitation of, 377 Elliptic geometry, double. 375 Elliptic geometry of three dimensions,
373
Elliptic pencils of circles. iMi' Elliptic plane, 371 ; double, 375 ; single,
371. 375
Elliptic plane geometry, 871 Elliptic points, 878 Elliptic polar systems, 218 Elliptic project! vity, 6, 171 Elliptic transformations, direct circular,
248
Emch, A., 230 End of a diameter, 161 Ends of a segment or interval, 45. 427 Enrique*. F., :;n-j Envelope of lines, 406 Equation of a conic, 202, 208 Equations of the affine and Euclidean
groups, 116, 185, 30.'» ; linearly inde- pendent, 466 ; and matrices, modular,
464
Equiaftine collineations, 105 Equiaftine group, 105, :.".il Equianharmonic cross ratio or set of
point*. •J-V.i
Equidistant ial curves, 856 Equilateral hyperbola, 169 Equivalence, of ordered point triads.
96, 288,290; of ordered t.-tra,!
with respect to a group, 89 Euclid, 360 Euclidean classification of conies, 186,
210 Euclidean geometry, 117, lis. Hi', i:;:,.
144, 287, 800, 8o-_' : a»mnptions fur.
.V.i. I II. :;o-j : as a limiting case of
n»n- Euclidean, 875 Euclidean group. 117, 118, 185. 14 ».
equations of. IK',. i::.'i. ;',ii:, Euclidean line, 68
Euclidean plane, 68, (MM;:;. 71 ; and com- plex line, ci.rre.sjMindence between.
._,._,._, .,;.s . j,1N,.rsj(in .;,-,,„,, jn the real,
i-e ill. (tl
Euclidean Hpaces, 58, 287 ; sense in. K\ Euler. I... *W, 387
1NDKX
F.ven polygons. 470, 482, 489; in the projects ve plane, 489
Even polyhedra, 482, 483
Expansion.
Extension. asMimptions of, 2
Kxterior. of ;ui angle. 4oU ; of a conic, 171, 174, 17»> ; of a polygon, 472 ; of an even polygon. 4'.M) ; of a qnadrir. 111!
Kxtrriur bisector, 179
Faces of a polyhedron, 474 ; oriented,
495
Family of points, continuous, 404 Family of sets of points, continuous, 405 Family of transformations, continuous,
400
Fano, G., 11, 285, 280 Feucrbach, 109, 233 Field, Galois, 85 Fine, H. B., 3, 18 Finzel, A., 309 Focal involution, 195 Focal properties of collineations, 201 Foci of an ellipse or hyperbola, 189 Focus, of a circle, 192 ; of a conic, 191 ;
of a parabola, 193 Follow, 13, 37, 47, 48 Forward, 303 Foundations, of complex geometry, 29 ;
of general projective geometry, 1 Fuliiiii, G., 302 Function plane, 208 Functions, trigonometric, 154 Fundamental circles, 254 Fundamental theorem of projectivity
for a chain, 22
Galois field, 35
Gauss, C.F., 40, 301
Generalization, by inversion, 231; by projection, 107, 231
Geometrical order, 40
Geometries, projective, 30
Geometry, aftine, 72, 147, 287 ; assump- tions for Euclidean, 59, 144, 302 ; com- plex, 0, 29 ; corresponding to a group, 70, 71, 78, 199, 285, 302 ; double ellip- tic, 375 ; elliptic, 371 ; Euclidean, 117, 118, 119, 135, 144, 287, 300, 302; Euclidean, as a limiting case of non- Euclidean, 375 ; foundations of gen- eral projective, 1 ; generalized, 285 ; history of non-Euclidean, 300 ; hy- perbolic plane, 850 ; inversion, 219 ; inversion plane and hyperbolic. ::".T : modular, 253 ; of nearness, 303 ; non- Euclidean, 350; parabolic metric group and. 119, 130, 135, 144. :.'<•:!; n-al inversion. 241; of reals, 140; three-dimensional elliptic. 373 ; three- dimensional hyperbolic, ;•{<!«.)
Grossman, II.. UiH, 290
( ira\ ity, center of, H4
Group, atline. 71. 7'J, 2H7. :io:i ; conju- gacy iimler. :;'.»; continuoii.s, loi;: of displacements, liilt; c<iuiattine. I0.">, °'.U ; equivalence with respect to. :'>'.>: Kiiclideaii. lie,. 117, 118, loo. 144,806; geometry correspondinir to. 70. 71. 7s. 'i '.'!•. 286,802; h ithetic, '.».->; inver- sion, in the real F.nclidean ])lane, 2'2.">, 2^( i ; one-parameter continuous, lor, ; parabolic metric, and lieometry. Mil, 130, 135, 144, 293; the projective, of a quadric, ii-V.t ; special linear. :.".»! : subgroujis of the aftine. 110
Groups, algebraic formulas for certain parabolic metric, 135; equations of the aftine and Euclidean, 110, loo. 305
Half turn, 299, 370
Half twist, 324
Ilalstuad, G.B., 301
Hamel, G., 28
Hamilton, W.R., 339
Harmonic homology, 257
Harmonic separation, 45
Harmonic sequence, 10, 33, 34; limit point of, 10
Hatton, J. L. S., 108
Heath, T. L., 300
Heine, E., 00
Hermitian forms, 302
Hess^, O., 284
Hilbert, D., 103, 181. 394
Homology, harmonic. 257
Honiothetic group. '.»")
Homothetic transformations, 95
Horocycle, 350
Horosphere, 370
Huntington, E. V., 3, 33
Hyperbola, 73 ; equilateral, 109 ; foci of, 189; rectangular. 109
Hyperbolic direct circular transforma- tions, 248
Hyperbolic displacements, parameter representation of, 380 ; types of. :!-V>
Hy])erbolic geometry, of three dimen- sions. o<;(.) ; and inversion plane. o">7
Hyperbolic lines, 350
Hyperbolic metric geometry in a plane, 350
Hyperbolic pencils of circles, 242
Hyperbolic plane, 350
Hyperbolic points. 350
Hyperbolic projectivity, 5, 171
Hyperbolic space, 309
Ideal elements, 71, 287
Ideal lines, 287. 350
Ideal minimal lines, 205
Ideal plane. 287
Ideal points. 71, 205, 208, 287, 850
Ideal space, 58 Imaginary circle. 1ST. 22H Imaginary elements, 7, 15(5, 182; con- jugate. 18*2
Imaginary ellipse. 1ST Imaginary one-dimensional form. 150 Imaginary lines, conjugate, 281, 2M.'. I 1 1 Imaginary points. H. 150 Imaginary sphere, rotations of, 335 Improper elements, 71 Incomplete symbol, 41 Independence of assumptions, 23; proofs
of. 24-29 Infinity, circle at, 293 ; lino at. 58, 71 ;
plane at, 287; points at, 71. 241. 20s.
2*7, 352 ; space at, 58 Inside, of a conic, 171; of aquadric. 344 Interior, of an angle, 432; of a conic.
171. 174, 170; of an interval or seg-
ment, 45 ; of a polygon, 472 ; of an
even polygon, 490 ; of a quadric, 344 ;
of a triangle, 389 Interior bisector, 179 Intermediate positions, 407 Interval, 45, 46, 47, 00, 450 ; directed,
484 ; ends of, 45, 427 ; oriented, 484 Intervals, complementary, 46 Intuitional description of the projective
plane, 67
Invariant subgroup, 39, 78, 106, 124 Invariants of a conic section, 207 Inverse matrix, 308 Inverse points, 162 Inversion, 162, 241, 266; generalization
by, 231 ; in a complex plane, 235 Inversion geometry, 219 ; real, 241, 268 Inversion group in the real Euclidean
plane, 225, 226 Inversion plane, 268; complex, 204. 20">:
hyperbolicgeometryand,357; real, 241 Inversor, Peaucellier, 229 Involution, absolute, 119; focal, 195;
order relations with respect to. 45 ;
orthogonal, 119; skew, 258; axes and
directrices of skew, 258 Involutoric collineations. 257 Involutoric projectivities. products of
pairs of. 277 Involutoric rotation. 299 Irrational points, 17, 21 Isogonality, 231 Isoinorphic, 8
Isotropic lines. 120, 125, 265, 294 Isotropic plane, 294 Isotropic rotation, 299 Isotropic translation, 317
Jordan. C., 4."):', Juel. C., 250. 251
Klein. F.. 71. 2 »'.i, 278, 284, 285,361,302, 374, :;:.-.. 4»o
Kline. .1. i;
(,.. ; ;•_•!, 339
Latu> re, mm of ;1 conic, 198 Left-handed Clifford parallel-.. :;:». HI Left-handed conjugate imaginary line.-,.
444 Left-handed doubly oriented line>. It.'
146
Left-handed elliptic congruence, lit Left-handed ordered pentad* ..f ixiints
Left-handed ordered tetrad of j«.int>,
442
Left-handed regnliis. 4 1:; Left-handed .-en>e-da». 407. 410 Left-handed triad of >kew line.-. 4 l:;. 447 Left-handed twist. 417. II.; Length of a circle. 1 4H Lennes. X. J.. 1H. 457 Lewis, <i. N.. HO, 138. 3U2 Lie. S.. :!41
Like sen.-e-classes of segments, 436, 437 Limit point of harmonic sequence, lit Limitini; points of pencils of circle-. I.V.i Lindeinann. F., 300. :;os. :;i;;i Line, of centers, 159; complex, 8;
doubly oriented, 440. 442. 445. 447.
449; Euclidean, 68. 0<> : hypvrMic.
350; ideal, 287, 350; imaginary 150.
at infinity, 58, 71 ; ordinary. 71. 2X7.
:i")() ; oriented. 420; real, 150; sides
of, 69, 392; similarly oriented with
respect to, 426; translation parallel
to, 288
Line pairs, measure of. lo:i Line reflections. Kin. 115, 258; direc-
trices. or axes of. 2"ih ; orthogonal,
120, 122. 120. 299. :!17. 352, 870 Linear convex regions. 47 Linear i:roiip. special, 291 Linearly dependent circles. 2 •">''• Linearly de|>eni!ent solutions of E,. 4HH Linearly independent columns of I . t-- Linearly independent eijnations (H,).
408
Line>. ln«. ken. 4.">» : .-oiii:nii-nce . 388; conjugate ima-inarv . H 444; cros>injs of pairs of. 270; en- velo}H- of. 4iK! ; ideal minimal. 205 ; meetings of pairs of. 270; minima) or i.Mitropic. 120. 125. 205. 2H4 ; negative l>airs of, 417 ; ordinary minimal. 205 : ortho-onal. 12".
pairs of. :,o. ]>;::-. parallel. 72. 2H7. :;:.! ; perpendleolar, ISO, 188, 898,800, 878; positive pairs of. 417; singular.
elementary traiisfi>nnati> triads of >kew, 447; right- and left- hainled triads of skew. 443; MiNli- vision of a |>!aue by, 61-63, 400-404 ;
506
LNDKX
L..l.ach.-\>ki. X. I.. :;t!l Logarithmic spirals, •-'•»'.• Lower side of a cut, 14 Loxodromic direct circular transfor- mations. •_' is Lliroth, J., 9
MaeGregor, H. H., 260
.Magnitude of a vector, 86, 147
Malfatti, U
Mamiing. H. 1'., 362
Matrices, algebra of, 838; modular equations and. 4f>4 ; sum of two, 333
Matrices E. and E2 for the projective plane, 484
Matrices H.. H,, and H8, 396, 398-400, 477
Matrix, inverse, 308 ; orthogonal, 308 ; rank of, 478 ; scalar, 334
Measure, of angles, 151, 153, 163; angu- lar, 163, 165, 231, 311, 313, 362, :-!(;.-, : of line pairs, 163 ; of ordered tetrads, 290 ; of ordered point triads, 99, 312 ; of a simple n-point, 104 ; of triangles, 99, 149, 312 ; unit of, 99, 140, 319
Median of a triangle, 80
Meetings of pairs of lines, 276
Menelaus, 89
Metric group and geometry, parabolic, 119, 130, 135, 144, 293
Metric properties of conies, 81
Mid-point, 80, 125
Milne, J. J., 168
Minimal lines, 120, 125, 265, 294
Minimal planes, 294
Minimal rotation, 299
Minimal translation, 317
Minkowski, H., 394
Mobius, A. F., 40, 67, 104, 229, 252, 292, 293
Model for projective plane, 67
Modular equations and matrices, 464
Modular spaces, 33, 35, 36, 253
Moore, E. H., 24, 36
Moore, R. L., 59
Morley, F., 222
Motion, rigid, 144, 297 ; screw, 320
Moved, 406
.AT-dimensional chain, 250
^-dimensional segment, 401
^-dimensional space, 68
jV-dimensions, generalization to, 304
Nearness, geometry of, 803
Negative ordered pairs of lines, 417, 418
Negative of an oriented segment or re- gion, 485
Negative points, 17
Negative relations between points and segments, 485
Negative rotations, 417
Negative sense-class, 407, 416
Negative translation, 416
Negative twist, 417
Negative of a vector, 84
Negatively oriented curve, 452
Negatively related sense-classes, 485, 491, 495
Net of rationality, 36 ; cuts in, 14 ; order in, 13
Neutral throw, 245
Nine-point circle, 169, 233
Nine-point conic, 82
Noncollinear points, 96
Nomlegenerate circle, 266
Nuiiclegenerate sphere, 316
Non-Euclidean geometry, 350 ; Euclid- ean geometry as a limiting case of, 375 ; history of, 360
NomiiMclular spaces, 34
Normal to a conic, 173
Normal curve, 286
Null vector, 83
X umbered angle, 154
Numbered point, 456
Numbered ray, 154
Numbers, complex, 219
Odd polygons, in a plane, 470, 482 ; in the projective plane, 489
Odd polyhedra, 482, 483
On, 440
One-dimensional form, imaginary, 166 ; order in, 46 ; real, 156
One-dimensional projectivities, 156, 170- 173; and quaternions, 331) ; repre- sented by points, 342
One-sided polygonal regions, 490
One-sided polyhedra, 493
One-sided region, 437
Open cut, 14
Opposite, 433
Opposite collineations in space, 438, 451
Opposite projectivities, 37, 38
Opposite to a ray, 48
Opposite sense, 61
Opposite transformations, 452 ; of a 2-cell, 452
Oppositely directed, 433
Oppositely oriented, 448, 450
Oppositely sensed, 245
Order, 40 ; assumptions of, 32 ; geo- metrical, 46 ; in a linear convex re- gion, 47; in a net of rationality, 13; in any one-dimensional form, 46 ; on a polygon, 456 ; of a set of rays, 432
Order relations, on complex lines, 437 ; in a Euclidean plane, 138; in the real inversion plane, 244 ; with respect to involutions, 45
Ordered .pair, of points, 268, 271 ; of rays, 139
Ordered projective spaces, 32
Ordinary lines, 71, 287, 350
INDKX
Onlinary niininial lines, 206
( >rdinary planes, 287
Ordinary points, 71, 205, 208, 287, 350
Ordinary space, 68
Orientation of space, 490
< iriented. oppositely. 448,460; similarly,
448 ; similarly, with respect to a line,
O6
Oriented broken line, 484 Oriented 2-cell, 462 Oriented 8-cell, 463 Oriented curve, 452 ( >riented edges of a polyhedron, 495 Oriented faces of a polyhedron, 495 Oriented interval. 4*4 Oriented line, 426 ; doubly, 440, 442, 445,
447, 449 ( triented points, 420 ; segments of, 420
< )riented polygon, 484 Oriented polyhedron, 495
( >riented projective space, 453
Oriented segment, 484
( (riented segment or region, negative of, 4,85
Oriented simple surface, 463
Oriented three-dimensional convex re- gion, 484
Oriented two-dimensional convex re- gion, 484
( >rigin of a ray, 48
Orthogonal circles, 161
Orthogonal involutions, 119
Orthogonal line reflections, 120, 122, 120, 299, 317, 352, 370; center of, 122 ; pairs of, 126
Orthogonal lines, 120, 138, 293, 350, 352
Orthogonal matrix, 308
Orthogonal plane reflections, 295
Orthogonal planes, 293
Orthogonal points, 352
Orthogonal polar system, 293
Orthogonal projection, 313
Orthogonal transformations, 308
Outside of a conic, 171
< hit.side of a quadric, 344 Owens, F. W., 59, 371
I'adoa, A., 44
Pairs of lines, 50, 163 ; crossing of, 270 ;
measure of, 163 ; meetings of, 27»i ;
negative, 417 ; negative ordered. 41 H ;
positive, 417; positive ordered, 417;
separation of plane by, 60 Pairs, of orthogonal line reflections. 1 'JH :
of planes, 60 ; of points, ordered, 208,
271 ; of points, unordered, 271 1'aolis, R. De, 362 1'appus, 5, 103, 118 Parabola, 73; axis of. !!•:; : directrix
of, 193 ; focus of, 193 ; Steiner. I'.M; ;
vertex of, 193
Parabolic metric group and geometry 119, 130, 136, 144, |
I'araliolic pencils of circle**. _'!.'
Parabolic projeeti vities. .'., 171
Parabolic direct circular tninxfonua- tions. 24H
Parallel to a line, translation, 288
Parallel lii;. :. 351
Parallel planes, 2H7
Parallelogram, 7-2
Parallels, Clifford, 374, 375, 377, 444
Parameter of a conic. ll»h ; continuous one-parameter famih "t >.•!>,, f points. 405; continuous (. iic paiaiiu-ter family of transformations. 406; continuous one-parameter group. 4<>«;
Parameter representation. 844 ; of cllij>- tic displacement-. :;77 . «i hyperlxilic displacements, 380 ; of parabolic dis- placements, 344
Para tactic, 874
Pascal, E., 186, 235, 279, 280
Path curve. 249, 350, 406
Peaucellier inversor, 229
IVirce, B., 841
Pencil, base point of, 242 ; center of, 4211. 433 ; of directions, 433 ; of lines, correspondence between the real Euclidean plane and a complex, 238 ; of rays, 429 ; of segments, 433
Pencils of circles, 157,159,242; limiting I « Tints of, 159
Pencils of projectivities, 343
Pentads of noncollinear points, right- and left-handed, 442
Permutations, even and odd, 41
Perpendicular bisector, 123 ; foot of a perpendicular, 123
Pt-rpendicular lines, 120, 138, 293, 869, 873
Perpendicular planes, 298, 369, 873
Perpendicular points. 3:>2, 869, 873
Perspective, doubly. 448
Perspective eorresjxmdeuce, 271
Pieri, M., 244
Pierpont, J., 8
Planar convex regions, 886
Planar region, 404
Plane, of analysis. 2<iH ; complex, 164; complex inversion. 2i»4 W* ; corre- spondence lietw.cn a complex lini- and the real Kudidean. 2'J- double elliptic. :;7~> ; elliptic. 371 Kudidean. •> «i:;. 71 ; function. 2«»* hyp.-rl>"lic. :;.">(>; hyptTlxilic geoinetr\ aiid inversion, :;">7 ; ideal, 287; at infinity. 2H7 ; intuitional description of the pn-jective, (i7 : inversion. 2»W ; inversion group in the complex Ku- clidean, 2H"> : inversion group in the real Ku.-lidean, 225, 230; i*o- tropic. 21'4 ; minimal. '-".'4 ; intMlel for
508
INDEX
project ive, i!7 ; order relations in ;i Kuclideau, 138 ; order relations in the ival imersi'.n, :'.\l; nnlinary, 287; orthogonal. 2'. '3 ; projrcthe. L'C.H ; n-al. 1 »o. !."><! : real inversion. 211. 2H* ; reflections, orthogonal. 2'.i"> ; sense in a Kuclideau, (U ; sides of. .V.I, 392; single elliptic. 371, 375; subdivision of a plane l>y lines. :,\. .">:), 4 < 50-464 ; of syiiiiiH-try, 2.'">
IManes. pairs of. ">U ; parallel, 2H7 : per- pendicular. •_".':;. :!•;;•. :;7:; ; suhdivision of space by, 50, 54, 475-477; vanish- in ir. 348
Plucker. .1.. 292, 326
Poincare', II., 341, 362, 489
Point pairs, congruence of parallel, 80; mid-point of, 80 ; separation of, 44-47
Point-plane reflection. 2">7
Point reflection, 92, 122, 300, 352, 414
Point triads, measure of ordered, 99; equivalence of ordered, 96, 288, 290 ; sum of ordered, 96
Points, complex, 8, 156 ; circular, 120, 1-V> ; double, of a projectivity, 5, 114, 177 ; elliptic, 373 ; equianharmonic set of, 259 ; hyperbolic, 350 ; ideal, 71, 265, 268, 287, 350 ; imaginary, 8, 156; at infinity, 71, 241, 268, 287, 352; inverse, 162; irrational, 17, 21; negative, 17; noncollinear, 96 ; num- bered, 456 ; one-dimensional projec- tivities represented by, 342 ; ordered pairs of, 268, 271 ; ordinary, 71, 265, 268, 287, 350 ; oriented, 426 ; orthogo- nal, 352 ; of a pencil, base, 242 ; of pencils of circles, limiting, 159 ; per- pendicular, 352, 369, 373 ; positive, 17 ; projection of a setv of, 291; rational, 17; real, 8, 156; rotations represented by, 342, 343 ; segments of oriented, 426 ; singular, 235 ; in space, corre- spondence between the rotations and the, 328 ; in space, cross ratios of, 55; ultra-infinite, 352 ; unordered pairs of, 271 ; vanishing. 86
Polar coordinates. 24'.i
Polar system, 215; absolute, 293, 373; elliptic, 218; orthogonal, 2!>:5
Polygon, 454-459, 480, 481 ; bounding, 470, 482; directed, 484; even, 470, 482, 489 ; interior and exterior of, 472, 490; odd, 470, 482, 489; order on, 456; oriented, 484 ; regions deter- mined by, 4U7; sum modulo 2 of, 481 ; unieiirsal, 470
Polygonal regions, 473 ; one- and two- sided, 490
Polyhedra, odd and even, 482, 483 ; one- and two-Aided, 493 ; oriented. 4'. 15 ; oriented edges of, 495 ; oriented faces of, 495 ; sum modulo 2 of, 482
Polyhedral regions, 473
Polyhedron, 474; bilateral, I'.U; con- nectivity of, 47"); edges of. 471 ; faces of, 474; one-sided, 494; oriented ed^es and faces of, 495 ; two-sided. »:i 1. 4'.«; ; unilateral, 494; vertices of. 171
Positions, intermediate, 407
Positive coordinate system, 407, 408, 416
Positive ordered pairs of lines, 417
Positive pairs of lines, 417
Positive points, 17
Positive relation between points and oriented segments. 485
Positive rotation, 417
Positive sense-class, 40. 407, 416, 491
Positive translation, 416
Positive twist, 417
Positively oriented curve, 4~>2
Positively related sense-classes, 485, 491, 496
Power, of a point with respect to a circle, 162 ; of a transformation, 87, 2:;<>
Precede, 13, 15, 37, 47, 48, 350, 387
Product, of pairs of involutoric projec- tivities, 277 ; of two vectors, 220
Projection, generalization by, Ki7, 2:!1 ; orthogonal, 313; of a set of points, 291
Projective classification of conies, I.M;
Project! ve correspondence, 272
Projective geometry, 36; foundations of general, 1
Projective group of a quadric, 2-V.i
Projective plane, 268; intuitional de- scription of, 67 ; matrices E. and K., for, 484
Projective space, collineations in a real, 2">2 ; sense i;i. ><l
Projective spaces, ordered, 32 ; sense- classes in. 418
Projectivities, bundle of, 342 ; cyclic, 258 ; direct, 37, 38, 407 ; double points of, 5, 114, 177; elliptic, 5, 171 ; hyper- bolic, 5, 171; one-dimensional, 170, 171; opposite, 37, 38; parabolic, 5, 171 ; pencil of, 343 ; powers of, 87 ; products of pairs of involutoric, '277 ; of a quadric, 273; real, 156, 170- 173; representation by points of one- dimensional, 342; representation by quaternions of one-dimensional, 339
Projectivity, assumption of, 2
Prolongation of a segment, 48
Proofs, independence, 24-29
Quadrangle, diagonals of, 72
Quadrics, absolute, 369, 373 ; axes of, 316; confocal system of, 348; direct collineations of, 260 ; interior and exterior of, 344 ; projertive group of, 2-V.i; projectivities of, 27:;; real, 202 ;
INDKX
509
ruled. L'.V.I; sides of, 844; sphere ;ni.l other. :;i.~> ; unruled, -.v.i (Quadrilateral, diameter of, 81 Quaternions, 887-841, 378 ; and the one- dimensional projecti\e -mnp, 339
Radical axis, 159
Hadii, transformation by reciprocal, 102
Hank, of H2, 479 ; of a matrix, 478
Rational curve, 280
Rational modular space, 35, 30
Rational points, 17
Rationality, net of, 35; order in a net of, 13
Ratios of colliiiear vectors, 85
Rays, 48, 00, 143, 350, 872, 387, 429; bundle of, 435; numbered, 154; oppo- site, 48 ; order of a set of. -l.'L' : oidered pair of, 139; origin of, 48
Real and imaginary elements and trans- formations, 150
Real inversion geometry, 241
Real inversion plane, 241, 208; order relations in, 244
Real line, 150
Real one-dimensional form, 150
Real plane, 140, 150
Real points, 8, 150
Real projective transformations, 150
Real quadrics, 202
Reals, geometry of, 140
Reciprocal radii, transformation by, 102
Rectangle, 123
Rectangular coordinates, 311
Rectangular hyperbola, 109
Reflections, axes of line, 258 ; center of orthogonal line, 122; directrices of line, 258 ; line, 109, 115, 258 ; orthogo- nal line, 120, 122, 120, 299, 317. :\:x. 370; orthogonal plane, 295; pairs of orthogonal line. li'O; point, 92, 122, 300, 852, 414 ; point-plane, 257 ; in a three-chain, 284
Region, convex, 385-394 ; negative of an oriented segment or, 485 ; one- sided, 437; order in a linear convex, 47 ; planar, 404 ; polygonal, 473; polyhedral, 473 ; sense in overlapping convex, 424 ; simply connected three- dimensional, 404 ; tetrahedral, 54, 398,399; three-dimensional, 404 ; tri- angular, 53, 389, 395; trihedral, :W7 ; two-sided region, 4:!7 ; vertices of a triangular, 63
Regions, bounded by a polyhedron, 4n:! ; determined by a polygon, 407
Regulus, right- and left-handed, 448
Restricted elementary transformations, 410, 414, 420, 430
Beye, T., 108
Rhombus, 125
Ricordi, K.. Riemann. II.. :;<;! Right angles, i:,;;
Right-handed Clifford parallel.,. :;;i. nt Right handed conjugate imaginary lin.-.-.,
4 1 I
Right-handed coordinate system, 408,410 Right-handed doubly oriented lii
ii:,
Right-handed elliptic ri>iigrneii< • Right-handed ordered pentad of iH,int>,
412
Right-handed ordered tetrad of points,
•41.'
Right-handed regnlus, 448
Kight.-handed sense-class, 4<>, 4o7. 4 Id II-
Right-handed triad of skew lines. 44:',. 447
Right-handed twist. 417. I).;
Rigid motion, 141. 297
Rodrigues, <)., 330
Rotation, angle of. :;-2'>. :i:.'7 : U 299; center of a, 122; involutor isotropic, 2!»'.t ; minimal. . live, 417; positive. 417 ; -
Rotations, 122, 128, 141, 299, 321. :J2K- 837; correspoinlence )>rt \\een the points of space and, 328 ; of an imag- inary sphere, 835 ; represented l>y poin'ts. ::.»i'. :;i.:
Ruled <madric, 2-V.i
Ruler-and-comp;uss constructions, 180
Russell, 11., 41
Russell, . I. W.. IliS, 2D1
Saccheri. (',.. :;iil
Same sense. <il
Scalar matrix. :;.'.{
Schillini:. M.. >n
Schweit/.er. A. R.. ::2. 415
Screw motion, :;-Jo
Se-ment, 45. 4(i. 47. tin. 3.V) ; »r inter- val, eoinpleinentary, 4»> ; di: 484; ends of, 4.'., "42" ; interior of interval or, 4"> ; ii-diineiisional. 4itl . oriented, 4H4 : iirolongation of, -Jl-
Segnieiits, huildle of, 4:it! ; of oriented points, 420; pencil of. I
m of. 4:;»;. 1:17
S.-LTI-. C., '•'. -'•"><>. -'"-1
Self-conju-ate sub-mnp, 39, 78, 100, 124 Sense, 82. 41. 01, :;*7. 4i:;; t-lockwis.-, 40 ; criteria for. 49; in a Kuclidean plane. 01; in Kuclidean spaces, 68; in a linear region, 47; more general theory of. 451 ; in a one-<limensional form, 40, 43 ; op^sitf, 01 ; in over lapping convex regions. UJ; positiv.-. 40, 407. 41U; in a protective >pa. .-. •i.:ht-handed. in. Jo7. 41f,. 4I'J ; of i..ta:ion. 1 !•_' ; same. »!1
510
INDEX
Sense-class, 61, 04, 60, 413, 414, 430, 431, 434, 487, 466, 460 ; of a 2-ceIl, 4:.i' , of a curve, 4.">2 ; left-handed, 407, 416 ; in a linear region, 47 ; negative, 407, 410, 4~>2 ; on a one-dimensional form, 40, 43; positive, 40, 407, 41«, 4;">2, 4'.'1 ; right-handed, 40, 407, 41ti, llu
Sense-classes, agree or disagree, 485; negatively related, 485, 491, 495 ; posi- tively related, 485, 491, 495; in pro- jective space, 418
Sensed, oppositely, 245; similarly, 245
Separated, 392, 432
Separation, algebraic criteria of, 55 ; harmonic, 45; by pairs of lines, 51 ; by pairs of planes, 61 ; of point pairs, 44,47
Sequence, harmonic, 10, 33, 34 ; limit point of, 10
Sets of points, connected, 404, 405 ; continuous family of, 405
Shear, simple, 112, 293
Sheffers, G., 841
Sides, of a circle, 245 ; of a line, 59, 392 ; of a plane, 59, 392 ; of a quadric, 344
Similar, 119
Similar and similarly placed, 95
Similar figures, 293
Similar triangles, 134, 139
Similarity transformations, 117, 119, 293 ; direct, 185
Similarly directed, 433
Similarly oriented, 448 ; with respect to a line, 426
Similarly sensed, 245
Similitude, center of, 162, 163
Simple broken line, 454
Simple curve, 401
Simple polygon. 454-457
Simple shear, 112, 293
Simple surface, 404 ; oriented, 453
Simplex, 401
Simply connected element of surface, 404
Simply connected surface, 404
Simply connected three-dimensional re- gion, 404
Singular lines, 235
Singular points, 235
Singular space, 68
Skew involutions, 258 ; directrices or axes of, 258
Skew lines, elementary transformations of triads of, 447 ; right- and left- handed triads of, 443
Smith, H. .1. S., 201
Somraerville, D. M. Y,, 362
Space, assumptions for a Euclidean, 59 ; collineations in a real projective, 252 ; correspondence between the rotations ' and the points of. 828 ; cross ratio of points in, 55 ; direct collineations
in, 438, 451 ; Euclidean, 68, 287 ; hy- perbolic, 869; ideal, 68; at infinity, 58; modular and nonmodulai. 36, 253; n-dimensional, ;">H ; opposite collineations in, 438, 451 ; ordered projective. 32 ; ordinary, 58 ; orienta- tion of, 496 ; oriented project! \ > polygons in, 480, 481 ; rational modu- lar, 35, 86 ; sense in a Euclidean. <;:: ; sense in a projective, 64 ; sense-classes in projective, 418 ; singular, 58
Spatial convex regions, 386
Special linear group, 2!>1
Sphere, center of, 31. ">; degenerate, 315; and other quadrics, 315 ; rotations of an imaginary, 335
Spirals, logarithmic, 249
Square. 12")
Statements, vacuous, 24
Stand!.. K. G.C. von, 9, 40, 251, 283
Steiner, J., 196, 229
Steinitz, K., 35, 69
Stephanos, C., 280. 324, 342, 344
Study, K., 40, 327, 341, 347, 302, 374, 416, 44C,
Sturm, R., 168
Subdivision, of a plane by lines, 61-53, 460-464 ; of a space by planes, 50, :>4, 475-477
Subgroup, self-conjugate or invariant, 39, 78, 106, 124
Subgroups of the affine group, 116
Sum, modulo 2, of polygons, 481 ; modulo 2, of polyhedra, 4H2 ; of ordered point-triads, 96; of two an- gles, 154 ; of two matrices, 333 ; of two vectors, 83
Surface, simple, 404 ; simply connected. 404; simply connected element of, 404
Symbol, incomplete, 41
Symmetric, 124, 297, 300, 352
Symmetry, 123, 124, 129, 138, 297, 300, 352, 373 ; plane of, 295 ; with respect to a point, 300
Tait, P. G., 341 Taylor, C., 168 Taylor, W. W., 82
Tetracyclic coordinates, 253, 254, 255 Tetrad, measure of an ordered, 290 ; of points, right- and left-handed ordered, 442
Tetrads, equivalence of ordered, 290 Tetrahedral region, 64, 398. :-5«.i!> Tetrahedron, 52, 397; volume of, 290,
311
Three-dimensional affine geometry, 287 Three-dimensional convex region. -'!H<> Three-dimensional directed convex re- gion, 484 Three-dimensional elliptic geometry, 378
INDIA
511
Three-dimensional Euclidean geometry. 2S7
Three-dimensional hyperbolic geometry, MM
Three-dimensional region. 404 ; simply connected, 404
Throws, 40 ; neutral, 245 ; similarly or oppositely sensed, '2 \'<
Torsion, coefficient of, 489
Touch, 158
Transference, the principle of, 284
Transformations, of a 2-cell, circular, 225; continuous family of, 406; direct. •JL'">. 452 ; direct similarity j 185 ; ellip- tic direct circular, 248; elementary, 409, 411-414, 418. 41'.'. 421, 423, 430, 431, 434-437, 447, 455, 450 ; homothe- tic, i>5 ; loxodromic direct circular, 248 ; opposite, 452 ; orthogonal, 308 ; parabolic direct circular, 248 ; power of a projective, 87 ; power of a circu- lar, 230 ; real and imaginary, 156; by reciprocal radii, 102 ; restricted ele- mentary, 410,414,420,430; similarity, 117, 119, 293
Translation, 74, 117. 122. 288, 321. 374, 414; axis of, 317: distance of, 825, :!^7; isotropic, 317; minimal, 817; negative, 41(5 ; parallel to a line, 288 ; positive, 416
Transposition, 41
Transversals, 91
Triads of lines, elementary transforma- tions of, 447 ; right-handed and left- handed, 443
Triangle, area of, 149, 812 ; interior of, 389; measure of, 99, 149,312; median of, 80 ; separation of a plane by, 52 ; unit, 99, 149, 312
Triangles, congruent, 134, 139 ; similar, 134, 139
Triangular region, 53, 389, 395 ; vertices of, 53
Trigonometric functions, 154
Trihedral regions, 397
Turn, half, 299, 370
Twist, 320, 321 ; axis of, 320 ; half, 324 ;
left-hand. -d. 417. 4) . . 417 ;
positive, 417; right-hand. •:. 1 17. II., Two-dimensional con\e.\ n-gioii, 886;
directed, 484
Two-sided jM.lygonal region*, 490 Two-sided ]M.l\li.-dr:i. Two-sided jMilyhrdnm, 41>4, 490 re-ion, 487
Ultra-infinite points, 352
I'nieursal polygons, 470
I'nilateral polyhedron, 494
Unit of distance, 147
Unit triangle, 99, 149, 812
Unit vector, 220
Unordered pairs of jxiints, 271
Unruled quadric. L' .".'.'
Up. 308
Upper side of a cut, 14
Vacuous statements. ^ t
Vailati, C... 44
Vanishing lines, 80
Vanishing planes, 848
Vanishing points, 80
Vector, magnitude of, 86, 147 ; nega-
tive of . 84; null, 83; unit. I
83, 220 Vectors, 82, 83, 85, 147, 219, 288 ; addi-
tion of, 84 ; collinear, 84 ; product of
two, 220 ; ratios of collinear, 85 ; sum
of, 83 Vertex, of a conic, 191 ; of a parabola,
193 Vertices, of a broken line, 464 ; of a poly-
hedron. 474 ; of a triangular region, 53 Volume, 290, 811
Whitehead. A. V. :I2. 41 Wiener, II.. (.»4. 2HO. :W2. 327 Wilson. E. H.. '.»•!. US, 188,862
Young, J. W.. iii, 260 Young, J. W. A., 146
Xennelo. E.. -'7
Zero vector, 83, 220
QA Veblera, Oswald
4.71 Protective geometry
v.2
Phy&ic*! & Applied Sci.
PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET
UNIVERSITY OF TORONTO LIBRARY