f
PROBLEMS
IN
DESCRIPTIVE GEOMETRY
FOR CLASS AND DRAWING ROOM
A COLLECTION OF OVER 900 DEFINITE PROBLEMS,
FOR STUDENTS IN ENGINEERING AND TECHNICAL SCHOOLS.
GENERAL PROBLExMS, SPECIAL CASES, APPLICATIONS,
WITH 85 PRACTICAL FIGURES.
BY
WALTER TURNER FISHLEIGH, A.B., B.S.
Department of Mechanism and Drawing
University of Michigan
OF THE
UNIVERSITY
OF
PUBLISHED
BY THE A UTHOR
ANN ARBOR, MICH.
Copyright, igio
By W. T. fish LEIGH
The Ann Arbor Press
PREFACE
GREAT changes have been made in recent years in the courses offered in
Technical and Engineering Schools, but in none has the advance been
more marked than in Descriptive Geometry. Engineers and instructors
early came to appreciate the practical value of a thorough course in Descriptive
Geometry and its applications, and at the same time to realize that a working
knowledge in such a subject could not be ^iven to the student without dozens, if
not to say hundreds, of original problems. The theory is comparatively simple,
the figures in the text easily comprehended, but for the original lay-out, the
special case, and the practical application, the student must be prepared by
constant problem practice. Thus, just as the modern Law School has in many
instances been led to adopt the "case method" of instruction, so has the Engineer-
ing School advanced to a method of instruction which might well be termed the
"Problem Method."
It is to meet the deinand for a complete set of original problems in
Descriptive Geometry, which might be had by the student at a low price, that this
book has been arranged. The aim of the author has been two- fold : first, to present
to the student for reference and study, a collection of definite problems under
each of the common general principles, including special cases and groups of
applications illustrating the use of these principles in practical problems ; second,
to furnish for the instructor a book of simple notation in which he can readily
find the particular sort of a problem desired, and of such scope and arrangement
as to be convenient for use in assigning problems for blackboard work, for home
study, or for drawing-room solution. In attempting to produce a thoroughly
usable book, special attention has been paid to the following: —
1. Lay-outs. Adopting a 12" X 18" sheet as standard, problems have been
carefully arranged for ^, ^, 3^, or 1 whole sheet, as the case may be. A
number in brackets following the regular problem number indicates at once what
part of a sheet is needed for the solution of each problem, so that several prob-
lems may be arranged conveniently on a large sheet, or each worked upon a
separate sheet of the necessary size. For class blackboard work, the student has
only to select some scale convenient for the problem at hand.
2. Notation. Points are located by co-ordinates which refer to the three
common co-ordinate planes, H, Y and P. A right line is located by giving the
co-ordinates of two of its points. A plane is determined by locating the point
of intersection of its H and V traces with the ground line, then giving the angles
which its traces make respectively with the ground line.
This system has been tried in classes at the University of Michigan with striking
success. The notation, being extremely simple and in strict accord with the notation ot
Solid Analytic Geometry and Trigonometry, is mastered by the student in one lesson. In the
use of these co-ordinates for a given lay-out, the student must "think in space" and get in
mind at the very outset the general location of points, lines and planes with respect to one
another and the planes of projection. He is aided in "seeing" his problem and understanding
"what he is about" by the very notation by which he locates given projections or traces upon
the drawing sheet. For the instructor, a problem is easily "read" and at a glance he can
select such a lay-out as he may desire.
3. Quadrants. Problems are given in all four quadrants, with a majority,
however, in the third and first in order to familiarize the student with the arrange-
ment of views for these two quadrants, which are commonly used in practice.
4. Practical applications. A large number of applications have been given
to interest the student in his work, to give him at first hand a knowledge of
294669
iv preijpace;
the way the different problems come up in practice, and to give him as soon as
possible a working knowledge of Descriptive Geometry through problems where
ingenuity and originality in applying the principles are as essential as the general
analyses and figures of the text. The trouble with ordinary engineering students
seems to be, not so much that they do not knozt' their Descriptive Geometry, as
that they cannot apply it. In reply to the possible criticism that a few of the
problems offered are somewhat more artificial than practical, the author has only
to say that they illustrate principles, that they interest the student more than mere
point-line-plane problems, and that pedagogically they have been found by experi-
ence to be of value.
5. Figures. A large number of practical figures are presented on blue prints
not only for use in the particular problems in this book, but with the hope of giving
to the student and instructor a suggestion of the many useful applications of
the subject in hand, and thus adding to his study an interest which perhaps
could be stimulated in no other way. It had been the intention of the author to
have plates made, but at the suggestion of practically all the instructors who have
used the book, the figures are retained in blue print form. They give to the
student at first hand, excellent examples of the kind of draughting which should
be aimed at in a drawing course, and make valuable reference sheets, as the work
progresses.
6. Arrangement and grouping. General grouping is made under (1) pre-
liminary problems on the point, line and plane, (2) general problems based
thereon, (3) representation of surfaces, (4) planes tangent to surfaces, (5) inter-
sections and developments. Under each of these general groups are arranged
smaller groups which cover general cases, special cases or variations, and practical
applications. Particular attention has been paid to arrangement and headings,
with a view to making the various problems easy of access and the book as a
whole, convenient for the assigning of problems with desirable variation and
sequence.
7. Scope. The scope of the book is thought to be sufficient for any element-
ary course in Descriptive Geometry.
The author would appreciate any suggestion as to additions or changes which
it might seem desirable to make. While a large number of the problems included
are original, he has felt free in consulting other works, and would mention the
following to which he is indebted for many valuable ideas and suggestions :
ELEMENTS OF DESCRIPTIVE GEOMETRY A. E. CHURCH.
DESCRIPTIVE GEOMETRY J. A. MOVER.
ELEMENTS OF DESCRIPTIVE GEOMETRY C. W. MAC CORD.
NOTES ON DESCRIPTIVE GEOMETRY J. D. PHILLIPS AND A. V. MILLAR.
PROBLEMS IN DESCRIPTIVE GEOMETRY G. M. BARTLETT.
CARPENTRY AND SHEET METAL WORK. . .AMERICAN SCHOOL OF CORRESPONDENCE.
DESCRIPTIVE GEOMETRY FILES UNIVERSITY OF MICHIGAN.
The author wishes also to acknowledge the kind assistance and continued encour-
agement of his colleagues, Messrs. F. R. Finch and D. E. Foster of the Univer-
sity of Michigan.
W. T. FiSHLElGH.
Ann Arbor, Mich., June, 1910.
CONTENTS
INTRODUCTORY
PRELIMINARY PROBLEMS.
TO FIND THE H AND V PROJECTIONS OF POINTS 3
TO FIND THE H AND V PROJECTIONS OF LINES 3
TO DETERMINE THE TRACES OF PEANES 5
TO ASSUME RIGHT LINES IN GIVEN PLANES 5
REVOLUTION OF POINTS AND LINES ABOUT GIVEN AXES 5
PROFILE PROJECTIONS OF POINTS G
PROFILE PROJECTIONS OF LINES G
PROFILE TRACES OF PLANES G
PROBLEMS ON THE POINT, LINE AND PLANE.
L TO FIND THE POINTS IN WHICH A GIVEN RIGHT LINE
PIERCES THE PLANES OF PROJECTION 7
TO FIND POINTS WHERE LINES PIERCE PLANES PARALLEL TJ THE PLANES
OF PROJECTION 7
TO FIND THE PROJECTIONS OF LINES WHICH PIERCE THE PLANES OF PRO-
JECTION AT GIVEN POINTS 7
APPLICATIONS 8
2. TO FIND THE TRUE LENGTH OF A RIGHT LINE JOINING
TWO GIVEN POINTS IN SPACE 8
TO FIND A POINT ON A GIVEN LINE A GIVEN DISTANCE FROM ONE END 9
TO FIND THE ANGLE A GIVEN RIGHT LINE MAKES WITH H AND V 9
TO FIND THE PROJECTIONS OF A RIGHT LINE MAKING GIVEN ANGLES WITH
H AND V 9
TO FIND THE THREE PROJECTIONS OF A POINT ON A GIVEN LINE, EQUIDISTANT
FROM BOTH H AND V 9
APPLICATIONS 9
3. TO PASS A PLANE THROUGH THREE GIVEN POINTS 10
TO PASS A PLANE THROUGH TWO INTERSECTING LINES 10
TO PASS A PLANE THROUGH TWO PARALLEL LINES LI
TO PASS A PLANE THROUGH A POINT AND A RIGHT LINE 11
APPLICATIONS 11
4. TO FIND THE TRUE SIZE OF THE ANGLE BETWEEN TWO
GIVEN INTERSECTING LINES AND TO FIND THE PROJEC-
TIONS OF ITS BISECTOR 12
APPLICATIONS , 12
5. TO ASSUME CERTAIN LINES IN GIVEN PLANES 13
TO LOCATE CERTAIN FIGURES IN GIVEN PLANES 14
APPLICATIONS 14
6. TO FIND THE LINE OF INTERSECTION OF TWO GIVEN
PLANES 15
APPLICATIONS , 15
Vi CONTENTS
7. TO FIND THE POINT IN WHICH A GIVEN RIGHT LINE
PIERCES A GIVEN PLANE 16
WHEN PLANE IS GIVEN P.Y TWO RIGHT EINES IN IT, TO BE SOEVED WITHOUT
FINDING TRACES 16
APPEICATlONS 17
8 THROUGH A GIVEN POINT, TO DRAW A LINE PERPENDIC-
ULAR TO A GIVEN PLANE, AND TO FIND THE DISTANCE
FROM THE POINT TO THE PLANE 18
THROUGH A GIVEN POINT, TO PASS A PEANE PERPENDICULAR TO A GIVEN
PLAN E 18
THROUGH A GIVEN LINE IN A GIVEN PLANE, TO CONSTRUCT A PLANE PERPEN-
DICULAR TO THE GIVEN PLANE 18
APPLICATIONS 18
9. TO PROJECT A GIVEN RIGHT LINE UPON A GIVEN PLANE. . 19
TO PROJECT GIVEN FIGURES UPON GIVEN PLANES 19
APPLICATIONS 19
10. THROUGH A GIVEN POINT, TO PASS A PLANE PERPENDIC-
ULAR TO A GIVEN RIGHT LINE 20
THROUGH A GIVEN POINT, TO CONSTRUCT A LINE PERPENDICULAR TO A
GIVEN LINE 20
APPLICATIONS 80
11. TO PASS A PLANE THROUGH A GIVEN POINT PARALLEL TO
TWO GIVEN RIGHT LINES 21
TO PASS A PLANE THROUGH ONE LINE PARALLEL TO ANOTHER 21
TO PASS A PLANE THROUGH A GIVEN POINT PARALLEL TO A GIVEN PLANE AND
FIND DISTANCE BETWEEN THE TWO PLANES 21
APPLICATIONS 21
12. TO FIND THE DISTANCE FROM A GIVEN POINT TO A GIVEN
RIGHT LINE 22
TO FIND THE PROJECTIONS OF A LINE THROUGH A GIVEN POINT PERPENDIC-
ULAR TO A GIVEN RIGHT LINE 22
TO FIND THE DISTANCE BETWEEN TWO PARALLEL LINES 22
APPLICATIONS 22
13. TO FIND THE ANGLE WHICH A GIVEN RIGHT LINE MAKES
WITH A GIVEN PLANE 23
APPLICATIONS 23
14. TO FIND THE ANGLE BETWEEN TWO GIVEN PLANES 24
ONE TRACE OF A PLANE BEING GIVEN, AND THE ANGLE WHICH THIS PLANE
MAKES WITH A PLANE OF PROJECTION, TO FIND THE OTHER TRACE 25
APPLICATIONS 25
15. TO FIND THE SHORTEST DISTANCE BETWEEN TWO RIGHT
LINES NOT IN THE SAME PLANE 26
APPLICATIONS 26
GENERAL PROBLEMS BASED ON POINT, LINE AND PLANE,
WITH APPLICATIONS 27-32
BUILDING UP SOLIDS, CONDITIONS GIVEN.
PYRAMIDS, CONES, PRISMS, CUBES 33-34
CONTENTS vii
REPRESENTATION OF SURFACES
HELICAL CONVOLUTES. assume i;le;ments; intersection with h or
OBEIQUE PLANE ; ASSUME POINTS ON SURFACE 35
HYPERBOLIC PARABOLOIDS AND CONOIDS, assume elements,
FIRST AND SECOND GENERATIONS; PI,ANE DIRECTERS OF BOTH GENERA-
TIONS ; ASSUME POINTS ON SURFACE 35
HYPERBOLOIDS OF ONE NAPPE, ETC. assume elements through
GIVEN POINTS 37
HYPERBOLOIDS OF REVOLUTION OF ONE NAPPE, assume Ele-
ments; SECOND generations; construction of meridian curves; as-
sume POINTS ON SURFACE 37
HELICOIDS. OBLIQUE AND right; ASSUME ELEMENTS; INTERSECTIONS WITH
H OR V; ASSUME POINTS ON SURFACE 38
APPLICATIONS 39
PLANES TANGENT TO SURFACES.
THROUGH POINTS ON THE SURFACE; THROUGH POINTS OFF THE SURFACE; PAR-
ALLEL TO GIVEN lines; through given lines.
1. RIGHT CYLINDERS 41
2. OBLIQUE CYLINDERS 41
3. RIGHT CONES 43
4. OBLIQUE CONES 43
5. HELICAL CONVOLUTES 43
6. HYPERBOLIC PARABOLOIDS 43
7. HYPERBOLOIDS OF ONE NAPPE 43
8. HELICOIDS 44
9. SPHERES 44
10. ELLIPSOIDS 44
11. HYPERBOLOIDS AND PARABOLOIDS OF REVOLUTION ... 45
13. TORI : 45
TO FIND POINTS WHERE LINES PIERCE SURFACES
RIGHT PRISMS AND PYRAMIDS 46
OBLIQUE PRISMS AND PYRAMIDS 46
RIGHT CYLINDERS AND CONES 46
OBLIQUE CYLINDERS AND CONES 46
WARPED SURFACES 47
DOUBLE CURVED SURFACES 47
Viii CONTENTS
INTERSECTIONS AND DEVELOPMENTS
I. INTERSECTIONS OE SURFACES AND PLANES, TRUE SIZES OF
INTERSECTION CURVES, DEVELOPMENTS, TANGENT LINES
TO INTERSECTION CURVES.
1. PRISMS CUT BY PI.ANES AND DEVFXOPMEINT 48
2. PYRAMIDS CUT I'Y PLANES AND DICVEILOPMENT 48
3. RIGHT CYLINDERS CUT BY PLANES, TRUE SIZE OF CURVE, AND DEVELOP-
MENT 49
4. OBLIQUE CYLINDERS CUT BY RIGHT SECTION PLANES AND DEVELOPMENT 50
5. OBLIQUE CYLINDERS CUT BY OBLIQUE PLANES AND DEVELOPMENT 51
6. RIGHT CONES CUT BY PLANES AND DEVELOPMENT 52
7. OBLIQUE CONES CUT BY SPHERES AND DEVELOPMENT 53
8. OBLIQUE CONES CUT PLANES AND DEVELOPMENT 54
9. SPHERES CUT BY PLANES 54
10. ELLIPSOIDS CUT BY PLANES, TRUE SIZE OF CURVE OF INTERSECTION .... 55
11. PARABOLOIDS AND HYPERBOLOIDS OF REVOLUTION CUT BY PLANES, TRUE
SIZE OF INTERSECTION CURVES, LINES TANGENT TO CURVES 55
12. TORI CUT BY PLANES, LINES TANGENT TO INTERSECTION CURVES 56
13. SQUARE RINGS CUT BY PLANES; TRUE SIZE OF CURVES OF INTERSECTION 56
14. HYPERBOLIC PARABOLOIDS CUT BY PLANES 56
15. HYPERBOLOIDS OF REVOLUTION OF ONE NAPPE, CUT BY PLANES 58
16. HELICOIDS CUT BY PLANES, TRUE SIZE OF CURVES OF INTERSECTION .... 58
SHORTEST DISTANCES BETWEEN POINTS ON SURFACES.
PRISMS AND PYRAMIDS 60
CYLINDERS AND CONES 60
SPHERES 61
II. INTERSECTIONS OF TWO SURFACES.
1. TWO CONES. INTERSECTION CURVES, LINE TANGENT TO SAID CURVES,
OR DEVELOPMENT OF ONE CONE 62
2. TWO CYLINDERS. INTERSECTION CURVES, LINE TANGENT TO SAID CURVES,
OR DEVELOPMENT OF ONE CYLINDER 63
3. CONE AND CYLINDER. INTERSECTION CURVES, LINE TANGENT TO SAID
CURVES, OR DEVELOPMENT OF ONE SURFACE 64
4. GENERAL INTERSECTIONS OF TWO SURFACES. INCLUDING COMBINATIONS
OF SINGLE-CURVED, WARPED, AND DOUBLE-CURVED SURFACES 67
III. GENERAL INTERSECTIONS.
INCLUDING THREE SURFACES 70
IV. GENERAL APPLICATIONS.
BASED ON INTERSECTIONS AND DEVELOPMENTS 71
BLim PRINT FIGURE INDEX 74
INTRODUCTORY.
Working Space. Adopting a 12" x 18'' sheet with ^" border as a standard,
problems have been divided into -1 classes: (1) those for whose solution a whole
sheet is necessary, with the ground line as shown in Fig. 1; (2) those which
can be arranged two on a sheet, with ground lines as shown in Fig. 2; (3) those
which can be arranged four on a sheet, with ground line as shown in Fig. 3 ;
(4) those for whose solution -J of a sheet is necessary, with ground line as shown
in Fig. 3. After the problem number, a number is given in brackets, thus [1],
[2] J [4], [8], to indicate whether 1, 2, 4 or 8 such problems can be worked upon
a standard sheet. Where such a number in brackets is given in a group heading,
any problem in this group can be worked in the space indicated. Thus in the
solution of problems in the drawing room or at home a standard sheet may be
used and a number of problems arranged thereon in accordance with the space
required, or each problem may be solved upon a separate sheet, using standard
sheet, half standard, quarter standard, etc., as the particular problem may require.
The Profile Plane and the Origin. Unless otherwise stated the reference
profile plane is assumed to be perpendicular to the ground line XX at the point
where the latter cuts the right hand boundary line of the working space. Thus,
in Fig. 4, the profile planes would be assumed to be perpendicular to the ground
lines XX at the points marked O. The right hand boundary line of each working
space would then represent the profile ground line. The point O is the origin
of co-ordinates and from it will be located all points, as per the paragraph below.
Where it is desirable to have the profile plane located otherwise for any particu-
lar problem, or group of problems, the distance of the point O from the right
hand boundary line of the working space will be given. Thus, a statement,
P at - 5", would locate the profile plane and origin O as shown in Fig. 5.
Location of points and right lines by co-ordinates. Adopting the usual
notation of Analytic Geometry, a point is located in space by its X, Y and Z
co-ordinates, these co-ordinates respectively giving its distance from the profile,
the vertical and the horizontal planes. A right line, likewise, is located by giving
the co-ordinates of two of its points. X distances are considered + when to the
right of the profile plane, and give in a particular case the distance from the pro-
file ground line, of the line joining the H and V projections of a point. Y dis-
tances are considered + when in front of the vertical plane, and give the distance
of the H projection of a point from the ground line. Z distances are considered
4- when above the horizontal plane, and give the distance of the V projection from
the ground line.
In short, distances are positive zuhen to the right of P, the front of V, and
above H. Thus a point M(- 4"*, + 2", - 3") is a point 4 inches to the left of P,
2" in front of V, and 3" below H, therefore in the 4th quadrant. To illustrate
the complete problem notation, suppose a problem reads : —
173. [2] P at - 3-J-". Find the three projections of the line joining the point
A(-2",-3",-l") with the point B(- 4", + 1", + 1"). The lay-out is shown
with dimensions in Fig. 6.
Location of planes. The traces of a plane are located by giving first the dis-
tance from O of the point of intersection of the traces with the ground line XX,
then the angle which the H trace makes with the ground Hne, then the angle
which the V trace makes with the ground line. Employing the usual trigono-
metric notation, angles measured counter-clockwise are positive, clockwise neg-
ative. The plane T(- 12'', + 60°, - -±5°) is shown located in Fig. 7.
In case a plane T is parallel to the ground line, both traces will be parallel to
the ground line and their point of intersection with the ground line is removed
to an infinite distance from O. The notation T( co, + 4", - 3'') here indicates
first that the traces are both parallel to the ground line, second that the H trace
is 4" in front of the vertical plane, third that the V trace is 3" below H. The
drawing board representation is given in Fig. 8. The notation S(oo,-3", oo )
indicates a plane S, parallel to and 3'' behind V.
Planes may also be determined by giving the co-ordinates of three points,
or of two intersecting or parallel lines therein. In special cases, when the above
notation for traces is not convenient, the traces may be located by co-ordinates,
considering them merely as right lines in H, V and P respectively.
Abbreviations which may be used.
alt. = altitude. pt. = point,
const. = construction, quad. = quadrant,
cyl. = cylinder. rt. = right,
dia. = diameter, tang. = tangent,
obi. = oblique. G. L. = ground line,
par. = parallel. H = horizontal plane,
perp. = perpendicular. V = vertical plane,
proj. = projection. P = profile plane.
Fig, = figure.
PRELIMINARY PROBLEMS.
1.— TO FIND THE H AND V PROJECTIONS OF POINTS.
1. Point A in 1st quad., 2" from V, 1" from H.
2. Point B in 2nd quad., f" from V, 2" from H.
3. Point C in 3rd quad., V from V, 1^ from H.
4. Point D in 4th quad., |" from V, 1^' from H.
5. Point E, in 4th quad., twice as far from H as from Y.
6. Point F, in 3rd quad., one-third as far from H as from V.
7. Point G, in 2nd quad., equidistant from H and V.
8. Point H, in 1st quad., equidistant from H and V.
9. Point J, 2" in front of V, f '' below H.
10. Point K, V behind V, 1'' below H.
11. Point L, ly in front of V, 2'' above H.
12. Point M in V, 1^^' above H.
13. Point N in V, 1" below H.
14. Point O in H, V in front of V.
15.
16.
17.
18.
Point Q, in the ground line.
Points R and vS, both in the ground line and f " apart.
Describe fully the location of the pts. A, B, E, H, M, in Fig. 9.
Describe fully the location of the pts. C, D, H, K, N, in Fig. 9.
19.
L
20.
L
21.
L
22.
h
23.
L
24.
L
25.
L
26.
L
27.
L
28.
Li
29.
Li
30.
Li
31.
U
32.
U
33.
u
34.
u
2.— TO FIND THE H AND V PROJECTIONS OF LINES.
ne joining pt. A in 1st quad, with B in 2nd quad,
ne joining pt. C in 2nd quad, with D in 3rd quad,
ne joining pt. E in 3rd quad, with F in 4th quad,
ne joining pt. G in 4th quad, with H in 1st quad.
ne joining pt. J in 2nd quad, with K in 4th quad,
ne joining pt. L in 1st quad, with M in 3rd quad.
ne joining pt. N in H with O in 1st quad,
ne joining pt. P in H with Q in 2nd quad,
ne joining pt. R in PI with S in 3rd quad,
ne joining pt. A in H with B in V.
ne joining pt. C in V with D in 1st quad,
ne joining pt. D in V with E in 3rd quad.
ne AB, par. to H, obi: to V, in 1st quad,
ne CD, par. to H, obi. to V, in 2nd quad,
ne EF, par. to V, obi. to H, in 3rd quad,
ne GH, par. to V, obi. to H, in 4th quad.
35. Line KL par. to ground line in 1st quad.
36. Line MN par. to ground line in 8nd quad.
37. Line OP par. to H and V in 3rd quad.
38. Line RS par. to H and V in 4th quad.
39. Line AB perp. to H in 1st quad.
40. Line CD perp. to H in 2nd quad.
41. Line EF perp. to V in 3rd quad.
42. Line GH perp. to V in 4th quad.
43. Line KL lying in H in front of V.
44. Line MN lying in V below H.
45. Line OP in the ground line.
46. Line RS in 1st quad, in a plane perp. to the G.L.
47. Line in a plane perp. to ground line, joining pt. A in 1st quad, to pt. B in
3rd quad.
48. Describe fully the positions of the lines AB, CD, EF, KL, MN, shown in
Fig. 10.
49. Describe fully the positions of the lines GH, CD, OP, KL, MN, shown in
Fig. 10.
Intersecting lines.
50. Lines AB and BC intersecting at a pt. B in the 1st quad.
51. Lines CD and DE intersecting at a pt. D in the 3rd quad.
52. Line FG, par. to H intersecting line GH at a pt. G in 2nd quad.
53. Line KL, par. to V intersecting line LM at a pt. L in 3rd quad.
54. Line NO, par. to the G.L., intersecting line OP at pt. O in 4th quad.
55. Line PQ, par. to the G.L., intersecting line QN at pt. Q in 3rd quad.
56. Line QR perp. to H, intersecting RS, par. to the G.L.
57. Line TU perp. to H, intersecting UV, par. to H.
58. Line YZ, perp. to V, intersecting ZM, par. to the G.L.
Parallel lines.
59. Parallel lines, obi. to H and V, through pts. A and C in 1st and 3rd quads.
respectively.
60. Parallel lines, obi. to H and V, through pts. B and D in 2nd and 3rd quads,
respectively.
6L Parallel lines, obi. to H and V, through pts. E and G in 3rd quad, and in H
respectively.
62. Parallel lines, obi. to H and V, through pts. F and K, in 1st quad, and in V
respectively.
63. Parallel lines, in 3rd quad., par. to H and obi. to V.
64. Parallel lines, both par. to the G.L., one in 3rd quad., one in H.
65. Parallel lines, both perp. to H, both in 4th quad.
3.— TO DETERMINE THE TRACES OF PLANES.
66. Plane R, obi. to both H and V.
67. Plane S, par. to H, and above H.
68. Plane T, par. to V, and behind V.
69. Plane U, par. to ground line, cutting across the 1st quad. '
70. Plane R, perp. to ground line.
71. Plane S, perp. to H, obi. to V.
73. Plane T, perp. to V, obi. to H.
73. Plane U, passing through the ground line.
74. Describe fully the planes R, S, U, W, Z shown in Fig. 11.
75. Describe fully the planes R, P, Q, T, Z shown in Fig. 11.
Parallel planes.
76. Par. planes, R and S, obi. to both H and V.
77. Par. planes T and U, perp. to H, obi. to V.
78. Par. planes V and W, perp. to V, obi. to H.
79. Par. planes R and S, perp. to ground line.
80. Par. planes T and U, par. to V.
81. Par. planes V and W, par. to H.
4.— TO ASSUME RIGHT LINES IN GIVEN PLANES.
In plane S, Fig. 11 ; (b) in plane U, Fig. 11.
In plane S, Fig. 11 ; (b) in plane U, Fig. 11.
In plane W, Fig. 11 ; (b) in plane Z, Fig. 11.
In plane W, Fig. 11 ; (b) in plane Q, Fig. 11.
5.— REVOLUTION OF POINTS ABOUT GIVEN AXES.
86. The pt. A, Fig. 12, is to be revolved about the axis MX in H. Find the 2
projections of the pt. after revolution into H.
87. The pt. B, Fig. 13. is to be revolved about the axis RS in H. Find the 2
projections of the pt. after revolution into H.
88. The pt. C, Fig. 14, is to be revolved about the axis PQ into V. Find the 2
projections of the pt. where it falls into Y.
89. The pt. D, Fig. 15, is to be revolved about the axis RS into V. Find the
2 projections of the pt. where it falls into V'.
Revolution of lines about given axes.
90. The line AB, Fig. 16, is to be revolved about the axis MN (par. to AB in H)
into H. Find its 2 projections in revolved position.
91. The line CD, Fig. 17, is to be revolved about the axis PC (par. to CD in V)
into y. Find its 2 projections in revolved position.
92. The line AB, Fig. 10, is to be revolved into H about its H projection as an
axis. Find its 2 projections in revolved position.
93. The line EF. Fig. 10, is to be revolved into V about its V projection as an
axis. Find its 2 projections in revolved position.
94. The line EF, Fig. 10, is to be revolved about the H projecting perpendicular
of the pt. E until it is par. to V. Find its 2 projections in revolved position.
95. The line OP, Fig. 10, is to be revolved about the V projecting perpendicular
of the pt. O until it becomes par. to H. Find its 2 projections in revolved
position.
82.
(a)
82.
(a)
84.
(a)
85.
(a)
6.— PROFILE PROJECTIONS OF POINTS.
The student is referred to the problems in Set I. After assuming a profile plane,
in each case, find the profile projections of the points given
96. In problems 1, 3, 7, 12.
7.— PROFILE PROJECTIONS OF LINES.
The student is referred to the problems in Set 2. Having assumed a profile
plane, in each case, find the profile projections of the lines
97. In problems 25, 31, 39.
98. In problems 27, 33, 41.
8.— PROFILE TRACES OF PLANES.
The student is referred to the problems in Set 3. Having assumed a profile
plane in each case, find the profile traces of the planes
99. In problems 6Cy, 67, 69.
100. In problems 66, 68, 73.
PROBLEMS ON THE POINT, LINE AND PLANE.
1.— TO FIND THE POINTS IN WHICH A GIVEN RIGHT LINE
PIERCES THE PLANES OF PROJECTION.
Find the pts. M and N in which the following lines pierce H and V
respectively.
General. [8]
101. Line joining pt. A(- 2^', - 1", - D with B(- 1^", - 2", - 1").
103. Line joining pt. C(-3", + lY', + V') with D(- U'', + i", + 2").
103. Line joining pt. EC- 3", - 2'', + J") with F(- 1^", - ^'', + ID-
104. Line joining pt. G(- 34", - l^", - 1") with H(- 1^", + |", + 2").
105. Line joining pt. K(- 2f", + U", + 1") with L(- U". + i". - !")•
Lines par. to H or V. [8]
lOG. Line joining pt. A(- 3", - f ', - 1") with B(- U/', - 1^', - 1").
107. Line joining pt. C(- 3'', + 1^', + f") with D(- iV', + i", + f")-
108. Line joining pt. E(- 3", - V' , - UJ') with F(- U'^-V, + H")-
109. Line joining pt. G(- 2f' , ^ 1", + 1") with H(- 1^", + 1", - 2'')-
110. Line joining pt. K(- 3", - Y', + f") with L(- 1|'', - 2", + f")-
Lines in plane par. to P. [8] P at - 2".
111. Line A(-l^-ir,-U") B(- 1", - T, - D-
112. Line 0(0'',- U", + i") D(0^-i",-l|'O.
113. Line ECO'', -4", -ID F(0", + H", + If')-
Find the H, V and P piercing points (M, N, O) of lines. [8] P at - 2".
114. Line A(-H",-D-2") B(- r. - D -i")-
115. Line C(-ir,-r', + li'0 D(-y', + r', + |'0-
116. Line E(- D -h 1", + f) F(+f", - f', - Y')-
Find the 3 projs. of the P piercing point (O) of given lines. [8] P at - 1|-".
117. Line A(-ir;-l", -D B(- D + 1", - i").
118. LineC(-2",-^",-li'0 DC-^-D-lD-
119. Line E(-2'', 0'^ -4") F(-D0'',-li'O.
Find the points where the foei.owinc lines pierce a plane parall5;l to h
and below h. [8]
120. Line M(-2f' , + r', + rO N(- 1'', + i'', - 2'').
121. Line O(-3",-D-2'0 P(- 1^', -^^ + H'')-
122. Line R(-3^-D-l|'') S(- D - 1", - i'O-
find the projections of the lines which pierce the PLANES OF PROJECTION
RESPECTIVELY, AS FOLLOWS. [8]
123. V at A(- V', 0", - I'O, H at B(- 1^' , - 2", 0").
124. V at C(- 3'', (r, - H'O,- H at D(- ij", + 2", O'')-
135. Pat -2". H at E(-1D-D0"), Pat F(0^-li^-rO.
APPLICATIONS.
126. [4] On the plans for the mill building' shown in Fig. 18, the line
A(-3^--f' ,-2r) B(-3'',--r,-J'0 gives the direction of the center line of
a vertical pipe. Find the right end view (P projection) of the building and the
3 projections of the points M and N where this line pierces the floor planes F
and S.
137. [4] On the plans for the mill building shown in Fig. 18, the line
C(-2'',-i'',-li'0 D(-2",-2",-l-l") gives the direction of the center line
of a telephone wire. Find the right end view (P projection) of the building and
the 3 projections of the points K and L where this telephone wire pierces the
walls parallel to V.
128. [4] On the plans for the mill building shown in Fig. 18, the line
E(-r',-f' ,-U'') F(-Y\-^'',~iy) gives the direction of the center line
of a shaft for an outside rope drive. Find the right end view (P projection) of
the building and the 3 projections of the point R where this line pierces the right
hand end of the building.
129. [8] In Fig. 32, find the point X where the tow-rope MB pierces the hori-
zontal plane (extended) of the tow-path.
130. [8] In Fig. 32, find the point where the tow-rope MB, if extended, would
pierce a vertical plane through the rear end of the canal boat.
131. [2] G. L. par. to long side of space. P at - 4". Suppose that the telegraph
wire in Fig. 30 were to be extended in the line PW. Find (1) the profile projec-
tion (right end view) of the line and factory building, (2) the 3 projections of
the point X where the line PW would pierce the horizontal plane of the floor
ABC, of the point Y where it would pierce the plane of the rear wall parallel
to ABFE, and of the point Z where the line PW would pierce the plane of the left
vertical end wall of the building.
132. [2] G. L. par. to long side of space. P at -4". Suppose that the tele-
graph wire in Fig. 30 were to be extended in the line PW. Find (1) the profile
projection (right end view) of factory building and line PW, (2) the 3 pro-
jections of the point X where the line PW would pierce the horizontal plane of
the roof of the building, of the point Y where the line PW would pierce a verti-
cal partition plane half way between and parallel to the front and rear walls of
the main part of the building, of the point Z where the line PW would pierce the
plane of the left end wall of the building.
»
2._TO FIND THE TRUE LENGTH OF A RIGHT LINE JOINING
TWO GIVEN POINTS IN SPACE.
General. [8]
133. Line joining pt. A(- 3|'', - 1^'', - H") with B(- If, - i", - Y').
134. Line joining pt. C(-3'', - 2", + i") with D(-1Y',-V'> + '^D-
135. Line joining pt. E(- S^', - U", - 1") with F(- 1^'', + Y', + I'O-
136. Line joining pt. G(- 3'', - Y', !- I'O with H(- 1^', - 1^. - I'O-
137. Line joining pt. K(- 2f'', + U", f I'O with L(- H", + 1", - I'O-
Lines joining points in H, V or P. [8]
138. Line joining pt. A(- 3", 0", - I'O with B(- 1^', - 2", 0")-
139. Line joining pt. C(- 3'', 0'', - 1^") with D(- 1", + 2", O'O-
140. P at - v. Line joining pt. E(- If", 0", - 1") with F(0", + H'', -f i'O-
Lines in plane perp. to G. L. [8] P at - 2".
141. Line joining pt. M(0", - i", - IV') with N(0", - 1", - i").
142. Line joining pt. 0(0", - V', - I'O with P(0'', + 1|'', - li").
143. Line joining pt. R(0",-1|'', f|") with S(0", - 1'', - 14").
TO Find a point on a given line; a given distance erom one End. [8]
144. Find pt. in Hne P(- 3^", - H", - 1^") R(- l^"^ _ i", _ i/') which is f"
from R.
145. Find pt. in Hne M(- 3", 0", - 1") N(- H", - 3", 0") which is H" from M.
146. Find pt. in Hne K(- 3|", - U", - 1") L(- li", + T- + I'O which is 1"
from K.
TO find the angle a given right line makes with h and v. [8]
147. Line M(- 3|", -^", 0") N(- 1^", - 2", - 1").
148. Line 0(-2f", + l|", + l") P(- 1^', + |", - 1").
149. Line D(- 3", + i", - l-J") E(- 1", H 2", - i").
TO FIND THE PROJECTIONS OF A RIGHT LINE MAKING GIVEN ANGLES
WITH H AND V. [8]
150. Find the projections of a Hne through the pt. 0(- 2|", + 1^", + 1") mak-
ing 30° with H and 45° with V.
151. Find the projections of a Hne through the pt. M(- 2", - f", - f") making
45° with H and 22^° with V.
152. Find the projections of a Hne through the pt. P(- 2^", - 1^", - |") mak-
ing 22i° with H and 30° with V.
TO FIND THE 3 PROJECTIONS OF A POINT M ON A GIVEN LINE, EQUIDISTANT FROM
BOTH H AND V. [8] P AT - 1^".
153. Line joining A(- 1|", - Y', - 2") with B(- I", - 1", - i").
154. Line joining C(- If", - 1", - U") with D(- I", + 2", + i").
155. Line joining E(- 2", - J", - i") with F(-^", - H", - i").
APPLICATIONS.
156. [4] Find the true length of the hip rafter AC of the cottage in Fig. 19.
157. [4] Find the true length of the valley rafter MN of the cottage in Fig. 19.
158. [4] Find the distance from the point B in the ridge, to the point P, Fig. 19.
159. [4] F'ind the angle which the rafter BD makes with a horizontal plane
through CD, Fig. 19.
160. [4] Find the angle which the hip rafter AC makes with a horizontal plane
through CD, Fig. 19.'
161. [2] In the cap for a ventilator shaft shown in Fig. 20, find the length of
the intersection between the planes A and B ; also the distance from the point N
on the ridge of the ventilator cap to the point O.
162. [8] Find the length of the tug-of-war rope in Fig. 31.
163. [2] Find the lengths of the guy ropes AD, and CD of the boom derrick
shown in Fig. 27 and the angle which CD makes with a horizontal plane through
C. Find also the length of the boom FE and the true distance between the points
A and C.
ro
164. [2] Find the lengths of the guy ropes BD and AD of the boom derrick
shown in Fig. 27, and the angle which BD makes with a horizontal plane through
B. Find also the distance from the end of the boom F to the top of the mast D,
and the straight line distance from E to B.
165. [8] In the lamp shade of Fig. 21, find the length of the piece E used on
the edge between the planes A and B ; also the distance from the point E to the
point H of the base of the shade. Scale, 1" = I'-O".
166. [4] In the skew bridge of Fig. 28, find the length of the members EI and
f ID of the portal bracing ; also find the distance from the point C to the point B,
and from the point M to L.
167. [4] In the .skew bridge of Fig. 28, find the length of the members IF
and CG of the portal bracing ; also find the distance from the point A to D and
from O to K.
168. [8] Find the length of the tow-rope MB of Fig. 32.
169. [4] What is the length of the telegraph wire PW of Fig. 30. How much
farther is it from P to the point F of the roof. Scale, 1'' = 20'.
170. [8] Two stations A(- 28', + 14', + 9') and B(- 10', + 4', + 1') are to be
connected by a telegraph line. Find the shortest line that could be constructed.
Scale, i" = 1'.
171. [8] The center line of a straight tunnel enters a hill at a point
E(- 50', - 25', - 90') and comes out at a point F(- 170', - 80', - 35') on the other
side. Find the length of the tunnel and its grade, that is, the angle it mak'es
with the horizontal plane. Scale, 1" = 50'.
172. [8] Two wireless telegraph stations are located at the points
W(- 500', + 800', + 400') and Wi(- 1800', + 200', + 400') respectively. How far
are they apart? Scale, 1" = 500'. •
3.— TO PASS A PLANE THROUGH 3 GIVEN POINTS. [4]
173. Pass plane through points M(- 4^", - y, - 1") N(- 4", -^", - 1^") and
p(-3r',-ir,-r).
174. Pass plane through points A(- 6", - i", - 1^") B(- 4", - 1", - i") and
C(-U", + l", + r).
175. Pass plane through points E(- 6", - f, - 1|") F(- l^", + 1^", - |") and
G(-2",0",-U").
TO PASS A PI^\NE THROUGH TWO INTERSECTING LINES.
General. [4]
176. Lines A(- 6", + 1", 0") B(- 4", - 2", + 1") and BC(- 2|", + i", - H").
177. Lines D(- of , + |", - U") E(- 3^", + If, -i") and EF(-4f , 0", + f ).
178. Lines G(- 6", 0", - 2") H(- 4", - 2", - |" ) and HK(- U", 0", 0").
One line parallel to H or V or G. L. [4]
179. Lines A(- 5", - 1", - 2") B(- 3", - 1", - f ) and BC(- 4", 0", 0").
180. Lines D(- 6". - If, - |") E(- 3", + 1|", - ^") and EF(- 2^", 0", + 2").
181. Lines G(- 6", - f , - If ) H(- 3", - 1", - If ) and HK(- 4", - If , 0").
II
One line in plane parallel to P. [4]
182. P at -3". Line A(-1^0",-ir) B(-l",-l^-r) and the line
183. Line D(- 5-^,+ J-'', - l.D Z(- 3V\ + 1^',- V) and the intersecting line
EF(-3r,-ir,+r)-
184. Lines A(- 3", -r 1'', - 2'') B(- 3", + 1'', - I") and BC(- 5'', 0", - 1").
TO PASS A PLANE THROUGH TWO PARALLEL LINES. [4]
185. Line A(- 5'^ - U", f V) B(- 3", 0", - 1.^0 and line par. to AB through
c(-2-, + ir,-r)-
186. Line D(- 5'', + r', - 4") E(- 4", - r, - 1") and hne par. to DE through
F(-4".-H", 0").
187. Lines par. to G. L. through K(- 6", -]-",- 2|") and M(- 4'', + f", - y').
TO PASS A PLANE THROUGH A POINT AND A RIGHT LINE. [4]
188. Right line F(- 6'', - 1", - 2") G(-2V',- 1V\ + V) and the point
H(-4|^ + ]", + ^").
189. Right line parallel to ground line through D(- 6", - ^", - l^") and point
E(-l",-l:r, 0").
190. Right line A(- 3'', + 1'', - 2") B(- 3'', + 1", - i'') and a given point
C(-5",-r,-l").
APPLICATIONS.
191. [4] Assuming a ground line somewhere between the two views of the
cottage in Fig. 19, find the traces of the roof planes U, V and W.
192. [4] Assume a convenient ground hne in Fig. 39 and find the traces of the
roof planes R, S, T and X.
193. [2] Using the derrick of Fig. 27, find (1) the plane of the 2 guy wires
DC and DB, (2) a plane passed through the pts. E, C and D, (3) the plane of the
boom FE and the point C.
194. [4] In the skew bridge of Fig. 28, find (1) the traces of the plane of the
portal, that is the plane ABDC, (2) the traces of a plane through the points
C, D and L.
195. [4] In the skew bridge of Fig. 28, find (1) the traces of the plane of the
lines AC and CD, (2) the traces of a plane through the 2 parallels MN and DJ.
196. [2] In the Gondola Car of Fig. 23, the several plates are shown by the
projections of their bounding edges. Having assumed a convenient ground line,
find the traces of the planes of the plates A, C, D and the door plate.
197. [2] G. L. par. to long side of space. P at -4". In Fig. 30, find the
three traces of the front wall of the wing of the factory bi\ilding ; also the traces
of the plane of the three points P, W and B.
198. [4] Three vertical wells are driven at points A(- 50', + 17', + 53')
B(- 95', + 35', + 44') and C(- 135', + 15', + 25') of a hill side, striking water at
depths of 32 ft.. 20 ft. and 22-^- ft. respectivelv. Find the traces of the water
plane. Scale, 1" = 20'.
12 I .
4.— TO FIND THE TRUE SIZE OF THE ANGLE BETWEEN TWO
GIVEN INTERSECTING LINES AND TO FIND THE PRO-
JECTIONS OF ITS BISECTOR.
General. [4]
199. Angle A(- 2-^^ - 1^'', + *'') B(- 6'', - 1", - 2") C(-4^^ + 1'', + i") be-
tween lines AB and BC.
200. Angle D(- 3^', - H", + f'O E(- 5f', + i'', - H") F(- 3|'', + 1^'', -I'O
between lines DE and EF.
201. Angle G(- 4'', - 2", + 1'') H(- 2|", + i", - 1|'') K(- 6", + 1'', 0'') be-
tween GH and HK.
One side parallel to H or V, or G. L. [4]
202. Angle A(-7",-r,-l") B(- 6", - r, - 1") C(- 4", - f, + |'') between
lines AB and BC.
203. Angle D(-3^-r',-D E(- 5'', - 1'', - 2'') F(-4^0",0'0 between
lines DE and EF.
204. Angle G(- 6''. - If ', - |") H(- 3", + If'', - f ') K(- 2^", 0", + 2") be-
tween lines GH and HK.
One side perpendicular to H or V. [4]
205. Angle A(- 3'', ^ H'', - 2'') B(- 3", + If ', - f ') C(- 5'', - f ', - If ') be-
tween AB and BC.
206. Angle D(- 5^, + f ', + 1") E(- 5f ', + 2", + 1'') F(- 4'', -f f ', + 2") be-
tween DE and EF.
207. Angle G(-6^ + r', + fO H(- 6", + 1", + 2") K(- 4", - If, - If ') be-
tween GH and HK.
APPLICATIONS.
208. [4] In Fig. 19 find the true size of the angle between the hip rafters CA
and EA.
209. [4] In Fig. 19, find the true size of the angle between the hip rafter EA
and the ridge AB. Also find the projections of a line on the roof bisecting this
angle.
210. [4] In Fig. 19, find the true size of the angle between the two valley
rafters NM and MP.
211. [2] In the ventilator cap of Fig. 20, find the angle between KM and MN.
Also find the projections of a line in the plane D which bisects this angle.
212. [4] In the roof of Fig. 29, find the angle between the hip rafters 10-14
and 11-14, and the projections of the line bisecting that angle.
213. [4] In the roof of Fig. 29, find the angle between the lines 1-7 and 1-5
and the projections of the bisector of this angle.
214. [2] In Fig. 27, find the angle between the derrick guy wires BD and CD.
If a third guy wire be located so as to bisect this angle find its projection.
215. [2] In the derrick of Fig. 27, find the angle between the boom FE and
the mast DE. The b(3om is to be raised to such a position as to bisect the present
angle FED. Find its projections when in such a position.
216. [4] In the skew bridge of Fig. 28, find the angle between the end post
AC and the portal strut CD. In repairing the portal bracing, a brace was run
from the point C to the end post BD so as to bisect the angle ACD. Find its
projections.
217. [4] In the skew bridge of Fig. 28, find the angle between the portal strut
CD and the diagonal DL. Also find the projections of the bisector of this angle.
13
218. [4] A canal boat B(- 4|'', + 2^', + 1'') is towed along a canal bv two
mules M(-3", + U", + 2'0 and k(- Q¥\ + V', + ¥'), one on each bank at differ-
ent elevations. Find the angle between the ropes. Assuming that the mules
exert equal forces, the course of the boat will be shown by the bisector of the
angle between the tow ropes. Find this course.
5.— TO ASSUME CERTAIN LINES IN GIVEN PLANES.
Piercing points given. [8]
219. In plane R(~ 1", + 150°, - 120°), find line AB which pierces H 1" behind V
and V 1" below H.
220. In plane S{- 'SV', + 3()° , + 15°), find line CD which pierces H 1" behind V
and V y above H.
221. In plane T(- 2^1", - 90°, -i 15°), find line EF which pierces H 1^" in front
of V and V 1" above H.
Planes par. to ground line. [8] P at - 2".
222. Plane R( x, - ^", - 1''). Find the 3 projections of the line that pierces
H li" from P,'and Y Y' from P.
223. Plane S( x, + .^", - 2''). Find the 3 projections of the line that pierces
H IV' from P and V V' from P.
224. Plane T(x,-1''', x). Find the 3 projections of the line that pierces H
V' from P and makes an angle of 00° with H.
Lines to make given angle with one trace. [8]
225. Plane R(- 3|'', + 22^, - 45°), find hne AB which pierces H 1" behind V,
and makes an angle of 30° with the H trace.
226. Plane vS(- 3", - 120°, - 45°), find line CD which pierces V 1" below H,
and makes an angle of G0° with the V trace.
227. Plane T(- 3", + 90°, - 22^ ), find Hne EF which pierces V f" below H,
and makes an angle of 45° with the V trace.
Lines parallel to one trace. [8]
228. In plane R(- 1'', + 150°, - 120°) find line AB which is par. to its H trace
and li" below H.
229. In plane S(- 3|", -^ 30°, ^ 45°) find the line CD which is par. to its V trace
and 1" behind V.
230. In plane T(- 2f", - 90°, + 45°) find line EF which is par. to its V trace,
and 1" in front of V.
Passing through G. L. at equal angles with H and V traces. [8]
231. In plane R(- 2", + 90°, - 90°) find line AB which passes through the G.L.
and makes equal angles with its H and V traces.
232. In plane S(- 2^", - 90°, + 45°) find line CD which passes through the G.L.
and makes equal angles with its H and V traces.
233. In plane T(- 3", + 120°", - 45°) find line EF which passes through the G.L.
and makes equal angles with its H and V traces.
Lines parallel to and at given distance from trace. [8]
234. Plane T(- 3^', + 22r, - 45°), find line par. to the H trace and V' there-
from.
235. Plane S(- 2", + 90°, + 135°), find Hne par. to the V trace and 1" therefrom
236. Plane R(- 3'', - 120°, - 45°), find line par. to the V trace and 1^' there-
from.
14
One projection given, to find the other. [8]
237. The point 0{-2",-V',z) is a point in the plane T(- 3^", + 45°, - 30°).
Find its unknown projection.
238. The point M(-2^ + l", z) is a point in the plane R(- 1'', + 90°, + 135°).
Find its unknown projection.
239. The point 0(-2^'', y, - f') is a point in the plane S( oo, - 1", - 1|"). Find
its unknown projection.
240. The triangle A(- 2", - V', z) B(- U", - If', z) C(- 1", - i", z) lies in the
plane R(- 3f', + 45°, - 30°). Find its V projection.
241. The triangle M(-2^ + l",z) N(-l^ + li^z) 0(-3^ + r',z) lies in the
plane S(- 1'', - 90°, + 150°). Find its V projection.
242. A parallelogram in the plane S( oo, - 1", - 1|'') has three vertices
A(- If', y, - V') B(- 3", y, - i") and C(- 3f ', y, - 1''). Find its projec-
tions.
243. A parallelogram in the plane T(- 1", + 150°, - 135°) has 3 vertices
M(-3^-f',z) N(-3f",-r',z) and 0(- 2", - |", z). Find its projec-
tions.
APPLICATIONS"
AND LOCATION OF GIVEN FIGURES IN GIVEN PLANES.
244. [4] Plane T(- 6'',-45°, + 30°). Find the projections of a 1" square,
lying in the plane above H, two of its sides par. to the H trace and when center
is at a pt. which is ^" from the H trace and 2" from the point of intersection of
the traces at the ground line.
245. [4] Plane S(- 3'', + 157|°, - 90°) is one side of a portable sheet iron cot-
tage. Find the projections of a square window, whose center is located IV' be-
low H and 2" from the vertical trace of the plane. The window is If square,
to the scale of the drawing.
246. [4] In plane T(- 3'', + 120°, - 135°), find the projections of a circle, of
IV diameter, lying in the plane below H and behind V, whose center is horizon-
tally projected at (- 4^, - 1'^ 0'').
247. [4] The inclined plane S(- 6", + 90°, -30°) is the top of a sheet iron
vat. A conveyor tube enters the vat perpendicular to the plane S. The section
of the pipe is a circle of 2" diameter, and the center line of this tube pierces the
plane at a pt. 1'' below H and If from the V trace, to the scale of the drawing.
Find the projections of the hole to be cut from the plane to admit the conveyor.
248. [4] The plane T(- 6", + 45°, - 30°) is the north roof of an observation
tower on a summer residence. A hole is cut in this roof for a fire-place chimney
which is to be built in the form of a regular hexagonal prism. The horizontal
projection of the hole is therefore a regular hexagon, and in the particular draw-
ing, each side of this hexagon measures 1", with its center at a point
0(-3", -]f , z) in the roof plane, and with 2 sides parallel to the ground line.
Find the V projection and the actual size of the hole.
249. [4] A hole 3 ft. square is to be cut in the roof plane U of Fig. 19 for a
vertical chimney. The hole is to have 2 of its sides parallel to EF and its center
located at a point 4 feet from EF and FB. Find the projections of this hole.
250. [2] A circular steam pipe of 2'' diameter passes through the roof
T(- 3f , + 135°, - 120°) of a factory extension, with its center line horizontal
and perpendicular to V, through a point 0(- 6", y", - If ) in the roof plane.
Find the 2 projections of the hole cut in the roof.
251. [2] Ornaments upon the glass sides A, B, etc. of the lamp shade in Fig,
21, are 3'' squares and radiating diagonals, located symmetrically as shown. Find
the projections of the ornaments.
^5
6.— TO FIND THE LINE OF INTERSECTION, AB, OF TWO GIVEN
PLANES.
General. [4]
253. Planes T(- 6^, + 45°, - 30°) and U(- 1'', f 150°, - 120°).
253. Planes R(- 6'', + 67^, ^- 120°) and S(- 3", + 135°, + 60°).
254. Planes P(- 6^-45°, + 67i°) and Q(- 3^ + 60°, - 45°).
Including planes perpendicular to H or V. [4]
255. Planes W(- 5^^-45°, -I 90°) and Q(- 2", + 135°, + 150°).
256. Planes R(- 6'', + 90°, - 30°) and S(- 3^", + 135°, + 45°).
257. Planes T(- 5", -90°, -30°) and V(- 3", + 135°, + 90°).
Including planes parallel to ground line. [4]
258. Planes W(- 6", -45°, + 45°) and Q( oo, + 2", + 1")-
259. Planes R(x, 00, + ly') and S(- 6^ + 30°, + 67i°).
260. Planes T( 00, + ir',-1") and U( oo, - IJ'', - y').
Including planes perpendicular to ground line. [4] P at - 3'\
261. Planes R(-3'', + 30°,-22i°) and P.
262. Planes S(- 2'', - 150°, -45°) and P.
263. Planes T(- 3'', + 90°, - 30°) and plane U par. to P and 1" to left of P.
Traces not intersecting within limits of drawing. [4]
264. Planes W(-6", + 67|°,-30°) and Q(- 2^ + 112^°, - 120°).
265. Planes R(- 4'', -90°, + 30°) and S(- 5^^-90°, + 67^°).
266. Planes T( 00,-1", + 2'') and U( 00, -1^-2'').
APPLICATIONS.
267. [2] In Fig. 20, find the traces of the planes A and B and then find their
line of intersection. (It should check with the line MO.)
268. [2] In Fig. 20, find the traces of the planes A and C and then find their
line of intersection, XY.
269. [4] In the cottage of Fig. 19, find the traces of the planes R and V and
then find their intersection. (It should check with the line MO.)
270. [2] In Fig. 29, find the intersection of the planes R and X.
271. [2] In Fig. 29, determine the lines of intersection 7-8 and 9-8 between the
roof planes T, U and S.
272. [8] In Fig. 32, find the projections of the edges of the banks, that is, the
intersections of the planes N and S with the horizontal planes of the tow-path
and the north bank.
273. [8] In Fig. 31, find the water lines on the banks, that is the intersections
of the banks N and S with the plane of the water.
274. [4] In the skew bridge of Fig. 28, find the intersection of the plane of the
portal ABCD and the plane determined by the diagonal members CN and CK.
275. [2] Fig. 34 shows in outline two dormers on the roof of a country hotel.
Find the valley lines of the dormers (that is, the lines of intersection between
the dormer roof planes and the planes of the roof).
276. [2] A vertical partition wall is to be built in the top story of the gymna-
sium in Fig. 29, through the point M and parallel to the hip rafter 11-14. Find
the lines in which the partition wall will meet the roof planes T, X, U, S and Q.
277. [2] A concrete dock wall has cross-section and dimensions as shown in
Fig. 35. A retaining wall meets it at an angle, as shown. Find all the intersec-
tion lines of the planes of these two walls, completing all views.
i6
378. [4] The plane T(- 3'', - 22|°, + 30°) is the top surface of a vein of coal,
to which an incHned conveyor shaft is being run. The floor of this shaft is de-
termined by the two Hnes A(- 4^^ - If", - l^'O B(- 7'', - r, -i'') and
C(- 4^'', - Y', - 3i'0 D(- 7'^ + V, - lY'). Find the hne XY in which the shaft
plane will cut the coal vein.
279. [3] The five planes Q( oo, + ^", oo ), R(- 1", - 135°, + 112^°),
S(- 8", -45°, + 674°), T(- 3^'', -90°, + 103-1°) and U(- 5|'', - 90°, + 77^°) are
the faces of a bridge pier, such as shown in Fig. 44, whose base is in H and whose
top is shown 2V' above H. Find the projections of the edges of the pier.
380. [2] The plane T(- 6'', - 60°, + 45°) is a side plane of a conveyor hopper.
A wooden shute feeding into this hopper has one of its sides determined bv the
lines A(-U'',-1^-1J'') B(-4^-^^-2i") and a line through the point
C(-3f", -If', -f'O parallel to AB. Find the line in which the hopper plane
is cut by the shute plane.
381. [2] Construct the projections of a regular square pyramid (whose base is
2" square, altitude 2") standing in the 1st quad, on H. Assume a convenient
oblique plane, and find the intersection of the plane with the pyramid.
7.— TO FIND THE POINT P IN WHICH A GIVEN RIGHT LINE
PIERCES A GIVEN PLANE.
General. [4]
283. Find pt. where line A(- 5", + |", 0") B(- 3|", + 2'', + 1'') pierces plane
T(-2i'',-135°, + 150°).
283. Find pt. where line C(- 5", + V\ + l") D(- 3", + ^", + 1'') pierces plane
S(-5^ + 120°, + 45°).
284. Find pt. where line E(- 6", - 1", - i") F(- 4", - 1", - 2") pierces plane
U(- 3", + 1121'', -150°).
Plane parallel to the ground line. [4]
285. Find pt. where line A(- 6", - 2", - 1-i") B(- 3", - 4", + i") pierces plane
T(x,-ir, ^).
286. Find pt. where line C(- 6", - |", - 2") D(- 4", - i", - ^") pierces plane
S(oo,-l",-ir).
287. Find pt. where E(- 6", + 1|", - 1^") F(- 5", + 1", - ^") pierces plane
u(<^,-ir,+r).
Plane perpendicular to H or V or both. [4]
288. Find pt. where line A(- 4", + |", 0") B(- 2^", + 2", + IV') pierces plane
T(-6",-30°,-90°).
289. Find pt. where line C (- 6", 0", + 1") D (- 3", - H", - 1^") pierces plane
S (-3|", + 90°, + 45°).
290. P at -3". Find pt. where line E(- 3", + 1", - ^") F(- 1", - ^", - 1")
pierces P.
Plane given by 2 right lines in it. [4]
291. Find the pt. P where the line M(- 5^", + 1^", 0") N(-4", 0", + H")
pierces the plane determined by the intersecting lines A(- 5^", + 2", + 1^")
B(-4i", + l^", f 11") and BC(- Sr', + |", + ^"). Solve without finding
the traces of the plane.
292. Find the pt. P where the line M(- 5^", - H", - If") N(- 3", 0", - i")
pierces the plane determined by the intersecting lines A(- 6", - 1", + ^")
B(-3",-l",-l") and BC(- 6", + 2", - 1"). Solve without finding the
traces of the plane.
17
293. Find the pt. P where the line M(- 6", - 1'', -|'0 N(- 34", + 2'', + i")
pierces the plane determined by line A(- 6'', - I", - ^") B(- 4'', - 1", - V)
and a line through C(- -4", + -i", - 2") parallel to AB. Solve without find-
ing the traces of the plane.
APPLICATIONS.^
294. [4] In Fig. IS a mill building is shown. It is proposed to run a telephone
wire A(-'64',-2',-10') B(- 28', - 22', - 10') to an office in the second floor.
Find the point X where this wire would go through the roof.
295. [4] In the mill building of Fig. 18, a brace is to be run to the roof from
a point on a heavy beam in the second floor. If the center line of the brace is
B(-32',- lG',-20') R(-10',-24',-2') find its ends in the roof and floor.
296. [2] In Fig. 34, find the valley lines of the dormers by finding the points
where the several hip lines pierce the roof planes, without finding the traces of
the dormer roof planes.
297. [2] Find the intersections of the faces of the 2 concrete walls of Fig. 35,
by finding where the edges of the lower wall pierce the slope plane of the higher
wall.
298. [2] In Fig. 30, find where the center line PW of the telephone wire, if
extended, would pierce horizontal planes through the bottom floor, top floor and
roof of the factory building, also all vertical wall planes of the wing and main
part.
299. [4] A sloping floor in a laboratory is given by its wall intersections
A(-6i", + r, + r) B(-3i", + f", + lf") and BC(- 3^, + ir> + T )• A steam
pipe with center line M(- 5^", -h If', + 1'') N(- 2f", 0", h- 1") passes through this
floor at a point X. Find X.
300. [4] The plane S(- GV', + 90°, - 30°) is the top surface of a coal vein,
to which an inclined bore-hole was driven in the line A(- 6", - 2^", - 1^")
B(- 3", -^", - :^") from the point A on the ground. Find the point X where
the hole strikes the vein, and the length of the hole.
301. [4] The plane S(- 6J-", - 30°, + 30°) is the top surface of a coal vein,
to which a vertical bore-hole was driven, from the point A(- 4", + |", + 2|").
Find the point where the drill strikes the coal, and the length of the hole, if scale
is ]" = l'-0".
302. [2] Scale, 1" = 10'. Vertical borings at three points A(- 67', + 22', + 23')
B(-42', + 6', + 20') and C(- 26', + 15', + 12') of a hiflside show an ore vein at
depths of 20', 5' and 11' respectively. It is proposed to run a shaft to this vein
with center line M(- 20', + 30', + 27') N(- 45', + 10', 0'). How long wfll this
shaft be?
303. [2] Construct the projections of a regular square pyramid (base 2"
square, altitude 3") standing on H in the first quadrant. Assume an oblique plane
and find the intersection of this plane with the pyramid. (Hint: Join the points
where the edges of the pyramid pierce the given plane.)
304. [2] Construct the projections of a 3" cube in the 3rd quadrant with its
upper base in H. Assume a convenient oblique plane and find the intersection of
this plane with the cube. (See hint in problem above.)
305. [4] The plane M(- 6^', - 30°, + 45°) is a mirror from which a ray of
light L(-5^", + l", + 2f") I(-3", + l", + i") is reflected at a point R. Find R.
306. [2] Two vertical poles are located on a hillside S(- 112', - 30°, + 45°) in
such position that their center line if extended would pass through the points
A(- 76', + 18', + 24') and B(- 32', ^-48',4 28') respectively. Find the projections
and length of a wire joining insulators on tops of these poles which are 20' above
the points where the poles enter the ground. Scale, 1" = 16'.
i8
8.— THROUGH A GIVEN POINT TO DRAW A LINE PERPENDICU-
LAR TO A GIVEN PLANE AND TO FIND THE DISTANCE
FROM THE POINT TO THE PLANE.
General. [4]
307. Point A(-3^-f',-rO and plane R(- 6", + 30°, - 45°).
308. Point B(-3^ 0", + i'O.and plane S(- 5", + 120°, + 45°).
309. Point C(-3'',-l'', -2'') and plane T(- 3'', + 120°, -150°).
Plane parallel to the ground line. [4] P at - 3''.
310. Point A(- V, - Y', - V) and plane R( x, - V, - 1^").
311. Point B(-l^-r',-rO and plane S( 00, - 2'', + ^'')-
312. Point C(-r%-i",-n") and plane T(oo,-r', 00).
Planes perpendicular to H or V or both. [4]
313. Point A(-5", + Y', + lV') and plane R(- 6", + 90°, + 30°).
314. Point B(-5'',-^",-l'') and plane S(- 3", + 90°, - 135°).
315. Point D(- 1", -V' , - l-^'O and profile plane P, at - 3'^
THROUGH A GIVEN POINT TO PASS A PLANE; PERPENDICULAR TO A GIVEN
PLANE. [4]
316. Point Ai-^'',-lY',-V') and plane R(- 6", + 30°, - 30°).
317. Point B(-2'', -! 1'',+ U'0 and plane S(- 3", - 120°, - 67^°).
318. Point C(-4^+U", + i-") and plane T(-6^ + 90°, -30°).
THROUGH A GIVEN LINE IN A GIVEN PLANE TO CONSTRUCT A PLANE T PERPEN-
DICULAR TO THE GIVEN PLANE. [4]
319. Line AB in plane S(- 6'', - 60°, + 30°), which pierces H \\" in front of
V and V l^' above H.
320. Line BC in" plane R(- 3", + 135°, - 150°), which pierces V %" below H and
is parallel to the H trace of plane R.
321. Line CD in plane U(- 6", + 45°, - 45°) which is parallel to the V trace and
y distant therefrom.
APPLICATIONS.
322. [4] A brace is to be run from an assumed point M in the floor S of
Fig. 18, perpendicular to the roof R. Find its center line projection and its length.
323. [4] In the skew bridge of Fig. 28 find the distance from the point N to
the portal plane, determined by the points ABDC.
324. [2] In the derrick of Fig. 27, find the distance of the end of the boom F
from the plane of the two guy wires DB and DC.
325. [2] In Fig. 27, find the distance of the top of the derrick mast D, from
the plane of the guv wire posts A, B and C.
326. [2] The 'points A(- 1^, + 3^', + r) B(- 7^', + 41", + ^D and
C(- 5V', + 1'', + 3f") are in the roof of a mill building. The force due
to the wind is normal to the roof plane. Draw through a point
M(- 3^", + 3f", + 44'') the projections of an arrow which will show the
direction of the wind pressure. Make the tip of the arrow just touch the
roof plane, and find the arrow length.
327. [4] The plane F(- 3'', - 135°, + 135°) is the plane of a fly wheel, whose
shaft is perpendicular to F. Assume the projections of the center line of the
shaft and find the projections of a point in the center line which is 2" from the
wheel plane F.
328. [4] The plane T(- 7'', - 22^°,'+ 60°) is one plane of the top of a grain bin.
A shute runs into this bin perpendicular to the plane T, and has a center-line
length of 2 feet. Assume its center line, and then find both ends thereof.
Scale, V = 1' - 0''.
19
339. [2] At three points A(- 40', + 3^', + 32') B(- 23', + 11', + 15') and
C(- 32^', + 24', + 14') vertical borings strike coal at 7', 5' and 12'. It is
proposed to run an inclined shaft perpendicular to the vein from a point
M(- 25', + 12', + 28'). i" = 1' - 0". Find the length of this shaft.
330. [2] Borings in a coal region show coal at the points A(- 25'. + 1^', + 3')
B(-84', + 9r, + 8') and C(- 14', + 3', + 14^'). Assuming that the surface of
the coal is a plane, find the length of the shortest shaft that can be driven from
the point M(-25', ! 9-i', ^-11') to the vein. Scale, ]" - I'-O".
331. [2] Vertical wells at points A(- 124', + 7', + 38') B(- 70', + 15', + 82')
and C(- 42', + 47', 4 61') on a mountain slope strike water at depths of 22', 10'
and 20' respectively. Assuming that these 3 points determine the plane of the
water, what is the direction and length of the shortest pipe that could be driven
from the point M(- 103', + 40', + 64') to strike water. Scale, 1" = 20'.
9.— TO PROJECT A GIVEN RIGHT LINE UPON A GIVEN PLANE.
General. [4]
333. Line A(-4r,-ir,-r) B(- 3", - f", - §''), plane R(- 6", + 30°, - 45°).
334. Line B(-5", + 2",+ l") C(- 3", 0", - i") and plane S(- 5", + 120°, + 45°).
335. Line C(- 3", - 1", - 2") D(- 5", - 1", - j"), plane T(- 3", + 120°, - 150°).
Plane parallel to ground line. [4] P at - 3".
336. A(- 3", - 4", - 2") B(- 1", - ]", -1") and plane R( oo, - 1", - 1^").
337. Line BC- 3", +-J", + U") C(- 1", - 1", - 1") and plane S( oo, - 2", + 1").
338. Line C(- 3", - U", - J") D(- 1", - i", - U") and plane T( oo, - 1", oo ).
Plane perpendicular to H or V or both. [4]
339. Line A(- 5", + i", + li") B(- 2^", + 2", + 1"), plane R(- 6", + 90°, + 30°).
340. .Line B(- 5'',--Y',- 1") C(- 3", - H", - 1"), plane S(- 3", + 90°, - 135°).
341. Line C(- 3", + i", + 2") D(- 1", - 1", - U") and profile plane. P at - 3".
TO PROJECT GIVExNT FIGURES UPON GIVEN PLANES.
342. [4] Find the projection ABC of the triangle M(- 6^", + 2", + If)
N(-5", + 2", + 2") 0(-5i", + f", + l") upon plane R(- 6", - 45°, + 45°).
343. [4] Find the projection ABC of the triangle M(- 5", - 1|", - 1")
N(- 3",- H",- 1") 0(- 5|",- yV U") upon plane S(- 3",+ 135°,- 135°).
APPLICATIONS.
344. [4] Asuming that rays of incident light are perpendicular to the mirror
plane M(- 7J", - 45°, J- 30°), find the reflection of a rectangular frame ABCD
of which A(-4",-^-f'^ + l") B(-4", + If", + 1") and BC(- 4", + If", 0") are
2 sides.
345. [41 The plane B(- 7", + 90°, - 30°) is the top plane of a refuse bin. A
triangular shute runs into the bin perpendicular to plane. If M(- 3", - 1^", - 1^"),
N(-3",-2", -i") and 0(- 3", - 1", - J") are respectively points in the 3 edges
of this shute, find the hole to be cut from the plane B.
346. [4] An engineer's draughtsman has assumed plane T(- 7^", + 30°, - 90°)
as his vertical plane of projection, upon which he has drawn the projection of a
triangle A(- 3f", - f", - i") B(- 3", - If", - 2") C(- 5-1", - 1", - U")- Find
the projections of his projection.
347. [4] The plane T(- 2", + 150°, - 150°) is a roof plane on a loading shed
for cars at a coal mine. A flat belt conveyor is to pass through this roof and per-
pendicular to it. If A(-5f",-l",-f") and B(- 6^", - 1^", - li"-) are points
20
in the two edges of the belt respectively, find the width of hole to be cut in the
roof to just allow the conveyor belt to pass through. Let V = 1'.
348. [8] An inclined mine shaft is to be run perpendicular to a coal vein
C('-7V;, + 90°,-60°). If the points M(- 3", -i'', - 2'') N(- 3'', - If', - 3")
0(-3'',-|",-^'') V{-^",-iy\-Y') are the corners of the shaft entrance,
find the true shape and size of the intersection of the shaft and the vein, if 1" rep-
resents 10',
10.— THROUGH A GIVEN POINT TO PASS A PLANE T, PERPEN-
DICULAR TO A GIVEN RIGHT LINE.
General. [4]
349. Point A(-4y',-H'',-l'0. Hne M (- 5|", - 2", - U'') N(- 4", - *", 0").
350. Point B(-4", -r',-r'), line 0(- 5", + 1'', - i") P(- 3", - U", - H").
351. Point C(-4",-^",-|") and line R(- 5", - 1^", + ^'') S(- 3", 0", + 2").
Line parallel to H or V or both. [4]
353. Point A (-4^ + 1", 4 1") and line M(- 6", - 2", - f ') N(-3^0,-f')-
353. Point B(-4", + l",+ r') and line 0(- 6'', - 2", + ^") P(- 3", - 2", - I'O.
354. Point C(- 4", - 1", - lY') and line R(- 6", + 2", + 1") S(- 3", + 2", + 1").
Line in plane perpendicular to ground line. [4] P at - 3".
355. Point A(-ir',-r', -r). line M(- 1", - ^", - i") N(- 1", - 1". - 1^")-
356. Point B(-3",-H",4l") and line 0(0", - 2'', - 1") P(0", 4-^'', + U")-
357. Point C(0'', - V, - U") and line R(0", - 1^", - ^'0 S(0", - \\", - 2'').
THROUGH A GIVKN POINT, TO CONSTRUCT A LINE PERPENDICULAR TO A
GIVEN LINE. [4]
358. Point M(-4J",-ir',--l"), line A(- 5^", - 2", - If ') EC- 4", - |", 0").
359. Point N(-4", + 1" , + 1") and line B(- 6", - 2'', - f ') C(- 3", 0", - V).
360. Point M(- W', - V, - V'), line C(- 4", - f ', - i") D(- 4", - 1", - If ')•
APPLICATIONS.
361. [4] The line A(- 3'', - f", - 2f') B(-7",-2r,-D is the center line
of a shaft, upon which a pulley is to be located at a point between A and B, and
H" from A. Find the plane of the pulley.
"362. [4] C(- 6'', + If, + ]") D(- 3", V If, + If) is the center line of a water
pipe which enters a tank at the point D. If the side of the tank is perpendicular
to the water pipe, find its traces.
363. [4] The line M(- Sf, - 2f , - 2f ) N(-5f, -1",-D is the center
line of a square conveyor-casing. Find the projections of the casing, whose sec-
tion is to the scale of the drawing, a If square with 2 sides parallel to the H
plane.
364. [4] A steam pipe E(- 3f , + f , + If ) F(- 7", + If , + 2f ) enters a
boiler, at the point E, perpendicular to its head B. Find the plane of B.
365. [2] A square 1" x 1" stick has the line A(-6", -f, -f )
B(-3'', - 4", - 3f ) as center line. Find the projections of the stick, if 3 of its
side faces are perpendicular to H.
366. [4] The plane R(- 5f , - 30°. + 30°) is a roof plane on a country hotel.
A tower is to be constructed upon this roof, with one side passing through the
line A(-6^-^2f , + lf ) B(- 4f , -h If , + If ) and perpendicular to the plane
R. Find the plane of this side.
367. [2] The plane of one face of a cube is determined by the edge
A(-6^-r',-2f ) B(-3f ,-3f ,-f ) and a point C(-4f , - 1", 0"). Find
the plane of the other face of the cube through the edge AB.
21
368. [4] A mine shaft, driven in line A (-41", + 1", + If") B(- 6", + 2", + |")
was found to be perpendicular to a coal seam C at the point B. Find the plane C.
369. [4] A telephone wire runs from a point A(- 6", + ly, + 2") in a pole
through a roof at a point R(- 3J-", + 1|", + f"). If the roof is perpendicular to
the wire, find the traces of its plane.
11.— TO PASS A PLANE, T, THROUGH A GIVEN POINT PARALLEL
TO 2 GIVEN RIGHT LINES. [4]
370. Point M(-4V',-r\-V') and lines A(- 7", 0", - 1") B(- 6", - 1^", 0")
and C(-3",-H", 0") b(- 1", - i", + 1").
371. Point N(- 4", -'y\ - 1") and lines D(- 5", - f", - 1") E(- 4", - U", - V)
and F(-3", + l",--l") G(- 2", + U", + 1").
372. Point P(-5", + l", + l"), lines G(- 6", -1", - IJ") H(- 5", - U", - U")
and K(-5",4--i", + r) L(- 2", + i", + ^").
To PASS A PI,ANF; THROUGH ONE LINE PARALLEL TO ANOTHER. [4]
373. Through line A(- 5", 0", + 1") B(- 4", - U/', 0") par. to C(- 3", - 1|", 0")
D(- 1", -V', + 1").
374. Through line C(- 5", -|", - 1") D(- 4", - 1|", -1") parallel to line
E(- 3", + 1", - i") F(- 2", + ir, + 1").
375. Through line E(- 6", - J", - U") F(- 5", - U", - U") parallel to
G(- 5", + i", + 4") H (- 2", + 1", i ^") .
TO PASS A PLANE THROUGH A GIVEN POINT PARALLEL TO A GIVEN PLANE, AND FIND
DISTANCES BETWEEN THE TWO PLANES. [4]
376. Pass plane through point A(- 6", - 1", - ^") parallel to plane
R(-3", + 112-r,-150°).
377. Pass plane through point B(- 5", - ly, + 1^") parallel to plane
S(-5i", + 90°, + 45°).
378. Pass plane through point C(- 6", - i", - 2") parallel to plane
T(x,-1",-U").
APPLICATIONS.
379. [4] The Hne A(- 35', + 23', + 5') B(- 25', + 10', + 17') and the line
BC(- 17', + 26', + 17') are the ridge line and valley line of a cottage roof. The
point M(-45', + 20', + 17') is a point in the ridge line of a parallel roof T on an
adjacent cottage. Find the trace of the plane T. Scale, 1" = 10'.
380. [4] A new building was designed for a lithograph company with a saw-
toothed roof, one roof plane being R(- 7", - 90°, + 30°). The points
M(-4", + U", H-4") and N(- 21", + l-i", + i") are points in the ridges of two
other roof planes parallel to R. Find these planes.
381. [2] The line A(- 80', - 30', - 17') B(- 20', - 88', - 82') is the outcrop of
a bed of gypsum on a hillside and a vertical bore-hole at point C(- 55', - 83', - 15')
showed the same bed at a depth of 20 feet. At a depth of 60 feet at the point C
another bed was struck which is surmised to be parallel to the first. Find this
plane. Scale, 1" = 20'.
382. [2] The hne A(- 90', - 42', - 21') B(- 54', - 24', - 5') and the line
C(- 90', -24', -12') D(-54',-6', + 4') determine the inclined roof of a tunnel
into a mountain. The point M(- 48', - 17', - 10') is in the floor of the tunnel,
which is parallel to the roof. Find the floor plane. Scale, 1" = 12'.
383. [2] The line A(- 7r, ^- 3^", + 1") B(- 5^", + 1", + 2f") and the line
BC(-3|:", + 3|", + ^") are two lines in a gable roof plane. The point
M(-3", + lf", + 1V') is a point in another gable plane T parallel to the first.
Find the traces of T.
22
12.— TO FIND THE DISTANCE FROM A GIVEN POINT TO A GIVEN
RIGHT LINE.
General. [4]
384. Point P(-4r,-U^-l'0, Hne A(- 5^'', - 2'', - H'O B(- 4'', - i", 0").
38.5. Point M(- 4", - V, - V') and line C(- 5'', + 1'' - i'') D(- 3" - 1^'- IV).
386. Point N(-r\-V\-V) and line K(- 5'',- 1^% + ^') F(- 3'', 0", + 3").
When line is parallel to H or V or both. [4]
387. Point P(-4", + 1", + 1") and line A(- 6^,-2'',- ^'') B(- 3", 0", -^'0-
388. Point M(-4", + r', + -r') and line C(- 6", - 2", 4-^) D(- 3'', - 2", - l").
389. Point N(- 4", - 1'', - iVO, line E(- 6", - 2'', -^ 1'') F(- 3^", + 2", + I'O-
Where line is in plane perpendicular to G. L. [4] P at - 3''.
390. Point P(-ir,-l^-i") andline A (- IV r,-D B(- 1", - 1", - 1|")-
391. Point M(-2",-ir, + l") and line C(0",-2",-r) D(0^ + 1", + 1^").
392. Point N(0", - 1", - 1^') and line E(0", - 1V',-D F(0'', - 1^'', - 2").
TO FIND THE PROJECTIONS OE A LINE THROUGH A GIVEN POINT PERPENDICULAR TO A
GIVEN LINE. [4]
393. Point P(-4",-l",-r), ^ne A( - 5", + 1", - T) B(- 3", - H". - ID-
394. Point M(-4", + l", + r') and line C(-6",-2", + D D(- 3", - 2", - l'')-
395. Point N(-r/',-U", + l"), line E(- 4", - 2", - 1") F(- 4", + ^", + IJ")-
TO FIND THE DISTANCE BETWEEN TWO PARALLEL LINES. [4]
396. Line A(- 6", - 1", - I'M B(- 3", + V', 0") and line parallel to AB through
point C(-4",-r, + r0.
397. Lines through points D(- 6", - 1", + 1") and E(- 4", + 1^'', + fO, parallel
to H and making 45 '^ with Y.
398. Lines parallel to the ground line, through points G(- 5", - J", - 1^") and
H(-4",-i-,-r).
APPLICATIONS.
399. [4] The Hne A(- 600', + 50', + 25') B(- 340', + 200', + 190') is a tele-
phone wire running from a valley town to a camp on the side of a mountain. A
house on the mountain at the point H(- 280', + 70', + 150') is to be connected to
the line AB by the shortest possible wire. Find the projections and length of the
wire. Scale, 1" = 100'.
400. [4] For supporting a boom derrick a wire cable is to be run from a point
M(- 130', + 100', + 90') on its mast to a point X in a beam A(- 260', + 100', + 55')
B(- 160', + 15', + 55'). Find the projections and length of the shortest possible
cable that can be used. Scale, 1" = 40'.
401. [4] In the cottage roof of Fig. 19, find the distance from the point N to
the hip rafter AC.
402. [4] In the cottage roof of Fig. 19, find the distance from the point C to
the hip rafter AE.
403. [2] In the ventilator cap of Fig. 20, find the distance from the corner O
to the lines KM and MN.
404. [2] In the derrick of Fig. 27, find the distance from the mast top D to
the boom, also show the projections of a wire running from the post C perpendic-
ular to the guy-wire DB.
23
405. [2] In the skew bridge of Fig. 28, find the distance from the point M to
the diagonal DL ; also from the point M to the portal post AC, and from the
point C to the portal post BD. Use scale, 1'' = 10'.
406. [4] The line A(- 5', - 2', - i') B(- 3-^, - i', - U') is the center line of
a gas pipe. A pipe for a light at the point L(- H', - f ', - If) is to connect with
the first pipe by a right angled elbow. Find the length of the latter, making no
allowance for the elbows. Scale, 1" = 1'.
407. [4] A soldier at target practice fired in the line A(- 6", + 2", + 1")
B(-3",-f 1'', + !"). By how much did he miss the center 0(- Sf, + 1", + y')
of a target?
408. [4] A mountain railroad takes the direction M(- 1|", + ^", 0")
N(- 7", + 2Y', + lY'). A hut on the same slope at the point H(- 5^", + f", + Y')
is how far from the railroad ? I<et V = ^ mile.
409. [4] A sewer pipe has the line B(- 37', + 18', + 7') C(- 25', + 3', + 4') as
a center line. What is the shortest waste pipe that could be put in from a drain
at the point M(- 40', + 6', + 16") ? Scale, ^" = I'-O".
410. [4] Theline A(-6", + 2", + r) B(- Sf', + T, + 3") is the center line of
a diagonal member of a roof truss, to be connected to point M(- 3|", + 1^", + 2")
by a member perpendicular to the first. Find its length.
411. [4] The line A(- 93', + 45', + 15') B(- GO', + 10', + 15') is the center
line of a mine shaft. It is proposed to connect with this shaft by a tunnel from a
point M(- 100', + 15', + 5') in another shaft. Find the length of the shortest pos-
sible tunnel. Scale, 1" = 20'.
412. [4] A stroke of lightning was estimated to have taken the line
A(-60', + 3', + 25') B(-35', + 16', + 2'). By how much was an ornament
0(-42', + 5',-^20') on a church spire missed? Scale, 1" = 10'.
13.— TO FIND THE ANGLE WHICH A GIVEN RIGHT LINE MAKES
WITH A GIVEN PLANE.
General. [4]
413. Line A(- 5", - V', 0") B(- 3", - 1^", - 2"), plane R(- 2", + 150°, - 150°).
414. Line C(- 6",- l",-!") D(- 4",- l",-2"), plane S(- 3",+ 112^°,- 150°).
415. Line E(- 4^",+ U" - 1") F(- 3", 4 i", + 1"), plane T(- 6",- 30°, - 60°).
When plane is perpendicular to H or V or both. [4]
416. Line A(- 4", + 4", 0") B(- 2^", + 2", + 1^"), plane R(- 6", + 30°, - 90°).
417. Line C(- 6", 0", + 1") D(- 3", - H", - H"), plane S(- 3|", + 90°, + 45°).
418. P at - 3". Line E(- 3", + 1", - i") F(- 1", - |", - 1") and plane P.
When plane is parallel to ground line. [4] P at - 3".
419. Line A(- 3", - i", - 2") B(- 1",- i", -i") and plane R( oo, - 1", - 2").
420. Line C(- 3", + If, - 1|") D(- 2", + 1", - J"), plane S( oo, - H", - f ).
421. Line E(-3",-2",-U") F(-f,-i", + f) and plane T( oo, - If, oo).
APPLICATIONS.
422. [2] P at - 3". An inclined guide pulley shaft has center line
A(-2',-2f',-3J:') B(-4',-^',-l^'). What angle does it make with the floor,
(H plane) the front wall (the V plane) and with a roof plane R( co, - 2f', - If).
Scale, 1" = 1'.
24
123. [4] Rays of light in the direction A(- 6'', + Y', + f") B(- 3i'',+ f",+ 2^")
are reflected from a mirror plane M(- IfJ", + 45°, - 30°). What is the angle of
incidence ?
424. [4] A vertical telephone pole stands on a hillside S(- 6^", - 60°, + 45°).
What angle does the pole make with the plane of the hill?
425. [2] The line A(- Gf, + 2^, - i') B(- 1-^', + 4', + 2f') is the center line
of a tie rod in a roof truss. What angle does the tie rod make with the floor
plane (H plane) and what angle with the roof plane determined by the members
M(-7^',f5', + lf') N(-4r, + ir, + 2f') and NP(- 1|', + 2f , - 1^')- Scale,
1" = 1' 0".
42f). [2] In the roof of Fig. 19. find the angle which the valley rafter MN
makes with the roof plane U.
427. [2] In the roof of Fig. 19, find the angle which the ridge AB makes with
the roof plane T.
428. [2] In the skew bridge of Fig. 28, find the angle that the portal post DB
makes with the plane of the diagonals DL and DM ; also the angle which the por-
tal brace EI makes with the plane of CD and CN. Use scale, 1" = 10'.
429. [2] In the roof of Fig. 29, find (1) the angle which the ridge 8-14 makes
with the roof plane S, (2) the angle which the same ridge makes with the plane
X. Use scale, Y' = I'-O''.
430. [4] In Fig. 31, find the angle which the rope AB makes with the south
bank plane S.
431. [2] Find the angle which the telegraph wire PW of Fig. 30 makes with
the walls A E F B, F B C G, the left end wall of the factory, and the roof.
432. [4] What angle does the tow-rope of Fig. 32 make with the river bank S?
433. [4] What angle does the tow-rope of Fig. 32 make with the floor of the
canal boat?
434. [2] In the reducer of Fig. 24, what angle does the edge between planes
A and B make with the plane C? Use scale, 1" = V -0".
435. [2] Fig. 27 shows a Boom Derrick. Find (1) the angle between the
boom FE and the plane determined by the guy wires BD and CD, (2) the angle
between the guy-rope BD and the plane determined by the points A, B and C.
14.— TO FIND THE ANGLE BETWEEN TWO GIVEN PLANES.
General. [4]
43G. Planes P(-6", -45°. + G7^°) and Q(-3", + 60°,-45°).
437. Planes R(- 6'', + 67^°, f 120°) and S(- 3", + 135°, + 60°).
438. Planes T(-6'', + 67r,-30°) and U(- 2", + 112|°, - 120°).
Including planes perpendicular to H or V, or both. [4]
439. Planes Pf- 6", 4- 90°, -30°) and Q(- 3^", + 135°, -135°).
440. Planes R(- 5". H 90°, -30°) and S(- 3", + 135°, + 90°).
441. Planes T(-r/', + 30°,- ■?2^°) and U(- 3'', - 90°, + 90°).
Including planes parallel to ground line. [4]
442. Plane P(- 6", -45°, + 45°) and Q( 00, + 2",^ 1").
443. Plane R( x, - 1^", - 2") and S( oo, + 1^", - 1").
444. Plane T(- 6'', + 45°-, -30°) and U( oo, - 1'', co).
When one of planes is a plane of projection. [4]
445. Plane R(- 6", + 45°, -30°) and H.
446. Plane S(-6", + 67r,-60°) and V.
447. Plane TC- 2", + 90°, -30°) and P at -4".
25
ONlv TRACE OF A PLANE BEING GIVEN, AND THE ANGLE WHICH THIS PLANE MAKES
WITH A PLANE or PROJECTION, TO EIND THE OTHER TRACE. [4]
4-18. The H trace of a plane R is given as (-6'', + 30°). The plane makes an
angle of 15° with H. Find the Y trace.
449. The V trace of a plane S is given as (-(>", + 45°). The plane makes an
angle of 60° with H. Find the H trace.
450. The H trace of a plane T is given as ( 20, - 1|")- The plane makes an an-
gle of 30° with H. Find the V trace.
b'
APPLICATIONS.
451. [8] In the cottage of Fig. 19, find the bevel angle of the hip rafter AC,
that is, the angle between the planes T and R. Also find the valley angle between
the planes R and V.
452. [2] In the cottage of Fig. 19, find the bevel angle of the hip rafter AH,
that is, the angle between the roof planes T and U. Also find the bevel angle for
the ridge rafter AB.
453. [2] In the skew bridge of Fig. 29, find the angle between the portal plane
AP)DC and the bridge floor. Also the angle between the portal plane and the
plane of one of the trusses of the bridge. Use scale, 1" = 10'.
454. [2] In the sheet metal reducer shown in Fig. 24, find the plane angles for
corner angle-irons, that is, the angles between the planes A and B, B and C, etc.
Use scale, 1" = V - 0''.
455. [2] In the house of Fig. 29 find the bevel angle for the hip rafter 1-5, that
is, the angle between the planes R and S. Also find the valley angle between
S and T, and the bevel angle for hip rafter between planes T and X, Use scale,
J/' = I'-O''.
456. [2] In the house of Fig. 29, find the bevel angle for the hip rafter 11-14,
that is, the angle between planes X and W. Find also the bevel angle for the
ridge rafter 5-6, and the vallev angle between planes S and T. Use scale,
Y' =r - 0".
457. [2] In the ventilator cap shown in Fig. 20, find the angles between the
planes D and B, between A and B, between A and the plane of the cap base,
between B and the plane of the cap base,
458. [2] In the gondola car body of Fig. 23, find the angles between plates
C and D, between A and C, between C and the door plate.
459. [4] The line A(- G'', + 1", + f") B(- 4^", + 2-|'', + 2^") and a line par-
allel to AB through the point C(- 4f", + J", + 1:J") are the back edges of a chan-
nel iron connected by a bent plate to a plane T(- 3if' , - 150°, + 120°). Find the
flare angle for the bent plate. See Fig. 41.
460. [4] A( -6", + U'',-f If") B(-4r, + lF', + r') and a line parallel to AB
through C(- 4f", + 2", + l:f") determine the web plane of an I beam which is
riveted by a bent plate connection to a horizontal bed plate. Find the flare angle
for the bent plate. See Fig. 40.
461. [2] The top edges of the slope planes of a dry dock form a rectangle
100' by 60'. These planes slope toward the center at an angle of 67^° with the
horizontal. Find the traces of the planes, their intersection with each other, and
their intersection with the bottom of the dry dock, 40' below the top rectangle.
Scale, 1" = 30'.
462. [2] The line A(- 70', + 35', 0') B(- 37', ^ 5', 0') is the H trace of a rail-
road embankment plane whose batir is 1 horizontal to 2 vertical, and whose ver-
tical height is 20 feet. Find the top line of the slope plane, the intersection of
26
this plane with a hillside H(- 48', - 90°, + 30°), and the angle between the em-
bankment plane and the hillside. Scale, 1'' = 10'.
463. [3] The line M(- 50', + 9', 0') N(- 17', + 40', 0') is the trace on the
ground of a wing wall slope plane whose batir is 1 horizontal to 2 vertical. The
top line of the wall is parallel to the ground trace and 25 feet above the ground.
Find this top line, the intersection of the wing wall plane with the embankment
plane T( cc. + 30', -h 15') and the angle between wing wall and embankment plane.
Scale, 1" = 10'.
464. [2] A tower roof is to be built in the same general shape and position as
shown in Fig. 20, for the ventilator cap. The planes D and B are to have a slope
of 1-^ vertical to 1 horizontal, while the planes C and A are to have a slope of 2
to 1. Find the projections of the roof and the angles between the planes C and
D, and between planes D and B.
15.— TO FIND THE SHORTEST DISTANCE, XY, BETWEEN TWO
RIGHT LINES NOT IN THE SAME PLANE.
General. [2]
465. Lines A(- 6", f 2", + 2") B(- 3", - 1^", + If ) and C(- 5", + If , + |")
D(-3",ff",-F2i").
466. Lines E(- 5J-", -" 2^', + 1") F(- 2f , + 1", - 2") and G(- 5^", - 3", 0")
H(- 2V, 0'', -^2^").
467. Lines 'l(- 6'', - f', - P/') J(- U/', -f V/', - D and K(- 3^, - U", - D
L,(_ 3^"^ _ 1''^ -1").
468. Lines "m(- 6'', -^2^,-1-1") N(- 3f , -!- -f, + 1") and 0(- 6", + f , + 2f )
P(- 3J.", + 2", + f ).
469. Lines Q(- :r, + 2", 1-41") R(- 5f , + 3|", + 3|"), S(-'2|", + 2i", + 2")
T(-r>+r,+'ir)-
Including lines parallel to the planes of projection. [2]
470. Lines A(- 51", - T, - D B(- 5^', - T, - U") and C(-4", + H,-r)
D(-3",-|",-l").
471. Lines E(- 6", - 2", - 2") F(- 5r, - T, " 21") and G(- 3^", - H", - D
H(-3r,-in-3").
472. Lines J(- 6^", - 2", - 2") K(- 2^", + 2^", + £") and L(- 4f", + If", + i")
M(-U", + i-2", ^4i").
APPLICATIONS.
473. [1] The line C(- 23', + 2', + 4') D(- 15', h- 7', + 4') is the center line of
a telegraph wire. It is proposed to cross this line with a high tension transmission
line whose center line is A(- 23', + 8', + 10') B(- 11', + 2', + 2'). If the safe dis-
tance between such wires is H feet, is this crossing safe or not, and by how
much? Scale, y' = l'-0".
474. [2] The center lines of two gas pipes which are to be joined by means
of one straight length of pipe and two right angled elbows are A(- 6', - 2', - 2')
B(-5i',-J',-2i') and M(- 3^', - U', - f) N(- 3^', - U', - 3'). Find the
shortest possible connection of this sort which can be made, allowing 1^" at each
end for the elbow. Scale, 1" = 1' - 0".
475. [2] The line L(- 9^', - 1', - 3^) M(- 3', - 9.', - 3^) is the center line of
the main shafting in a certain shop. It is proposed to run an electric light wire in
the direction J(- 13', + 4', 4 4') K(- 5', - 1|', - 4|'). Find the distance between
the two at their closest point. Scale, |" = I'-O".
27
GENERAL PROBLEMS BASED ON POINT, LINE AND PLANE,
WITH APPLICATIONS.
480. [2] A 2V^ cube stands upon H in the first quadrant, with one of its faces
making 22^° with V. Find (1) the H and V projections of the cube, (2) the
true length of a diagonal, (3) the distance from one corner to the plane of the
three adjacent corners.
481. [2] A horizontal trapeze bar 21 feet long is hung by vertical ropes 4^
feet long attached to its ends. How far would this bar be raised by turning it
through^)0° ? Use scale, !'' = !'- 0".
482. [1] Locate P at -54", and ground line parallel to shorter edges of sheet.
The point M(- f", -!-|", + 1") is revolved about the line A(+ |-", + 3^'', + 4i")
B(+3'', + 1", 0"). Find (1) the points where this point pierces H, V and P,
(2) the H, Y and P projections of the point M after 180° revolution from its
original position.
483. [1] P at - T". Scale, i'' = 1' - 0". Revolve the point A(- 5V, ~ If, - l^')
about the line E(- 41', - 5', - o|') C(- V, - V, -V) as an axis and find the points
where it pierces H, V and P in this revolution.
484. [1] P at -7V\ Scale, l" = l'-0''. Let H, V and P represent respec-
tively the ceiling and end and side partitons of a factory room. The point x\ on
the rim of a guide pulley revolves about the line BC which is the center line of
the pulley shaft. Find the points where this point A pierces H, V and P in the
revolution. Also find the projection of A after 30° revolution.
A(- 4'5", - l'(r, - 1'21") n(~ 3'1", - 4'0", - 4'24-") C(- O'Ci", - PO", - 0'5").
485. [1] (Ground line parallel to shorter edges and If below middle of sheet).
P at-5|". Scale, f = I'-O''. The three planes H, V and P represent respec-
tively the floor, and two partitions of a power house. The point X, 4'-0''' above the
floor, 4'-0" in front of V, and 2'- 5V' to the right of P, is a point on the circum-
ference of a guide-sheave for a wire rope drive. The center line of the axis of
the wheel pierces H 6'' in front of V and 4'-5'' to the right of P and pierces V
I'-O'' below H and 5'-10i" to the right of P. Find the points where the point X
cuts through the floor and partitions as it revolves about the axis. Also find the
projections of the point X when it has revolved through an angle of 60°.
48G. [1] Scale, J/' = I'-O". The line A(- 48', + 16', + 8') B(- 28', + 2'. + 8')
is the axis of the shaft of a fly wheel in a power station whose floor and wall are
represented by H and V. The point 0(- 30', h 11', + 18') is in the center line of
the rim of a fly wheel to be located upon the shaft. The width of the fly vv^heel
face parallel to the shaft axis is to be 2', that is, 1' on each side of O. Find
(1) the radius and projections of the proposed fly wheel, (2) the holes which
must be cut from the wall and floor to allow a clearance of 6" all round the fly
wheel for safety.
487. [2] Given the line M(- 6", + 1", + i") N(- 3", + 3", + 2") and the line
A(-7", + 2y', + 2i") B(-4", + 4",-U"). Find the projections of AB after it
has been revolved about MN as an axis through an angle of 75°.
488. [1] Pass a sphere through the four points B(- 10", + li", - 1")
D(-8f , + 3", + 2J:") G(-7i-",+ li", 0") and W(-7|", + r',-f").
489. [2] Circumscribe a sphere about the triangular pyramid whose vertices
are the points A(- 5", + 24", + 2") B(- 6§", + 3|", 0") C(- 3§", + 3f", 0") and
D(-4r,+ino").
490. [2] The following four points M(- 5", -f 24", + 2") N(- 6f". + 3|", 0")
O(-3|", + 3§",0") P(_4i", + li", 0") are the vertices of a tetahedron. Find
the inscribed sphere.
28 GENKRAl, PROBLEMS RASED ON POINT, UNE AND PLANE
491. [2] The points A(- 7", - 3", 0") B(-5",-r,0") C(- 2^'', - 2^^ O'O
and D(-4^'', - If", - 2|") are the vertices of a tetrahedron. Find the inscribed
sphere.
492. [2] Let H, V and T(- 2", - 120°, + 135°) represent respectively the floor,
the wall, and the roof plane in a corner of a garret. A perfectly elastic ball is
thrown in the direction of the line A(- 7^", + 2", + 2") B(- 5", + 2", + 2"),
strikes the roof plane T, bounds off and strikes the floor H, then bounds off of H
and strikes the wall Y. Find the 3 points X, Y and Z where it strikes these planes.
493. [2] The plane T(- 6|", - 45°, + 60°) and the planes of projection H and
V are three mirrors, from which a ray of light A(- 2^", + 2-|", + 2-')
B(- 4^", f 2|'', h-J") is successively reflected at the points X, Y and Z. Find
these points.
494. [2] P at - 2". Construct the H and V projections of a cube, in the first
quadrant, one of whose faces is in the plane F(- 3|", - 30°, + 60°) with the side
A(-2r,+r,z) B(-r,+ir,z).
495. [2] Find the H and V projections of a regular square pyramid whose ver-
tex is at the point 0(- 3", - 3^", - 2^") and whose 1|" square base is in the
plane T(- 2i", + 150°, - 120°).
496. [1] Find the projections of a regular hexagonal pyramid having its base
in a plane through A(- 10:^", + If, + 2-f ') and par. to lines B(- 12'', + ^'% + 4")
C(-6i", + r', + 3r) and D(-9r, + r. + 'ir) E(- 6^, + 4", + ^'0- One side
of the base lies in the line of intersection of the plane of the base and a plane
through A perpendicular to line BC. Vertex of the pvramid at point
0(-5", + 4i'', + 5^'')-
497. [2] Construct a line XY, parallel to the plane T(- 6^', - 30°, + 60°) and
1'' therefrom, which cuts the lines M(- 6'",+ 2f ,+ 2f") N(- 4^",+ 1", f |'') and
o(-4'',+ir,+2D p(-ir",+ir,+D-
498. [2] Given the points A(-6", + l", 0") B(-3a", + f, 0") and
C(-2f , + 3|'', 0") as the vertices in H of the base of a triangular pyramid,
whose vertex is at a point F, such that the lateral edges AF, BF, and CF are
4^', 4f and 4" in length respectively. Find the H and V projections of the
pyramid.
'499. [2] Scale, 1" = 50'. Vertical drill holes at points A(- 260', + 60', + 180')
B(- 310',+ 175',+ 110') and C(- 200', + 55', + 110') of a hillside strike an ore
vein at depths of 170', 210' and 70' respectively. Find the line of outcrop of the
vein, that is, the intersection of the ore plane with the plane of the hillside.
500. [2] Find a line XY which touches the three lines A(- 6", -i", - 2^")'
B(-5",-2",-i"),c(-4r,-3",-2r) D(-3r,-r,-r),andE(-3",-ir,-r)
F(-l",-3",-3").
501. [2] through the point 0(- 4|", - 1^", -1^") pass a line parallel to planes
T(-6i", + 30°,-50°) and S(- 1", + 115°, - 140°).
502. [4] Pat -I". The vertical line A(- 51", + li", + 2") B(- 5^", + li",0")
is a body diagonal of a cube in the first quadrant with its lowest corner on H at
the point B. Find the three projections of such a cube.
503. [2] Through the point M(- 6|", - 1", - lJ:")construct the line MXY
touching the lines P(- 6", - 2", -3") Q(- 5", - i", - ^") and R(- 4", - |", - 3")
S(-2",-3",-f").
504. [2] Through the point GC- 5f",- 1",- 1|") construct a line GXY touching
the lines M(- 4^", - 3", - 2^") N(- 3^", - i", - i") and 0(- 3", - 1^", -i")
P(-l", -3",-3").
505. [2] Through the line M(- 4f", - 2|", z) N(-2",-^", z) in the plane
R(-7i", + 45°,-30°) pass a plane T making an angle of 30° with R.
GENERAI, TROBLTvAIS BASED ON POINT, LINE AND PLANE 29
506. [2] Through the Hne 0(-6", y,-V') P(-2r', y, + H") in the plane
S(-tV, - 150°, + 130°) pass a plane U making an angle of 45° with S.
507. [2] Find line CD, parallel to A(-6r,-r,-r) B(- of', - T, - ^D
and intersecting the line M(- 6.^, - 2'', - 3") N(- 5i", -i", -^'0 and the
line P(- 4r, - r, - 2r) Q(- H'', " 3", -T )•
508. [2] Find a plane R which bisects the dihedral angle between planes
S(- 11'', -60°, + 30°) and T(- T-f, + -iO°, - 45°). .
509. [2] Find the point K in line A{- 1^', + !^, + H") ^(- H'', + V', + D
which is equidistant from planes S(- 7'', - 60°, + 30°) and T(-f'', + 45°, - 60"°).
510. [2]Through point C)(- 4'^ + 3", + 2|") pass a plane T making an angle
of 60° with H and 67-^° with V.
511. [1] Construct the projections of four spheres of radii 1^", 2", 2^" and 1"
respectively, so located that each sphere is tangent to the other three.
512. [2] Given two points 0(- H", - ^i", - D and K(- 2r, - i", - 2D
and a mirror plane M(- 7^", f 30°, - 45°). Find the projections of a ray of light
which emanates from O and after being reflected from M passes through the
point K.
513. [2] If the direction of rays of light is given parallel to the line
I/_6i", + 2'', + 2^") M(-4", + 1'', + |'0 find the shadow cast upon the hori-
zontal plane of projection by a certain triangle A(- 6|", + 4-|-", + f")
B(- 4'^ + 2r, + 3^0 C(- 4^', + 5r, + 1").
514. [2] A ray of light passes through the points A(-iy', + 4^", + 31") and
B(- 33/', + If', + 2") and is reflected from a mirror M(- |""+ 60°, - 45°). Find
the projections of the incident and reflected rays, and the angle of incidence.
515. [1] P at -8". The top of an inclined draughting table is a rectangle,
with three corners at the points A(- 7^", - 1", - 4f") B(- 7|^ - 5'', -3^")
C(-2'',-5", -3rO- An electric light at the point L(- 4f ', - 3f', - 1'') is
guaranteed to light satisfactorily at this distance within a cone of maximum
intensity rays, whose elements make an angle of 30° with the vertical. Find the
three projections of sixteen of these rays and then approximately the curvf
bounding that portion of the table within the cone of maximum intensity.
516. [2] Two pulleys are located with their pitch circles in the same plane
T(- 12', - 45'^, + 60°) and are 2' and 3' in diameter respectively. The face 01
each pulley is 12", their shafts are 6' apart, and they are connected by an open belt
9" wide. Find projections of pulleys and belt, and length of the latter, making
no allowance for slack. vScale, Y' = I'-O".
517. -[1] Scale, J" = I'-O''. Two pulleys of 3' diameter are located at points
A of shaft A(- 24'", - 6', - IV) B(- 24', - 6', - 6') and C of C(- 8', - IV, - 7^')
D(- 8', - 6', - 7|'). They are to be connected by a belt running over two guide
pulleys of 3'diameter, so located that the belt will be as short as possible and will
run in either direction. Find its length and center line projections, also the pitch
line projections of the guide pulleys. (Hint. A belt must always be delivered in
the plane of the pulley toward which it is running. .See Fig. 33.)
518. [1] G. L. par. to short edges of sheet. Scale, 4" = I'-O". Assuming the
point D as per dimensions in Fig. 33, find the projections of the main and guide
pulleys and belt, assuming all pulley faces to be 9", width of belt 6", shaft diame-
ters 4". (See hint in problem above.)
519. [1] G. L. par. to short edges of sheet. Scale, 4" = I'-O". In Fig. 33,
locate a guide pulley as indicated, but in such a position as to give the shortest
possible belt. Show only pitch line of guide pulley and of belt. Find length
of belt.
30 GENERAL PROBLEMS BASED ON POINT^ LINE AND PLANE
520. [1] Scale, |'' = I'-O". Find the projections of two pulleys, diameters 3'
and 4', to be located at points A and B respectively of shafts A(- 25', - 4^', - 3')
C(-32',-U',-l') and B(- 11^, - n', - 4') D(- 13^, - 5^', - i'), and to be
connected by a belt running over two guide pulleys of 2^' diameter, so located
that the belt will be the shortest possible and run in either direction. (Hint: A
belt must always be delivered in the plane of the pulley toward which it is run-
ning.)
521. [1] On a certain set of plans, the projections of a ventilator cap are as
shown in Fig. 20. Find (a) the angle between planes D and B, that is, the flare
angle for the ridge angle-iron, (b) the angle between planes A and B, that is, the
flare angle for the hip angle irons, (c) the angle which the ridge makes with the
plane A, (d) the angle which the planes D and C make with a horizontal plane
through the base of the cap, (e) the pattern (true size and shape) for the cap,
allowing 3" for lap where necessary.
522. [2] Fig. 21 represents in outline the projections of a lamp shade. Find
the following values, which would be used in its construction: (1) Angle be-
tween planes A and B, (2) angle which planes A and B make with a horizontal
plane through the base of the shade, (3) angle which planes A and B make with
a vertical plane through the edge FG, (4) angle which edges, as EF, make with
the top T, (5) angle which edge EF makes with the base edge FG, (6) size of
glass planes, allowing I'' inside of extreme edges as EF and FG.
523. [1] Scale, ^" = I'-O". The sketch in Fig. 22 is a steel tank for an elevator
boot, the plates to be bent and riveted as shown. Dimensions of open top are
5'_0'' by 3'-0", of bottom plate without laps 3'-0" by 18". Height of boot is
2'-0". Find (1) angle between side and end plates, (2) angle between side and
bottom plates, (3) angle between end and bottom plates, (4) angle which edge
between side and end makes with bottom plate, (5) dimensioned pattern for all
plates, allowing laps of 3" for riveting as shown.
524. [1] Scale, ^" = I'-O". The steel plate body of a Gondola car is shown
in Fig. 23. Show the 3 views of the car and find the plate dimensions, lengths
and angles, as follows :
Plate A= X Plate B= x
Plate C= X and Plate D = x . . . ; . . and
Door Plate = x Angle between A and C = ° '.
Angle bet. B and D = ° '. Angle bet. C and D = ° '.
Length of intersection between C and D =
525. [1] In the sheet metal reducer shown in Fig. 24, find, the following:
(a)angle between planes A and B, and between planes B and C, (b) angles be-
tween planes A and G, and B and F, (c) pattern for transition section of the
reducer, dimensioning same in full.
526. [1] Scale, ]|" = I'-O". The plan and elevation of a metal ash shute are
given in Fig. 25, the dimensions locating all lines except MN and NO which are
to be found. Lay out and dimension the top plate A, bottom plate C, and one
side plate B of the transition connection. Also find the following angles:
(1) Angle between A and B, (2) angle between F and E, (3) angle between B
and G.
527. [1] Scale, V = I'-O". A stone flat arch over a window is shown in Fig.
26. Find the angles between the various faces of the keystone K, and develop
this stone, showing dimensions in full.
528. [1] Scale, V = I'-O". In the flat window arch of Fig. 26, find the angle
between the various faces of stone E and develop this stone, showing dimensions
in full.
GENERAI, PROEI.EMS BASED ON POINT, LINE AND PLANE 3I
Note.— Figure 29 shows in plan and elevation, a hip and valley roof, whose solution
may be taken as typical of the work necessary in the design of structural steel roofs. The
following are the important angles: (i) the angles which planes as R, S, X, etc., make
with a horizontal plane, (2) the angles which the lines 2-r, lo-ii, etc., make with vertical
hip-web planes through the hips 1-5, 11-14, etc. respectively, (3) the angles which the hip
or valley rafters, as ic-14, 2-5, and 7-8 make with a horizontal plane, (4) the angles in a
roof plane which the main rafters 5-6 and 8-14, etc. make with the hip or valley rafters
1*5. 7-8, etc., respectively, (5) the angles between a vertical line and the trace of a purlin
web upon the hip web plane (the purlins have their web planes perpendicular to the roof
plane, as indicated in figure — the line AB then would be the trace of a purlin-web plane
on the roof plane R — the hip-web plane is a vertical plane through the hip), (6) the angle
between the roof plane and the back of hip, (7) angle in the purlin-web between a normal
to the center line of the purlin and the trace of the hip web on the purlin-web, (8) angle
between purlin-web and hip-web, (9) angle in back of hip between a line normal to the hip
web and the trace of the purlin-web on back of hip, (10) angle between traces of hip-web
and back of hip, (11) angle between back of hip and purlin-web.
529. [2] Scale, y = I'-O''. Find the above angles for the plane R in Fig. 29.
530. [2] Scale, Y' = I'-O". Find the above angles for the plane S in Fig. 29.
531. [2] Scale, Y' = I'-O''. Find the above angles for the plane T in Fig. 29.
532. [2] Scale, y = I'-O". Find the above angles for the plane X in Fig. 29.
533. [1] A carpenter's saw-horse is constructed as shown in Fig. 36. Find
the correct projections of two legs as shown, then detail one of the legs, showing
all angles which would be used in cutting. Use scale, 3" = I'-O''.
534. [1] Fig. 37 shows the center line of a 4" x 4" brace to be run between
two 5" X 5'' beams, so that its top and bottom faces are perpendicular to V.
Find the projections of this brace, showing its intersection lines with each of the
beams and then detailing said brace, showing all angles which would be used in
cutting it. Use scale, |" = 1".
535. [1] Fig. 37 shows the center line of a 4" x 4'' brace to be run between
two 5" X 5" beams, so that its two side faces are perpendicular to H. Find the
projections of this brace showing its intersection lines with each of the beams, and
then detailing said brace, showing all angles which would be used in cutting it.
Use scale, |" = V.
536. [2] Construct the projections of a jack rafter B as shown in Fig. 38,
showing in detail its intersections with the hip and valley rafters. Then find the
angles which would be used in marking these rafters for cutting.
537. [2] Construct the projections of a house rafter A as shown in Fig. 39,
showing in detail its intersection with the ridge rafter and the plate. Then find
the angles which would be used in marking these rafters for cutting.
538. ri] Construct in detail the 3 projections of the auditorium floor joist A
shown in Fig. 42, finding the angles which would be used in marking out ready
for cutting.
539. [1] Construct in detail the 3 projections of the Auditoirum floor joist B
shown in Fig. 43, then detail the piece showing all angles which would be used
in marking out ready for. cutting.
540. [2] Assuming the base lines in H for the face planes of a bridge pier as
indicated in Fig. 44 and the respective batirs for said faces, find the projections
of the pier.
541. [2] The line B(- 6'', + 2'', 0'^ A(-2Y', + ¥', + H'') and a point
M(- 3'', + 3'', -1- 2'') determine the plane of the back of a channel iron used on a
hoist boom, the channel dimensions to the scale of the drawing being W = ly ,
F= f, as in Fig. 41. Find (1) a point C in the opposite edge of the channel
back, (2) the projections of the intersection of the channel iron with a horizontal
base plate, (3) the angle for a bent plate connection between the channel back
and the base plate as shown. Then detail the bent plate, allowing y clearance
all around.
32 gf:ne;rai, problems based on point, line and plane
543. [2] The line B(- 3'', + 3^", 0") A(- 5", + 2f", + 3f' ') and a point
M(-2f , f 1'', + 3'') determine the plane of the web of an inclined I-beam con-
nection in a steel structure, the beam dimensions to the scale of the drawing be-
ing W- 1^-", F = I'', as in Fig. 40. Find (1) a point C in the opposite edge of
the back of the I-beam, (2) the projections of the I-beam, (3) its intersection
with a horizontal base plate. Then detail a bent plate connection for the same,
allowing ^" clearance all round, and find the angle at which it is to be bent.
543. [1] A square butt-jointed hopper has dimensions as shown in Fig. 71.
Find all the bevel angles necessary for its construction, and in a separate figure
show detailed views of one side of the hopper, giving angles and lengths,
544. [1] Fig. 95 shows a tin furnace pipe transition piece. Find the angles
which its various sides make with each other and with the sides of the vertical
pipes connected. Then develop the transition piece, dimensioning in full.
545. [2] Draw the projections of the cottage of Fig. 19, and, assuming that
rays of light are parallel to the line R(- 2'', + 1^", + 1^") L(- l", + F', + i"), find
the shadow which the cottage casts on the ground, also the shadow which the
gable casts on the roof. Use scale, 1'' =40'.
546. [4] A 1|'' cube stands in the 1st quadrant on H, with t'gie edges of its
base oblique to the ground line. If line R(- 2", + 1^", + 1^") L(- 1", + V\ + i")
gives the direction of rays of light, find its shadow upon H.
547. [2] A regular pyramid (base V square) is located in the 3rd quadrant,
with its vertex V' below H and its base in a plane parallel to H and 3^" below H.
If the line R(- 2", -:- 1|", + 1^'') L(- 1", + 1", + *") gives the direction of rays
of light, find the shadow of the pyramid on the plane of its base.
548. [2] Find the shadow cast by the pyramid of Problem 303 upon a hori-
zontal plane V below H. The line R(- 2", + 1^", + H") L(- l", + r> + ¥')
gives the direction of light rays.
549. [2] Find the shadow cast by the pyramid of Prob. 730 upon the plane T
there given, also the shadow upon H. Direction of light rays is given bv line
R(-2", + u", + ir) h(-r', + V', + ¥')-
550. [1] The point 0(- 10", + 4^", + 3^") is the vertex of an oblique cone,
and the point M(- 13^'', -!- 2", 0") the center of its circular base of 2^" diameter
in H. Find its shadow upon H and the plane T(- 9", - 112^°, + 30°) if light ravs
are parallel to the line R(- 2'', i U", + H'') L(- 1", + i", f ¥')-
551. [2] An oak block, cut 2|" square and 5'' deep, lies upon H in the 1st
quadrant, with its vertical faces at 60° and 30° with V respectively. On top of-
this block, is a regular pyramid, altitude 2^", base 1^" square, center of base at
center of block. If the 'line R(- 2", + 1^"', + U") L(- 1", + F'» + i") gives the
direction of rays of light, find the shadow of these objects upon H and upon each
other.
552. [1] P at -6". Assuming that rays of light are parallel to a line
R(_3''/+l.|';, + l|'') L(_i'', ^.|", + ^"), find the shadow of the Boom Derrick
and guy ropes of Fig. 27 upon the horizontal plane.
553. [2] A 2'' cube stands on H with two of its vertical sides making 60° and
30° with V respectively. On top of this cube is a block 3" square and V' deep,
with its center above that of the cube and with its edges parallel to the cube
edges. If light rays are parallel to line R(- 2", f H", + 1^-") L(- V, + V',-+ V'),
find the shadow of these objects upon H and upon each other.
33
BUILDING UP SOLIDS, CONDITIONS GIVEN.
Pyramids.
559. [1] Find the H and V projections of a regular triangular (or square or
hexagonal) pyramid whose vertex is at the point V(- Sf", + 4-]-''', + 5|") and
whose base is in a plane T passed through the point 0(- 10;^'', + If, + 2|'') par-
allel to K(-12", + 4r, + -i") L(-6i", + f", + 2i'') and M(- 91", + ^", + 43")
N(- 6f , + 4", + 1"), one side. of the base being the line of intersection of said
plane T with a plane passed through the point O perpendicular to the line KL.
5G0. [1] P at - Ti". Construct the 3 projections of a regular square pyramid
whose vertex is at the point V(- If", - 5:]^", - 4f") and whose base, in a plane T
passing through point A(- G", - 23", ~ If") par. to the lines B(- 7f", - 4", - 4f")
C(-2",-3r,-r) and D(- Sf", - 41", - T ) E(- 21", - T, - 4"), has as one
side the line of intersection between the said base plane and a plane passing
through A perpendicular to the line BC.
561. [1] Find the H and V projections of a regular triangular pyramid whose
vertex is at the point V(- 5J", + 4f", + 5f") and whose equilateral base is in a
plane T passed through the point 0(- IQi", + If'. + 2f") parallel to the Hue
K(-12", + 4f",-^4") L(-6r, + r, + 2i") and the line M(- 9f", + i", + 41")
N(- Gf", + 4", + 1"), one side of the base being the line of intersection of said
plane T with a plane passed through the point O perpendicular to the H trace of
the plane T.
562. [1] P at -8". A regular hexagonal pyramid, altitude 5" has its base in
the plane B(- 8", + 60°, - 60° ) with its center at the point 0(- 4|", - 3", z). Each
side of the base is 2" in length and its lowest side makes an angle of 45° with the
V trace of the plane B. Construct the H, V and P projections of the pyramid.
563. [1] A regular hexagonal pyramid has vertex at point A(- 11|",+ 5",+ 5")
and its base in the plane B(- 8^", - 135°, + 150°). The diameter of the circle
which circumscribes its hexagonal base is 3" and two sides of said base are par-
allel to the H trace of plane B. Find the H and V projections of the pyramid,
its altitude, and its pattern, including the base.
564. [1] P at - ?4". A regular pyramid with a 1^" square base and 3^" alti-
tude is located in the third quadrant. The square base is in the plane
T(-2|", + 150°,-120°), with two sides parallel to the V trace and with its cen-
ter at a point O whose vertical projection is at the point (- 5", 0", - 1^").
Find the H, V and P projections of the pyramid, showing visible and invisible
edges.
565. [1] P at -8". Find the 3 projections of a regular square pyramid whose
vertex is at the point K(- 1^", - If", - 1") and whose square base in the plane
T(- 6", + 60°, - 45°) is inscribed in a circle of 3" diameter with center at
Oi-H'',-2Y',z), with two of its sides parallel to the H trace of plane T.
566. [1] P at -6". Scale, j%-'' = l'-0'\ The lines A(- 35', - 18'. - 5')
D(- 18', - 27', - 5'), a line parallel to AD through B(- 25', - 2V, - 20'), and AB
are respectively the ridge, eave and front rafter lines of a cottage roof. A tower
with a pyramidal roof is to be built thereon, the base of the tower on the roof
to be an 8'-0" square, whose sides are respectively parallel to the lines AB and
BC and whose center is at distances 14'-3" and 12'-0" respectively therefrom.
The top of the pyramidal roof is to be 12'-6" directly above this center, and the
4 vertical walls of the tower rise to a horizontal plane 7'-6" above the center of
the square in the roof. The four corner tower rafters are to be 9'-0" in length.
Find the 3 projections of the roof and tower theron, the distance of the top of
the tower from the plane of the roof and the angle which the corner tower rafters
make with each other.
34 BUIIvDING UP SOIJDS, CONDITIONS GIVEiN
Cones.
567. [2] Construct the H and V projections of a right circular cone whose
base of IV' diameter is in the plane T(- 5^'', - 60°, + 30°) with center at the
point 0(- 3f' , + If", z). Altitude of cone is 3".
568. [2] The point A(- 6'', - 4|", - 5") is the vertex of a cone of revolution
whose axis is perpendicular to, and whose base of 2^^ diameter is in, the plane
T(- 6", + 45°, - 45°). Find the H and V projections of this cone. What is its
altitude ?
569. [1] P at-7|". A cone of revolution with a base of 2"' diameter and
3|'' altitude is located in the 3rd quad. Its base is in plane T(- 2^', + 150°, - 120°)
with its center at the point 0(- 5", y, - 1^'')- Find the 3 projections of this cone.
Prisms.
570. []] P at - SV\ Construct the 3 projections of a regular hexagonal prism
in the 3rd quadrant 3^" high, the plane of whose base makes an angle of 30°
with V and 75° with H. Each side of the base is 1" long and two edges are par-
allel to V. The center of the base is 4" to left of P, f " behind V and 1^" below H.
571. [1] P at - 2". Construct the H, V and P projections of a regular square
prism -in the 1st quadrant, altitude 3^", each side of base If", center of base at
point C(-9i", + l|'', + f")- Plane of base makes an angle of 30° with H and
75° with V, two edges of base being parallel to H.
572. [1] P at - 8y'. A regular hexagonal prism in the third quadrant, of 2"
altitude, one hexagonal base in the plane T(- 4^", t- 45°, - 60°), with center at
point 0(- 2'', - 1^", z") and each side ly, is hollowed out by a right circular
cylinder of 1^" diameter whose axis coincides with that of the prism. Construct
the H, V, and P projections of the hollow prism.
Cubes.
573. [1] P at -3". A 2'' cube stands on the plane T(- 8", - 30°, + 60°) in
the first quadrant. The center of its base when revolved into H about the H
trace of plane T is at 0(- 7|", -(- 2|'', 0") and two of the sides of the base make
angles of 36° with this trace. Find the 3 projections of the cube.
574. [1] G. L. par. to short edge of sheet. P at -54''. A cube in the third
quadrant has its upper face in the plane T(- 4^", + 60°, - 45°), one side of this
face being the line A(- 2V', y", - If') B(- i", y", - 3f") in the plane T. Find
the 3 projections of this cube,
575. [1] The points A(- 11", + |", 0"), B(- 8", + i", 0"), C(- 8", + 3^", 0")
are three vertices of the base of a cube standing in the 1st quadrant on H. Re-
volve the cube about the H trace of the plane T(- 12^", - 45°, + 30°) until it
stands on T, showing its H and V projections in this position.
576. [1] G. L. par. to short edge of sheet. A 3" cube stands in the first quad-
rant with one of its diagonals perpendicular to H, and another parallel to V.
Find the projections of the cube, and the distance from one vertex to the plane of
the 3 adjacent vertices.
REPRESENTATION OF SURFACES.
HELICAL CONVOLUTES.
Assume elements; intersection with H or oblique plane; assume points on
surface.
578. [2] A Helical Convolute is formed by drawing tangents to a helix of 3''
pitch in the first quadrant whose axis is perpendicular to H through the point
0(- 6^", + 11", 0") and whose generating point starts at pt. M(- 7^'', + 1^", 0")
moving so its H projection appears to rotate counter-clockwise. Find 16 elements
in the first convolution of the surface, the line in which the surface thus determined
cuts H, and the projections of a point K of the surface not on one of the above
elements.
579. [1] G. L. par. to short edges of sheet. A Helical Convolute has as its
directrix a helix whose axis is M(- 7|", + 2", 0") N(-7i", + 2", + 5"), whose
pitch is 12'', whose generating point starts at K(- 9", + 2", 0") and moves so that
its H projection appears to rotate counter-clockwise. Find the intersection, (be-
tween H and the point where the helical directrix pierces T), of the helical con-
volute and a plane T(- 1-^', - 120°, + 150°).
HYPERBOLIC PARABOLOIDS AND CONOIDS.
Assuming elements, first and second generations; plane directors of both
generations; assuming points on surface.
580. [4] Plane T(-3f',- 15°, + 224°) is plane directer, A(- 7t'', + f", + 2f' )
B(-6Y', + 2V\0'') and C(-4r, + 2f', + 3") D(- Sf", + f", 4 2'') the directrices
of a Hyperbolic Paraboloid. Find the projections of an element of the surface
parallel to the line E(- 2|", 0", z) F(- 1V\ + i", z) lying in the plane directer.
581. [4] In the Hyperbolic Paraboloid of Prob. 580, find an element of the
surface through the point K(- 7|'', y, z) on the directrix AB.
582. [4] A conoid has plane T(- 3^', + 30°, - 30°) as a directer and line
^(-^V',-i'',-2^/'-) B(-6'', -2", -I") and arcs of 2^' radius struck to the
right of the projections of B as centers for directrices. Find an element of the
conoid parallel to line C(- 3", - -J", z) D(- f , - f , z) lying in the plane directer.
583. [4] In the Conoid of Prob. 582, find an element of the surface through
the point K(- 7", y, z) on AB.
584. [2] Given the plane directer T(- 3", + 18°, - 110J-°) of a Hyperbolic Par-
aboloid, with right line directrices A(- 7f' ', - 2'', - 3") B(- 6f' , - Y', - 1") and
C(-5V',-lV',-^¥') D(-4",-2f' ,-|"). Find the 9 elements of the surface
which divide the directrix AB into 8 equal parts. Find a point E on the surface
whose H projection is the point (- 5'', - 2'', 0").
585. [2] The lines M(- 6", + f ', O'O N(- 4'', + 2'', + 2''), P(- 2'', + 1'', 0'')
Q(- 1", - IJ'', + If") are the directrices of a Hyperbolic Paraboloid whose plane
directer is H. Asume 10 elements of the surface and find a point E on the sur-
face whose H projection is the point (-4:1", + 1|", z").
586. [2] In the Hyperbolic Paraboloid of Prob. 585, find the plane directer of
36 REPRESENTATION OF SURFACES
the second generation, ten elements of the second generation, and a point A on
the surface whose Y projection is (- 3|'', y, + f).
587. [1] The two right Hnes M(- 13^", + U", + 3^') N(- 11^", + 1", 0") and
P(-8J'', 0", + 3") Q(-5i", + 5i", 0") are the directrices of Hyperbolic Parabo-
loid, whose plane directer is D(- I31", - 150°, + 120°). Find (1) five elements of
the first generation through points M, A, B, C, N which divide the directrix MN
into 4 equal parts, (2) plane directer of the second generation, (3) five elements
of the second generation.
588. [1] The two right lines M(- 13i", + 4i", + 3i") N(- 11^", + 1", 0") and
P(-8|''", 0", f 3") Q(-5|", + 5y', 0'') are the directrices of a Hyperbolic Para-
boloid, whose plane directer is D(- K3^'', - 150°, + 120°). Find (1) nine elements
of the first generation through points on the directrix MN which divide it into 8
equal parts, (2) the vertical projection of a point O in the surface whose H pro-
jection is at the point (- 11", + 2", 0"), (3) an element of the 2nd generation
through this point O.
589. [1] The two right lines M(- 12", - 5f', - |") N(- 6^", - 3", - 4^") and
P(-llJ:",-l",-4") Q(-Tf' ,-l",-ir) are the directrices of a Hyperbolic
Paraboloid. The lines MQ and PN are elements of the first generation. Find
(1) eight elements of the first generation through points P, A, B, C, D, E, F, Q
which divide the directrix PQ into 7 equal parts, (2) plane directers D^ and Dj
of the first and second generations, (3) an element of the second generation
through a point K(- 8", y", - 3^") on the surface.
590. [1] The two right lines M(- 12^", - 5", - 3") N(- 12^", 0", - 3") and
P(- 8|", 0", - IV') Q(- 5i", - 5", - 4^") are the directrices of a Hyperbolic Par-
aboloid whose plane directer is V. Find (1) five elements of the first generation
through points M, A, B, C, N which divide the directrix MN into 4 equal parts,
(2) plane directer for the second generation, (3) an element of the second gen-
eration through a point on the surface whose H projection is at the point
(-9r,o",-2r).
591. [1] Lines M(- 12", 0",-l") N(- 9", - 3^", - 3^") and P(- 7", 0", - 2")
Q(- 5^", - 3i", - 2Y') are the directrices of a Hyberbolic Paraboloid whose plane
directer is V. Find (1) the 9 elements of the first generation which divide the
directrix MN into 8 equal parts, (2) the plane directer of the second generation,
(3) the H projection of the point K(- 9^", y, - 1|") on the surface, (4) an ele-
ment of the 2nd generation through this point K.
592. [2] Lines A(- 6^", - J", - i") B(-5",-3V4") and C(-3ViV3i")
D(-y, -4", - 2") are directrices of a Hyperbolic Paraboloid of which BC and
AD are elements of the first generation. Find 13 elements of each generation,
and a plane directer for each generation, through convenient points.
593. [2] Lines A(- 6|", - y', - i") D(- i", - 4", - 2"), B(- 5", - 2", - 4")
Cf- 3", - i", - 3|") are directrices of a Hyperbolic Paraboloid of which AB and
CD are elements. Find seventeen other elements of the same generation, plane
directers for the first and second generation through convenient points, the ver-
tical projections of a point M(- 3|", - If", z) on the surface, and an element of
the second generation through this point M.
594. [2] Lines A(- 6|", - ^", - D B(- 5",- 2",- 4^') and C(- 3".- i",- 3^")
D(-^'',-4", - 2") are directrices of a Hyperbolic Paraboloid of which AC and
BD are elements. Find thirteen other elements of this same generation, a plane
directer for the second generation through the point K(- 1|", - i", - ^"), and the
point on the surface whose V projection is at (- 3j[", 0", - 2f").
HYPERBOLOIDS OF ONE NAPPE, ETC.
Assuming elements through given points.
595. [2] A Hyperboloicl of one Nappe has as directrices the three given
right Hnes A(- 6", - |", - 2^'') B(- 5'', - 2'', - ^'0 and C(- 4^", - 3", - 2^")
D(-3|",-i",-r) and E(-3^-ir,--r) F(- 1", - 3", - 3")-" Find an ele-
ment of the surface through the point G(- 5i^", y", z") on AB ; also an element
through the point F.
596. [2] A Hvperboloid of one Nappe has as directrices the 3 right lines
K{-n",+ \"&n"^ B(-.3i'',+ 2Vi"), C(-4", + 3", + 2i") D(-5", + i", + i")
and E(- 5-^", + 1]", -i- \") F(- 7|-'', + 3", + 3''). Find an element of the surface
through M(- IJ'', y, z) on CD; also an element through the point F.
597. [4] The directrices of a Hyperboloid of one Nappe are the right lines
A(-7^i2Vi'0 B(-6",+ iVir) C(-5r,+ HVlD D(-5", + r, + |")
and E(-4f' , 0", + 2f'') F(- 4", -;- 2", + ^'0- Find an element of the surface
through the point M(- ^\" , y,z) on AB.
598. [4] A warped surface has as directrices the right line A(- 4f",- 2|",- ^")
B(-5^'', - f', -2|"), arcs of ?.\" radius struck to the left with the projections
of B as centtrs, and arcs of 2|" radius struck to the right with the projections of
A as centers. Find an element of the surface through the point C(- 5^", y" , z")
in line AB.
599. [4] A warped surface has as directrices right lines K{- \\" ,\'iV\^%"^
B(-4r', + li", + ir) and C(- 6^', + 2f', + l-J") D(-7r,^-r. + D and arcs
of 2|" radius struck to the right with the projections of A as centers. Find an
element through point B.
HYPERBOLOIDS OF REVOLUTION OF ONE NAPPE.
Assume elements; second generations; construction of meridian curves;
assume points on surface.
GOO. [1] The line A(- 9", - 3^', - 2]") B(- 9", 0", - 2^") is the axis of a
Hyperboloid of Revolution of one Nappe, located in the third quadrant with its
base in Y. The generatrix in its initial position is M(- 11", 0", - If")
N(- 9", - If", - If"). Find (1) thirty-two elements of one generation of the
surface, which pierce V at equal intervals around the circumference of its base,
(2) the horizontal projection of the medirian section parallel to H, (3) an ele-
ment of each generation through a point K whose V projection is not in the V
projection of any of the above- 32 elements.
601. [1] The line A(- 11", + 2f", + 5") B(- 11", + 2f", 0") is the axis of a
Hyperboloid of Revolution of one Nappe which stands in the first quadrant with
its base in H. The generatrix in its initial position is M(- 134", + 2", 0")
N(- 11", + 2", + 2i"). Find (1) sixteen elements of each generation of the sur-
face, which pierce H at equal intervals around the circumference of its base,
(2) the vertical projection of the meridian section parallel to V, (3) the vertical
projection of a meridian section making 45° with V.
602. [2] The line A(- 5", - 2", - 4") B(- 5", - 2", 0") is the axis of a Hyper-
boloid of Revolution of one Nappe, located in the third quadrant with its upper
and lower bases respectively in H and in a horizontal plane 4" below H. The gen-
eratrix of this surface is, in its initial position, the line M(- 6|", - 1|", - 4")
N(- 5", - \V' , - 2"). Find (1) thirty-two elements of one generation of the sur-
face which pierce H at equal intervals around the circumference of its base,
38 REPRESENTATION OE SURFACES
(2) the vertical projection of the meridian section parallel to V, (3) the vertical
projection of a meridian section making 45° with V.
603. [3] A Hyperboloid of Revolution of one Nappe stands in the first quad-
rant with its base of 4'" diameter in H with center at the point 0(- 4|'', + 2^'', 0").
The circle of the gorge, of 2V' diameter, is on a horizontal plane 1^' above H,
with its center directly above O. Find (1) thirty-two elements of the surface,
(2) the vertical projection of the meridian section parallel to V, (3) an element
of each generation through a point K whose H projection is not on the H projec-
tion of any of the thirty-two elements assumed above.
604. [2] A portion of a Hvperboloid of Revolution of one Nappe is generated
by the revolution of the line'C(- 4^'', + 4", + 2|'0 D(- 2", + 2^", 0'') about the
axis A(- 4", + 2V', 0") B(- 4'', + 2|", + 4|")- Show 32 elements of this surface
and determine accurately the projections of the meridian curve cut from the sur-
face by a meridian plane parallel to V.
605. [1] G. L. par. to short edge of sheet. A portion of a Hyperboloid of Rev-
olution of one Nappe is generated by the revolution of line M(- 7|", + If", + f )
N(-5J-", + 4^'',-! 5D about the line A(- 5i'', + 3", + f 0 B(- 54", + 3", + 5^:")
as an axis. Find 16 elements of each generation of the surface and determine
accurately the projections of the meridian section parallel to V.
HELICOIDS.
Oblique and Right; assume elements; intersections with H or V; assume
points on surface.
606. [2] The line A(- 4", + 2", + 4") B(- 4", + 2", 0") is the axis of an
oblique Helicoid ; the generatrix in initial position is M(- 5^", + 2", 0")
N(-4", + 2", + 1|") which moves in such a way that its horizontal projection ap-
pears to rotate counter-clockwise. The pitch of the helix which is generated by
the point M is 2". Find (1) sixteen elements of the surface generated during one
complete movement of the generatrix about the axis, (2) the intersection of the
surface thus determined with the H plane of projection, (3) the two projections
of a point K which is not on any of the elements assumed above.
607. [2] The line A(- 4^, - If", - 4") B(- 4^, - IF, 0") is the axis of an
oblique Helicoid, whose generatrix in initial position is M(- 5^", - If", - 4")
N(-4y, - If", - If"). This generatrix moves in such a way that its H projec-
tion appears to rotate clockwise and the point M generates a helix whose pitch
is 2|". Find (1) sixteen elements of the surface generated during one complete
movement of the generatrix about the axis, (2) the intersection of the surface
thus determined with a horizontal plane 4" below H, (3) the projections of a
point K of the surface not on one of the above elements.
608. [2] The line A(- 5", - 4", - 2") B(- 5", 0", - 2") is the axis of a Right
Helicoid whose generatrix in its initial position is determined by the points
M(- 3|", 0", - 2") and N(- 5", 0", - 2"). This generatrix moves in such a way
that its V projection appears to rotate clockwise, and the point M generates a
helix whose pitch is 2". Find (1) thirty-two elements of the surface, generated
while the generatrix swings about the axis twice, (2) the projections of another
helicoid, exactly similar to the first and 4" above it.
609. [2] The line A(- 4", - 2V', 0")"B(- 4", - 2^", - 5") is the axis of a Right
Helicoid, whose generatrix in its initial position, is determined by the points
M(-6|",-24",-5") and N(- 4", - 2^', - 5"). This generatrix moves in such
a way that its H projection appears to rotate counter-clockwise, while the point
M generates a helix whose pitch is 2^". Find (1) sixty- four elements of the
surface, generated while MN swings about the axis twice, (2) the projections of
OP THE
I^NIVERSITY
OF
TSENTATION OF SURFACES 39
another right hehcoid, with same axis, same pitch, generatrix starting from same
initial position, but being only 1^" in length.
610. [1] G. L. par. to short edges of sheet. An oblique Helicoid is in the first
quadrant, with a vertical axis through the point 0(- 5^', + 2", 0"). Its genera-
trix moves from its initial position A(- 7", - 2", 0") B(- 5^", - 3", + 2V') so that
its H projection appears to rotate counter-clockwise, while the point A generates
a helix of 8" pitch. Find (a) 12 elements of the surface generated in f of a rev-
olution, (b) the intersection of the surface thus determined with H, (c) the pro-
jections of the helices generated by the two points which divide the generatrix
into 3 equal parts.
611. [2] The line M( - 3f ', - If , 0") N(-3r', + ir.^ ^") is the axis of an
oblique Helicoid whose elements make an angle of 45° with H. In initial position
the generatrix pierces H at K(- 2^", + If ', 0") and K appears to rotate clock-
wise. Find (1) 1() elements of the surface formed in one revolution, (2) the in-
tersection of the surface thus determined with the H plane.
612. [2] An oblique Helicoid has as axis the line M(- 4|", 0", - 2") and
N(- -ll'', - 4^", - 2"), its elements making 30° with this axis. In initial position
the generatrix pierces H at the point K(- 6", 0", - 2) and in moving K generates
a helix of 4" pitch. Find 16 elements of the first convolution, and the curve in
which the surface thus determined cuts V.
613. [4] An oblique helicoid has as axis vertical line through N(- 4", + 1", 0''),
its elements making 22^° with the H plane. In initial position the generatrix
pierces H at the point K(- 3", f 2", 0") and in moving K generates a helix of
2" pitch. Find 16 elements of the first convolution of the surface.
APPLICATIONS.
614. Fig. 50 shows a square threaded screw, formed by the motion of a square
about a line in its plane as axis. Each point in the square generates a helix. The
top and bottom thread surfaces are right helicoids, limited by the blank and root
cylinders. Find the projections of such threads, showing clearly the method of
construction, with the following dimensions : —
1. [8] D = ir- p = y'.
2. [8] D = 2 ". P = f".
3. [4] G. h. par. to short edge of space. D = 3 ". P = 1^".
4. [4] G. L. par. to short edge of space. D = 2Y'. P = f .
615. In Fig. 51 is shown a "V" thread. The two surfaces are oblique heli-
coids and the threads may be thought of as having been generated by the motion
of a triangle about a right line in its plang as axis, each point generating a helix.
Such threads or modifications thereof are common on bolts, screws, etc. Find
the projections of such threads, showing clearly the method of construction, with
the following dimensions : —
Thread angle = 60°.
Thread angle = 90°.
Thread angle = 60°.
Thread angle = 90°.
616. In Fig. 52 is shown a round helical spring generated by moving a circle
about a line in its plane in such a way that all its points generate helices of the
same pitch. Find the projections of such a spring, using the following dimen-
sions : —
1. [8] D = 1V'. d = r. P=l ".
2. [8] D = 2"". d = y'. P = H".
3. [2] D = 4 ''. d = l". P = 2 ".
1.
[8]
D = ir.
P = 1"
2.
[8]
D = 2 ".
p = f :
3.
[2]
D = 2y\
p = r
4.
[2]
D = 3 ''.
P = 2",
40 ri;pre;sentation of surfaces
G17. In Fig. 53 is shown a square helical s])ring, formed by the motion of a
square about a line in its plane as an axis, each point generating a helix of the
same pitch. Find the projections of such a spring, using the following dimen-
sions : —
1. [8] D = i-Y'. p = f ''. s = -r.
2. [8] D = 2 ''. P = 1 ''. S = V'.
3. [2] D = 3 ''. P = 1J". S = r.
618. In Fig. 54 is shown a helicoidal conveyor, whose flights are right heli-
coidal- surfaces. Neglecting the thickness of the flight, find the projections of
such a conveyor, using the following dimensions, and scale, 1" = I'-O".
1. [8]'d = 15''. d = 3'^ P = 9''.
2. [8] D = 2'-0^ d = 6". P = 12".
3. [2] D = 2'-(r'. d = 12''. P = 2'-0''.
619. In Fig. 56 is shown a wood screw thread formed by 2 helical convolute
surfaces, the large spaces being left between threads to make up for the small
shearing strength of wood as compared to the metal used. Find the projections
of such a screw thread, using the following dimensions : —
1. [8] D = 2''. d = li''. P = H''.
2. [8] D = r\ d = lV'. P = l'''.
3. [2] D = 3^ d = 2X P = 2 ''.
620. [1] Fig. 57 shows a right helicoidal stairway built in a square tower, with
a cylindrical sheet metal well. Assume reasonable dimensions and a convenient
scale, and construct the detailed projections of such a stairway, showing all nec-
essary construction lines.
621. [1] Fig. 58 shows a right helicoidal stairway designed for a cylindrical
corner tower in a gymnasium, with a cylindrical well. Asume reasonable dimen-
sions and a convenient scale, and construct the detailed projections of such a
stairway, showing all necessary construction lines.
PLANES TANGENT TO SURFACES.
Through points on surface ; through points off the surface ; parallel to given
lines; through given lines.
1.— PLANES TANGENT TO RIGHT CYLINDERS.
625. ["3] Pass a plane tangent to the right cylinder whose base is a ?>" circle
with center at 0(- 4-^", + 2^', 0") and lying in the H plane.
(a) Through the point K(- 3^", y, + 2^'') on the surface.
(b) Through the point M (- hV , + f ", + 3'') , not on the surface.
( c ) Parallel to the line Q ( - 6'', + 2|'', + f ' ) R ( - 2|'', + 4f' , + 4'' ) •
620. [2] Pass a plane tangent to the right cylinder whose base is in the V
plane and is the ellipse whose major axis is 3" long, makes -45° with the ground
line and is intersected at point B(- 4:|", 0", - 2|") by the minor axis, which is
2'' long.
(a) Through a convenient point M on the surface.
(b) Through an assumed point N not on the surface.
(c) Parallel to the line G(- 6f ', - 31", - 3'0 H(- 3^'', - 1", - l^'')-
627. [4] P at - 4". Pass a plane tangent to the right cylinder whose base is
a ?^\" circle lying in P with center at K(0", - \\" , - \\").
(a) Through point \^{- \\" , - \" , z) on the surface.
(b) Through point N(- 3'', + f"'',-^''), off the surface.
(c) Parallel to line 0(- 2", - f', - ^'0 Q(-|", - H", - 1|").
628. [2] Pass a plane tangent to the right cylinder whose base is a 2^' circle
lying in the H plane with center at M(- 5:1:", - 3^'', 0").
(a) Through a convenient point A on the surface.
(b) Through point R(- 2f", - V\ - \\"), off the surface.
(c) Parallel to line S(- 6^'^ - 4", - 3i") T(- 3-]-", - l^'', - I").
2.— PLANES TANGENT TO OBLIQUE CYLINDERS.
629. [2] Pass a plane tangent to the cylinder having line A(- 3|'', + 3", + 2^")
B(- 5'', + 2^'', 0") for its axis and its base in H, a circle of ?>" diameter with
center at ?j.
(a) Tbrough the point 0(- 3'', !- 2f' ', z) on the surface of the cylinder.
(b) Through the point C(- 1^", + 3'', + If"), not on the surface.'
(c) Parallel to the line Q(-l-K', + 2f', 0'') R(-2'^ + ll'^ + 3'')•
630. [2] Pass a plane tangent to the oblique cylinder whose axis is the line
K(-5i", 0'',-U-") L(-2i",-2r,-3'') and whose base in V is an ellipse
whose major axis is 4'' long, makes an angle of - 30° with ground line and is in-
tersected at pt. K by the minor axis which is 2" long.
(a) Through a point A on the surface.
(b) Through point M(- If", - If", - If '), not on the surface.
(c) Parallel to line N (- 6f", - 2|", - |") R(- 4V', - 1^", - 3").
631. [2] P at - 3^'. Pass a plane tangent to the cylinder whose base is a 2"
circle lying in P with its center at A(0", - If , - 1^") and having
AB(-4i",-2t", -23") for its axis:
(a) Through point C(- 2^1", - 1-J", z) on the surface.
(b) Through point D(- 1^", - 3", - 1^").
(c) Parallel to line E(- If", - 2^", + f") F(- 3|", - \l" , - 2f' )•
42
632. [2] Pass a plane tangent to the cylinder whose base is a 2V circle lying
in H with its center at G(- H'',- 2^, 0"), whose axis is CxH(- If'',- 4^,- 3f").
(a) Through point K(- 3i", y, - If") on the surface.
(b) Through point L(- U", - 2'', - 2f").
(c) Parallel to line M(- 6^, + 1^ - 1") N(- 3r, - 3r, - iD-
3.— PLANES TANGENT TO RIGHT CONES.
633. [4] Pass a plane tangent to a right cone whose base is a If circle lying
in H with center at 0(- 5a", + If", 0") and whose altitude is 2^".
(a) Through point M(- 5^', y, + f") on the surface.
(b) Through point N (- 4^", + f ", + |"), not on the surface.
(c) Parallel to line K(- 31", + 2", + 2i") L(- H". + 1^, + D-
634. [4] P at - 34". Pass a plane tangent to a right cone whose base is a If'
circle lying in a plane perpendicular to the ground line with center at
R(_li''^_li", -11'') and whose altitude is 3|" :
(a) Through a point A(- 2f ', - 1", z") on the surface.
(b) Through a point B(- 2f", - ^", - -^"), not on the surface.
(c) Parallel to line S(- 3^", - T, - 2") T(- f", + f", - !")•
635. [2] Pass a plane tangent to a right cone having for its base the ellipse
whose major axis is 4i" long- and makes +30° with the ground line, and whose
minor axis is 2f" long and intersects the major axis at point 0(- 4|", - 2|", 0").
Its base is in the H plane and its altitude 4^".
(a) Through point D(- 3f", y, - l^") on the surface.
(b) Through point Q(- 6", - 3f", - 1|"), off the surface.
(c) Parallel to line R(- 4i", - f", - D S(- 6", - If", - 31").
4.— PLANES TANGENT TO OBLIQUE CONES.
636. [2] Pass a plane tangent to the cone whose base is the 2-^" circle lying
in a plane parallel to H with center at A(- 2|", - 3", - 4^") and whose vertex is
at B(-6r,-n-r):
(a) Through point C(- oV', - If", z) on the surface.
(b) Through point D(- 3|", - 1|", - 3"), not on the surface.
(c) Parallel to line E(- 2^", - 2^", - 3") F(- 6^", - 2|", + 3").
637. P at - 3^'. Pass a plane tangent to the oblique cone whose base is a 2|"
circle lying in P with center at L(0", - 1/^", - 2-|"), and whose vertex is at
M(-3r,-3J",-4f")-
(a) Through point N(- 1^", y, - 3^") on the surface.
(b) Through point 0(-^", - 2", - 1").
(c) Parallel to line R(- f", - f", - i") Q(- 3|", - Y', - ^D-
638. [2] Pass a plane tangent to the cone whose base is the ellipse with major
axis A(-6", + 3", 0") B(- U", + 3", 0") and minor axis C(- 3f", + 4f", 0")
P(_3|'', + iy', 0") and whose vertex is E(- 7i", + 4-i", + 4^").
(a) Through a point M on the surface.
(b) Through a point N not on the surface.
(c) Parallel to the line F(- 2f", + f", + 1") G(- 4i", + 3^", + ^'0-
639. [2] Pass a plane tangent to the oblique cone having 0(- li", + 4f , + 1^')
for its apex and the 3f" circle lying in V with its center at N(- 6J", 0", + 3")
for a base :
(a) Through the point Q(- 2J", + 2|", z) on the surface.
(b) Through the point R(- 2", + 2^", + 2f ") .
(c) Parallel to the line S(- 4^, + -if", + 3^") T(- 7", + f", + 3f").
43
5.— PLANES TANGENT TO HELICAL CONVOLUTES.
640. [2] A convolute is generated by a line moving tangent to the helix gen-
erated by the point M(-2^-", + Ig", 0'') moving in a negative direction about. line
N(-4i^ + ir, + 2r) O(-4J", + ir,0'') with a pitch of 2". Pass a plane
tangent to the convolute :
(a) Through point Q(- 3f", -t- 3f", z) on the surface.
(b) Through point R(-7p, + 3", + f'), not on the surface.
(c) Parallel to line S(- 2f' , + f", + 1") T(+ 5f", + If ', + 2-D.
641. [2] G. L. :[" above lower border of working space. A convolute is gen-
erated by a line moving tangent to the helix generated by pt. A(- 6f", - 7f , 0'')
moving in a positive direction about line B(- 5f ', - 7f ', 0") C(- 5f", - 7f ', + 4'')
with a pitch of 3|". Pass a plane tangent to the convolute.
(a) Through point D(- 7", - of", z) on the surface.
(b) Through point E(- 31'',- 51'', + li'O. off the surface.
(c) Parallel to line F(-5-J",- 91", + 21") G(- 44-", - Sf", + f').
642. [2] A convolute is generated by a line moving tangent to the helix gen-
erated by pt. A(- 6f", + 4|", 0") moving in a negative direction about line
J(- 5i", + 43", 0") K(- Sf, + 4r', -u 4") with a pitch of 4^". Pass a plane tan-
gent to the convolute :
(a) Through point L(- 6", + 2|", z) on the surface.
(b) Through point M(- 3f", ^ lf", + li"), not on the surface.
(c) Parallel to line N(- 2", + If", + 21") 0(- 4i". + H", + 3|").
6.— PLANES TANGENT TO HYPERBOLIC PARABOLOIDS.
643. [2] Pass a plane T tangent to the hyperbolic paraboloid of Prob. 585
through the point 0(-4", + iy, z) on the surface.
644. [1] Pass a plane S tangent to the hyperbolic paraboloid of Prob. 587 at
a point R on the surface whose horizontal proj. is at the point (- 11", + 2", 0").
645. [2] Assume P at + 5". Pass a plane U tangent to the hyperbolic para-
boloid of Prob. 589 through the point Q on the surface.
646. [1] Pass a plane T tangent to the hyperbolic paraboloid of Prob. 590 at
a point O on the surface whose V projection is at the point (- 9|", 0", - 2f").
647. [2] Assume P at + 4|". Pass a plane R tangent to the hyperbolic para-
boloid of Prob. 591 through that point O on the surface which is verticallv pro-
jected at (-9i", 0",-li").
648. [2] Pass a plane S tangent to the hyperbolic paraboloid of Prob. 592 at
the point K(- 2§", y, - 2") on the surface.
649. [2] Pass a plane R tangent to the hyperbolic paraboloid in Prob. 584
through the point O on the surface whose V projection is at (- 6f , 0", - 2f ).
650. [2] The right lines C(- 7", - 2i", - 3") D(- 5i", - f'^ _ i^^) and
A(-2f", -1", -1") B(-2-2". -1", -3-}") are directrices of a Hyperbolic Para-
boloid whose plane directer is PI. Find a plane directer of the 2nd generation,
and a plane R tangent to the surface at a point K(- 5", y", - 2^").
7.— PLANES TANGENT TO HYPERBOLOIDS OF ONE NAPPE.
651. [2] Pass a plane T tangent to the hyperboloid of Prob. 602 through the
point 0(- 4f , - If", z) on the surface.
652. [2] Pass a plane T tangent to the hyperboloid of Prob. 603 through the
point R on the surface whose H projection is at the point (- 4f", + f", 0").
653. [2] P at +4-^". Pass a plane T tangent to the hyperboloid of Prob. 600
through the point T(- 8f , y, - 2|") on the surface.
44
8.— PLANES TANGENT TO HELICOIDS.
65i. [2] Pass a plane tangent to the helicoid of Prob. 607,
(a) Through the point 0(- 3^'', y'', z") on the hehcal directrix.
(b) Through the point R(- 3^'', -1^'', y'') on the surface.
655. [1] G. L. parallel to short edge of sheet and 2" below middle of sheet.
Pass a plane tangent to the helicoid of Prob. 610,
(a) Through the point A(- 6|", y", z'')on the helical directrix.
(b) Through the point B on the surface which is horizontally projected
at (-5'',-2r,0'').
656. [2] Find the traces of a plane tangent to the helicoid of Prob. 611.
(a) Through the point A(- 4f'', y'', z'') on the helical directrix.
(b) Through the point B(- 3", + ly, z") on the surface.
657. [2] Pass a plane tangent to the right helicoid of Prob. 608.
(a) Through the point R(- 4i", y'', z") on the helical directrix.
(b) Through the point S(- 5^", y", + 3'') on the surface.
658. [1] Find the traces of a plane tangent to a helicoid of 3" pitch, whose axis
is C(-9i", + 2^", 0") D(-9^'', + 2|", + 4"), initial position of generatrix
£(_7i''^ + 2^'', 6") F(-9-i-", + 2^', + 2^'') which moves so that its H projection
rotates clockwise.
(a) Through the point A(- lOf'', y , z") on the helical directrix.
(b) Through the point B on the surface which is projected at
(-8r,+3r,z'0-
9.— PLANES TANGENT TO SPHERES.
659. [2] A sphere of 4" diameter is located in the first quadrant with its cen-
ter at the point 0(- h\" , + 2^", + 2^"). Find a plane T, tangent to the sphere at
that point P on the bottom portion of the surface whose H projection is at
660. [2] Find a plane S tangent to the above sphere through the right line
A(-3-, + 2r,+ir) B(-r,+r, + 3r)-
661. [2] A sphere of o" diameter rests upon a horizontal plane in the 3rd
quadrant at the point M(- 3J", - 2", - 3fO- Fintl a plane T tangent to the
sphere at a point P on the upper portion of the surface whose H projection is
(-2f'',-l'',0'0-
662. [2] Find a plane U tangent to the above sphere through the right line
c(- 4r, - 5-, + r ) D(- 7i-, - \'\ - 3r ).
663. [2] P at -3i". A sphere of 2^" diameter is located in the third quad-
rant with its center at the point 0(- \\" , - \\" , - \%"). Find the tangent plane
T to the sphere at the point A on the surface whose H projection is at
10.— PLANES TANGENT TO ELLIPSOIDS.
664. [2] An ellipsoid of revolution is formed in the third quadrant by the rota-
tion about its vertical major axis of an ellipse whose axes are respectively 3''
and 2". Find a plane tangent to the surface at some convenient point R.
665. [2] An ellipsoid of revolution is formed in the 3rd quadrant by the rota-
tion about its vertical minor axis, of an ellipse whose axes are 3" and If" respec-
tively. Find a plane tangent to the surface at a convenient point A.
666. [2] Find a plane tangent to the above ellipsoid through a convenient
line EC.
667. [2] An ellipsoid of revolution is formed in the 3rd quadrant by the rota-
tion about its vertical major axis of an ellipse whose axes are 4'' and 2|'' respec-
tively. Find a plane tangent to the surface at a point \" from the top.
668. [2] An ellipsoid of revolution, axes 6'' and 3" respectively, has its center
at the point 0(- 41:'', + 2", 0"), and its long axis perpendicular to H. Find a
45
plane tangent to the ellipsoid at a point B on the surface horizontally projected
at point {-3V', + 1Y',0'').
669. [21 Find a plane N, tangent to the above ellipsoid and through the line
A(- 6f ^ + If", + r ) B(- 5-, + 4r, + 3^') .
11.— PLANES TANGENT TO HYPERBOLOIDS AND PARABOLOIDS
OF REVOLUTION.
670. [3] A Hyperbola, in a plane parallel to V, has its foci at the points
F(-5i",-2r'.-2i'') and G(- H'',-2V',-2V') and its vertices at the points
V(- 4|", - 2V', - 2V'), W(- 3^', - 2f ', - 2i")- This Hyperbola spins about its
vertical conjugate axis and generates a Hyperboloid of Revolution. Find a plane
T, tangent to the surface at the point K on the surface which is projected at
(-3r,y,-i").
671. [3] The line A(- 5f ', - 3^, - i'') B{- r',-2Y\- V') is the directrix
and the point F(- 3^", - 3|'', - 1") the focus of a parabola which is the genera-
trix of a paraboloid of revolution formed by rotating the parabola about its axis.
Find a plane T tangent to the surface at a point K on the surface.
673. [3] Pass a plane T tangent to the above paraboloid through the line
K(- 6^' - t", - 4i") M(- 4i", - Sr, - D.
673. [3] A hyperbola, in a plane parallel to V, has its foci at the points
F(-4i",-3|",-lf'0 and {^{-A^,",-2\",-'i\"), and its vertices at the points
V(-4^':, -2i",- 3") and W(- 4|^ - 21'', - 3"). This hyperbola spins about its
vertical transverse axis and generates a Hyperboloid of Revolution. Find a
plane S tangent to the surface through the point O on the surface whose V pro-
jection is at the point (- 4|", 0", - 3]'').
674. [3] P at +3". The line A(- 81'', - 2'', - 4") B(- 8^", - 2", 0") is the
directrix and the point F(- 7^", - 2", - 2") the focus of a parabola which is the
generatrix of a paraboloid of revolution with AB as its axis. Find the plane T
tangent to the surface at a point P on the surface which is horizontally projected
at p(-8",-l|", 0").
675. [3] P at + 3". Find a plane U tangent to the above surface through line
K(-3r,--3r, + r) N(-5r/-r, + ir)-
12.— PLANES TANGENT TO TORI.
676. [3] A Torus is formed by the revolution of a circle of W diameter
about the axis A( - 4 J", -! 3f' , + 4") B(- 4-3", + 3^", 0"). The circle In its initial
position is parallel to V with its center at the point 0(- 6|", + 3f", + If"). Find
a plane T tangent to the surface through the pt. M on the bottom of the surface
horizontally projected at M(- 3|", + 4", 0").
677. [3] A Torus is formed bv the revolution of a circle of If" diameter about
the axis A(- 3^", - 3f", - 3") b'(- 3^", - 3f", 0"). The circle in its initial posi-
tion is parallel to V with its center at the point 0(- 4f", - 3f", - If"). Find a
plane U tangent to the surface through the line M(- 4|", 0", - 3f")
N(:-6r,-ir,-r)-
678. [1] A Torus is formed by the revolution of a circle of 3" diameter, about
the axis A(- 6^", - 1-J", - 4") B(- 6|", - U", 0"). The circle in its initial posi-
tion is parallel to V with its center at the point 0(- 7f", - 1|", - 2"). Find a
plane S tangent to the surface at a point on the lower portion of the surface,
whose H projection is at (- 8|", - f ", 0").
679. {1] Find a plane R tangent to the surface of Prob. 678, through the line
K(-8f",-31",-^") L(-10f",-f",-3f").
680. [1] (Ground line parallel to short edge of sheet.) A Torus is formed
by the revolution of a circle of 3^" diameter about the axis A(- 3f", - 5", - 4y )
B(- 3f", 0", - 4^"). The circle in its initial position is parallel to H with its cen-
ter at the point 0(- 6|", - 3", - 4^"). Find a plane Q tangent to the surface
through the point R(- 6^", y", - 6") on the upper part of the surface.
TO FIND POINTS WHERE LINES PIERCE SURFACES.
Right Prisms and Pyramids.
685. [2] Assume P at -f 9. Find the points where line A(- 12", - 1^", - 2-|")
B(-15f", -2-^", -f ) pierces the right prism of Prob. 725.
686. [2] Assume P at h 0. Find where the line C(- 12i", - 1^", - SJ")
D(- 15", - If, - If") pierces the right prism of Prob. 726.
687. [2] Find the points where Hne E(- 2f ", + 2|", + 1") F(- 5^", + 1", + If)
pierces the right pyramid of Prob. 730.
688. [2] Find where the line G(- 2f", + If", + If ) H(- of , + 3", + .}")
cuts the right pyramid of Prob. 730.
Oblique Prisms and Pyramids.
689. [2] Find where the line J(- 3^", + f", + 3") K(- 5f", f 2f". + ^") pierces
the oblique prism of Prob. 727.
690. [2] Find the pts. where line L(- 3f , + 3^", + 1") KC- 6f , - f , + If")
pierces the oblique prism of Prob. 727.
691. [2] Assume P at 1-9. Find where the line A(- llf", - ^", - 3|")
B(- 13^", - 2f", - 1-^") pierces the oblique pyramid of Prob. 731.
692. [2] Assume P at +9. Find the points where line C(- llf , + If , + |")
D(-13f , + 2", + 3") pierces the oblique pyramid of Prob. 732.
Right Cylinders and Cones.
693. [2] Assume P at f 9. Find the points where the right cylinder of Prob.
734 is pierced by the line A(- llf , + 3f , + 3|") B(- 15^", + f , + If ).
694. '[21 Find where line. C(- 2", - 1", - 3f ) D(- 5f , - 1^', - f ) pierces
the right cylinder of Prob. 736.
695. [2] P at - 3f . Find where line E(- V', - f", - 3^") F(- 3^", - 3^", - i")
pierces the right cylinder of Prob. 738.
696. [2] Assume P at -i 8i". Find where the line G(- llf , + 1", + f")
K(-15f , + 3",-:-2f ) pierces the right cone of Prob. 747.
697. [2] Assume P at + 8J/'. Find where the right cone of Prob. 748 is pierced
by the line A(- 11", + f , - V') B(- 14^", + 4^", -^ 2f ).
698. [2] P at - 2". Find where line C(- f ,- 3f ,- 2f") D(- 4f",+ 1",- 3f )
pierces the right cone of Prob. 752.
Oblique CyHnders and Cones.
699. [2] Assume P at + S". Find where line E(- lOJ", - i", - 1")
F(-13f , -2f , -44") pierces the oblique cylinder of Prob. 740.
700. [2] P at 0". Find the points where the oblique cylinder of Prob. 741 is
pierced by line A(- 2^", - 1", - 4f ) B(- 6f , - 4", - 1").
701. [2] Assume P at + 9". Find the points in which the oblique cylinder of
Prob. 743 is pierced by line C(- 11", + 4f , + 3") D(- 15", + If , + f )'.
702. [2] Assume P at +8". Find where line A(- 9f , + f , - 1^")
B(-12f , - 2f , -3f ) pierces the oblique cone of Prob. 763.
703. [2] Assume P at + 8f . Find where line C(- lOf , - 1^", - 4")
D(-14f , + f , -If ) pierces the oblique cone of Prob. 764.
704. [2] Assume P at +8^". Find the points in which the oblique cone of
Prob. 765 is pierced by the line E(- lOf", - f , - 3") F(- 13|", - 3", - 4").
47
Warped Surfaces.
705. [2] Assume P at + U'\ Find where the line A(- 5^", - Sf", -4f'0
B(-10f' , -§", -2'') pierces the hyperbolic paraboloid of Prob. 795.
706. [2] Assume P at +5-]:''. Find where the Hne C(- 10|", - 3f ', - 4i")
D(- 12-1", -4", -3'') pierces the hyperbolic paraboloid of Prob. 793.
707. [2] Find where the hvperboloid of Prob. 806 is pierced by line
E(- 6r, - v, - 5'') F(- 2f^ - ir, + 1|").
708. [2] Assume P at H- 7". Find where the hyperboloid of Prob. 805 is pierced
by the line A(- 8^', + ^'', + f ) B(- 13|", + f", + 2f")-
709. [2] Assume P at +74". Find where line C(- 9^", + 1", - i")
D(- 14i", + 3|", ^ 4") pierces the helicoid of Prob. 811.
710. [2] Find where the helicoid of Prob. 812 is pierced by the right line
A(- ir.. - r, - 1") B(- 6r, - sr, - sf).
Double Curved Surfaces.
711. [2] Find where line A(- 3", + 1", + l^") B(- 6i", + 2^', + 4^") pierces
the sphere of Prob. 767.
712. [2] Find where the sphere in Prob. 768 is pierced bv the right line
c(-r,+r, + ir) d(-5",-4",-3").
713. [2] Find where the ellipsoid of Prob. 770 is pierced by a line conveniently
assumed.
714. [2] Find the points in which the ellipsoid of Prob. 773 is pierced by the
line E(- ir, + 3-1", + 3|") F(- 61", + 2-J", + f").
715. [2] Find where line A(- 2i", - T, - 3|") B(- 5^", - 2^, - |") pierces
the paraboloid of Prob. 775.
716. [2] Find where the torus of Prob. 779 is pierced by the right line
C(-2^", + 4|", + li") D(-7", + |", + 2f").
717. [2] Find the points in which the given right line K(- 1|", - J", - J")
M(- 6f", - 4f", - 2f") pierces the torus of Prob. 780.
I. INTERSECTIONS OF SURFACES AND PLANES.
1.— PRISMS CUT BY PLANES, AND DEVELOPMENT.
725. [1] The line M(- 14'', - If^ -4'^ N(- 14", - If , 0") is the axis of a
regular hexagonal prism, whose upper base is in H with center at the point N.
Each side of the base is ly, and two sides are parallel to the ground line. The
prism is cut by a plane T(- 11|-", f 120°, - 135°). Find the intersection of the
plane and the prism, the actual size of the section when swung into H, and the
development (pattern) of that part of the prism between the planes H and T.
Develop prism base along the ground line, beginning at a point (-9") with the
shortest element and developing to the right and below the ground line. (State
problem in upper right portion of sheet.)
726. [1] The line M(- 13|'', - 1^', - 4") N(- 13|", - 1^', 0") is the axis of
a regular square prism, whose base is in a plane 4'' below H and parallel to H.
The center of the square base is at the point M, each of the sides is 1|" long, and
two of them when extended make angles of +60° with the ground line. The
prism is cut off by the plane T(- 9f' , + 135°, - 150°). Find the intersection of
the plane and the prism, the actual size of the section when swung into V, and
the development (pattern) of the prism, between T and the plane 4" below H.
Develop the prism base on the line 4" below the ground line, beginning with the
shortest edge at a point 4|" to the right of m' and developing to the right and
above the base line. (State problem in upper right hand portion of sheet.)
727. [1] The line M(- 6", + H". + 3|'') N(- 3|", + 2", 0") is the axis of a
triangular prism whose base is an equilateral triangle in H. This triangle is in-
scribed in a circle of 2V' diameter and center at N, with its side nearest to V
parallel to the ground line. The prism is cut by the plane T(- 7^", - 45°, + 30°).
Find the intersection of the plane and prism, the true size of the section when
swung into H, and the development (pattern) of that part of the prism between
T and H. (Develop the prism in the upper left hand portion of the sheet, and
state the problem in the lower left hand portion.)
728. [2] Find the intersection of that portion of a plane included between
the right lines M(- 6i", + T, + D N(- 4^'', - t", + 3f'0 and 0(- 5^", + 2^", O'O
P(-3J:", + f' , + 4") with the prism made up of lines A(- ^'', + V', + 2^'')
E(- IV, + V, + 3D and the lines B(- 5f',+ 1",+ f") F and C(- 4|",+"2^",+ f")
D, drawn parallel to AE through the points B and C.
729. [2] The line M(- 5", 0'', - 2") N(- 24'', - 3|", - 1^") is the axis of a
square prism whose base is in V, center at point M, with 2 of its 3" sides parallel
to the G. L. Find the intersection of this prism with an opaque plane
T(- 1", + 150°, - 135°), indicating visible and invisible edges.
2.— PYRAMIDS CUT BY PLANES AND DEVELOPMENT.
730. [1] The line M(- 4", i- 2", + 3^") N(- 4", + 2", 0") is the axis of a reg-
ular hexagonal pyramid. Its base in H has sides 1|" long, two of these sides
being parallel to V. The point M is the vertex. The pyramid is cut by the plane
TC- 8^", -45°, + 30°). Find the intersection of the plane and the pyramid, the
actual size of this section when swung into H, and the development (pattern)
49
of the pyramid, showing the intersection. (Develop pyramid in upper left hand
portion of the sheet, and state problem in lower left hand portion.)
731. [1] The point M(- lU'', - J", - 1") is the vertex of a pyramid which
is cut by a plane T(- 16",-!- 30°, - 45°). The base of the pyramid is a 2" square
in a plane 4" below H and parallel to H. The center of the square is the point
0(- 12V', - lY', - 4") and two of its sides are parallel to the H trace of plane T.
Find the intersection of the plane and the pyramid, the actual size of this section
when swung into H, and the development (pattern) of the pyramid showing
the intersection. (Develop pyramid in upper right hand portion of the sheet, and
state problem in lower right hand portion.)
732. [1] The point 0(- 12", - 3", - U") is the vertex of a pyramid which is
cut by a plane T(- 9^", + 150°, - 120°). The base of the pyramid in V is an
equilateral triangle inscribed in a circle of 2" diameter, whose center is at the point
M(-13r', 0", -1-r'). The upper side of this base is parallel to H. Find the
intersection of the plane and the pyramid, the actual size and shape of the sec-
tion when swung into V, and the development (pattern) of the pyramid, show-
ing the intersection. (Develop the pyramid in the upper right hand portion of
the sheet, and state the problem in the lower right hand portion.)
733. []] P at - 7". Find 3 projections and the intersection of the right square
pyramid, having an altitude of 5" and having line A(- 5^", - f", z)
B(- 7", - 21", z) in plane T(- 6^", 4 120°, - 67J-°) for one side of its base which
is in plane T and does not cross its trace, with the opaque parallelogram
E(-3",-U",-3") F(-5",-l", -f ) G(-6",-4i", -li") H.
3.— RIGHT CYLINDERS CUT BY PLANES.
Curve of intersection, true size, and development.
734. [1] The line M(- 14", + 2", 0") N(- 14", -h 2", + 4") is the axis of a
right cylinder, whose circular base of 3" diameter is in H. The cylinder is cut
by the plane T(- 11^", - 112-|°, h 135°). Find the curve of the intersection of
the plane and the cylinder, the true size of this curve when swung into H, and
the development of that part of the cylinder between H and T. Develop the
cylinder base along the ground line, beginning at a point (- 10^') with the
shortest cylinder element, and developing to the right and above the ground line.
(State the problem in lower right hand portion of sheet.)
735. [1] The line M(- 14", - 2", - 4") N(- 1-4", - 2", 0") is the axis of a
right cylinder whose circular base of 3" diameter is in a plane parallel to H and
4" below H. The cylinder is cut by the plane T(- 15", + 120°, - 45°). Find
the curve of intersection of the plane and the cylinder, the true size of this curve
when swung into H, and the development of that part of the cylinder between T
and the horizontal plane 4" below H. Develop the cylinder base on a line parallel
to the ground line and 4" below it, beginning at a point 10|" to the left of the
right hand border line. Begin with the shortest cylinder element and develop
to the right and above the base line. (State the problem in upper right hand
portion of sheet.)
736. [1] The line M(- 4", - IJ", - 4") N(- 4", - IJ", 0") is the axis of a
right cylinder whose upper base is an ellipse in H. The major and minor axes
of the ellipse intersect at the point N and are respectively 2|" and 1|" in length.
The major axis is parallel to the H trace of the plane S'(- ^i". + 30°, - 22^°)
which cuts the cylinder. Find the curve of intersection of the plane and the
cylinder, the true size of the curve when swung into H, and the development of
that part of the cylinder between H and T. Develop the cylinder base along the
ground line, beginning at a point (- 9f ") with the shortest element and develop
50
to the left and below the ground line. (State problem in upper left hand portion
of sheet.)
737. [1] The line M(- 3", - 4'', - 2") N(_ 3"^ o", - 2'0 is the axis of a right
cylinder, whose circular base of 2'' diameter is in V. The cylinder is cut by a
plane T(- 6'', + 45°, - 60°). Find the curve of intersection of the plane and the
cylinder, the true size of this curve, when swung into V, and the development of
that part of the cylinder between V and T. Develop the cylinder base along the
ground line, beginning at a point (-SI") with the shortest cylinder element and
developing to the left and above the ground line. (State problem in lower left
Iiand portion of sheet.)
738. [1] Profile plane at -10''. The line M(- 5'', - 2", - 2'') N(0'', - 2",- 2'')
is the axis of a right cylinder, whose circular base of 2J" diameter is in P. The
cylinder is cut by a plane T(- %'', + 120°,- 120°). Find the curve of intersection
of the plane and the cylinder, the true size of the curve when swung into H, and
the development of that part of the cylinder between P and T. Develop the cylin-
der base along the ground line, beginning at a point (-8^'') with the shortest
element, and developing to the right and above the ground line. (State problem
in lower rieht hand portion of sheet.)
4.— OBLIQUE CYLINDERS CUT BY RIGHT SECTION PLANES AND
DEVELOPMENT.
739. [1] (Place ground line 4i" below top edge of sheet.) The point
N(- 12'', + 2|:", 0") is the center of the circular base in H of an oblique cylinder
in the first quadrant. The diameter of this base is 3". The elements of the cylin-
der are such that their PI and V projections when extended makes angles of - 30°
and 4-60° respectively with the ground line. The cylinder is cut by a right section
plane R whose traces meet the ground line at (- 7^", 0", 0"). Find the H and
V projection of the curve of intersection of plane and cylinder, the true size of
the right section when swung into H, and the development of that portion of the
cylinder between H and T. (Develop cylinder in upper right hand portion of the
sheet, and state problem in lower right hand portion.)
740. [1] The point N(- 13", 0", - 1|") is the point of intersection of the major
and minor axes of the elliptical base in \ of an oblique cylinder in the third quad-
rant. These axes are 4" and 2" in length respectively and when extended make
angles of - 30° and - 120° with the ground line. The horizontal and vertical pro-
jections of the elements of the cylinder make angles of + 45° and - 30° with the
ground line respectively. The cylinder is cut by a right' section plane R, whose
traces intersect the ground line at (- 9", 0", 0"). Find the H and V projections
of the curve of intersection of plane and cylinder, the true size of the right sec-
tion when swung into V, and the development of that portion of the cylinder be-
tween V and T. (Develop cylinder in upper right hand portion of sheet. State
problem in lower right hand portion.)
741. [1] P at - 8|". The point N(- 5i", - 1^", - 5") is the center of the cir-
cular base of 3" diameter of an oblique cylinder. This base is in a horizontal
plane 5" below PI and the cylinder stands in the third quadrant. Its elements are
such that their H and V projections make angles of + 30° and - 135° respectively
•with the ground line. The cylinder is cut by a right section plane R whose traces
meet the ground line at (-4", 0", 0"). Find the H, V and P projections of the
cylinder and curve of intersection of the cylinder and plane, the true size of this
section when swung into H, and the development of that part of the cylinder
between its base and R. (Develop cylinder in upper right hand portion of sheet.
State problem in lower right hand portion.)
5.— OBLIQUE CYLINDERS CUT BY OBLIQUE PLANES AND
DEVELOPMENT.
743. [1] The line M(- 14", + l^", 0'') N(- IQi", + 3^", + 3i'0 is the axis of
an oblique cylinder, whose right section is a circle of 2" diameter. The cylinder
is cut by a plane T(- (i", - 13.5°, + 135°). Find the base of the cylinder in H,
the curve of intersection of the plane T and the cylinder, and the development of
that part of the cylinder between H and the plane T. ( Develop cylinder in upper
right hand portion of sheet; state problem in lower right hand portion.)
743. [1] The point N(- 12'', + If", 0") is the center of the circular base in
H of an oblic|ue cylinder. The diameter of this base is 3" and the cylinder stands
in the first quadrant. The H and V projections of the cylinder elements make
angles of - 150° and + 120° respectively with the ground line. The cylinder is
cut by two planes, one the plane T(- 15^", - 90°, + 45°) and the other a right
section plane R through the same point in the ground line. Find the projection
of the curves of intersection of the planes and the cylinder and the development
of that portion of the cylinder between T and R. (Develop cylinder in the upper
right hand portion of the sheet; state the problem in lower rig'ht hand portion.)
744. [1] The point N(- 13", - 1^", - 4") is the center of the circular base of
2" diameter of an oblique cylinder. This base is in a horizontal plane 4'" below
H and the cylinder stands in the third quadrant. The H and V projections of the
cylinder elements make angles of +45° and -120° with the ground line respec-
tively. The cylinder is cut by a plane T(- 15", + 90°, - 45°) and by a right sec-
tion plane R(- 13", +■ 135°, - 30°). Find the projection of the curves of inter-
section of the planes and the cylinder, and the development of that portion of the
cylinder between T and R. (Develop cylinder in upper right hand portion of
sheet; state problem in lower right hand portion.)
745. [1] The point N(- 11", - l-J", 0") is the point of intersection of the axes
of the elliptical base in H of an oblique cylinder which is located in the third quad-
rant. The major and minor axes of the top base are respectively 2" and 1^" in
length, the former being parallel to the ground line. The H and V projections of
the elements of the cylinder make angles of + 135° and - 120° respectively with
the ground line. The cylinder is cut by a plane T(-l()", + G0°, - 45°) and by a
right section plane through a point (- 14|-", 0", 0") in the ground line. Find the
curves of intersection of these planes and the cylinder, and the development of
that part of the cylinder between T and H. (Develop cylinder in upper right
hand portion of sheet, stating problem in lower right hand portion.)
746. [1] P at -9". llie line M(- 7", - 3r', - 3-^") N(0", - 1]", - 1^") is
the axis of an oblique cylinder situated in the third quadrant, whose base in P
is a circle of 2" diameter. This cylinder is cut by the plane T(- 7", + 60°, - 60°)
and by a right section plane R through a point (- 3^', 0", 0") on the ground line.
Find the curves of intersection between these planes and the cylinder, and the
development of that part of .the cylinder between T and P. (Develop cylinder
in upper right hand portion of sheet, state problem in lower right hand portion.)
52 .
6.— RIGHT CONES CUT BY PLANES AND DEVELOPMENT.
747. [1] The line M(- 12'^ + 2^, + ^D N(- 12'', + 2^", 0") is the axis of a
right circular cone in the first quadrant, whose vertex is at M and whose base of
4^" diameter is in H with its center at N. This cone is cut by a plane
T(- 9", - 90°, f 150°). Find the H and V projections of the curve of intersec-
tion of the plane and cone, the true size of this curve, and the development of the
cone, showing the intersection.
748. [1] The line M(- 13^", + 2^", + Sf") N(- 13^", + 2^", 0") is the axis of
a right circular cone, whose vertex is at M and whose circular base of 3" diameter
is in H with its center at N. The cone is cut by a plane T(- lO-]", - 112^°, + 150°).
Find the H and V projections of the curve of intersection of the plane and the
cone, the true size of this section when swung into H and the development of that
portion of the curve between the plane T and H. (Develop the cone in the right
hand portion of the sheet, placing the vertex at a point (-6^") on the ground
line. State problem in upper central part of sheet.)
749. [1] The line M(- 13|", + 2", + 4^") N(- 13|", + 2", 0") is the axis of a
right circular cone, whose vertex is at M and whose circular base of 3^" diameter
is in H with its center at the point N. The cone is cut by the plane
T(-10^", - 120°, + 150°). Find the H and V projections of the curve of inter-
section of the cone and the plane, the true size of this section when swung into
H, and the development of that part of the cone between H and T. (Develop
the cone in the right hand portion of the sheet, placing the vertex at a point
(- 5^") on the ground line. State problem in the upper central part of sheet.
750. [1] P at -or. The line M(- 4^", - 4i", - 1^") N(- 4^, 0", - If') is
the axis of a cone whose vertex is at the point M and whose base is an ellipse in
V, with axes 2|" and 1|" respectively in length, intersecting at the point N. The
major axis of this base is parallel to the ground line. The cone is cut by a plane
T(+ 1", + 157|°, - 150°). Find the three projections of the cone, and the curve
of intersection of the cone and plane, the true size of the section when swung into
V, and the development of that part of the cone between the vertex and T. (De-
velop cone in upper right hand portion of sheet, stating problem in lower right
hand portion.)
751. [1] P at -7i". The line M(- 6", - 3J", - 2") N(- 6", 0", - 2") is the
axis of a right circular cone whose vertex is at the point M and whose circular
base of 3f diameter is in V at the point N. The cone is cut by a plane
T(- 3",+ 150°, - 120°). Find the three projections of the cone, and the Hne of
intersection of the cone and plane, the true size of the section when swung into
H, and the development of that part of the cone between the vertex, the plane
T, and V. (Develop cone in upper right hand portion of sheet, stating problem
in the lower right hand portion.)
752. [1] P at -51". The line M(- 2f", - 2", - i") N(- 2g", - 2", - 4") is
the axis of a right circular cone, whose vertex is at the point M and whose circular
base of 3|" diameter is located in a horizontal plane 4" below H with its center at
the point N. The cone is cut by the plane T(- 5f , + 60°, - 45°). Find the
three projections of the cone and the line of intersection of the cone and the
plane, the true size of the section when swung into H, and the development of
that portion of the cone between its base and T. (Develop the cone in the left
hand portion of the sheet, placing the vertex at a point (- 6f ) on the ground
line. State problem in upper right hand part of sheet.)
753. [1] P at-9J/'. The line M(- 2f , - H", - J") N(- 2|", - 1^", -4") is
the axis of a right circular cone whose vertex is at the point M and whose circular
base of 2" diameter is located in a horizontal plane 4" below H with its center at
53
the point N. The cone is cut by the plane T(- 6'', + 45°, - 45°). Find the 3
projections of the cone and curve of intersection of cone and plane T, the true
size of this section when swung into H, and the development of that part of the
cone between its base and the plane T. (Develop cone in upper right hand por-
tion of sheet. State problem in lower right hand portion.)
754. [1] P at -11". The line M(- 3^'', - 3", - If'') N(- 2^', 0", - If") is
the axis of a right circular cone, whose vertex is at the point M and whose cir-
cular base of 2^" diameter is in V with its center at the point N. The cone is
cut by the plane T( oo', - 3-|", - of"). Find the 3 projections of the cone and the
curve of intersection between the cone and the plane, the true size of the curve
of intersection when swung parallel to P and the development of that part of the
cone between its base and the plane T. (Develop the cone in the upper right hand
portion of the sheet. State the problem in the lower right hand portion.)
755. [1] P at -11". The line M(-'2i", - If", 0") N(- 2-^", - If", - 4") is
the axis of a right circular cone, whose vertex is at the point M and whose circu-
lar base of 3" diameter is located in a horizontal plane 4" below H with its center
at the point N. The cone is cut by the plane T( co, - f", ^ 2"). Find the 3 pro-
jections of the cone and curve of intersection of cone and plane, the true size of
the section when swung into the plane of the cone base, and the development of
that part of the cone between its vertex, the plane T and the plane of its base.
(Develop the cone in the upper right hand portion of sheet; state problem in
lower right hand portion.)
756. [1] Profile plane at - 11". A right circular cone has its base of 3" diam-
eter in the profile plane, and its axis parallel to the ground line through the ver-
tex 0(-5",-2", -If"). It is cut by the plane T(- 5", + 60°, - 45°). Find
(1) the 3 projections of the curve of intersection of cone and plane, (2) the de-
velopment of the cone, showing this intersection.
757. [1] Profile plane at - 10". A right circular cone has its base of 3" diam-
eter in the profile plane, and its axis parallel to the ground line through the ver-
tex 0(-5". -2-i", -2^'). It is cut by the plane T(- 4^", + 60°, - 45°). Find
(1) the 3 projections of the curve of intersection of cone and plane, (2) the de-
velopment of the cone, showing this intersection.
7.— OBLIQUE CONES CUT BY SPHERES AND DEVELOPMENT.
758. [1] The point 0(- 10", - 3|", - 4^') is the vertex of an oblique cone and
the point M(- 13f", 0", - 2") the center of its circular base of 2^" diameter in
v. The cone is cut by a sphere of 6i" diameter whose center is at the vertex of
the cone. Find the curve of intersection and develop that part of the cone be-
tween the sphere and V. (Develop cone in upper right hand portion of sheet.
State problem in lower right hand portion.)
759. [1] The point 0(- 14", + 2f", + 2f") is the vertex of an oblique cone and
the point M(- lOf", + 3", 0") is the point of intersection of the axes of its ellip-
tical base in H. These axes are respectively 4f" and 3-|" in length, the major
axis being parallel to the ground line. The cone is intersected by a sphere of 5"
diameter whose center is at the vertex of the cone. Find the line of intersection,
and develop that part of the cone between the sphere and H. (Develop cone in
upper right hand portion of sheet; state problem in lower right hand portion.)
760. [1] The point 0(- 10", - 4^", - i") is the vertex of an oblique cone, and
the point M(- 13", - 2", - 4") the center of its circular base of 3" diameter, in
a horizontal plane 4" below H. The cone is cut by a sphere of 6" diameter
whose center is at the vertex of the cone. Find the curve of intersection and
develop that part of the cone between the sphere and its base. (Develop in upper
right hand portion of the sheet. State problem in lower right hand portion.)
54
8.— OBLIQUE CONES CUT BY PLANES AND DEVELOPMENT.
761. [1] The line M(- 8^, + 21'', + 4'') 0(- llf", + If", 0'') is the axis of
an obHque cone, whose vertex is the point M and whose circular base of 2^'
diameter is in H and its center at the point O. The cone is cut by a plane
T(- 15", - 55°, + 20°). Find the projections of the curve of intersection, the
true size of the section, and the development of the cone showing the curve of
intersection.
763. [1] The line M(- 12", + 2", + 3") 0(- 12|", + 2",-0") is the axis of an
oblique cone whose vertex is the point M and whose circular base of 3^" diameter
is in H with its center at the point O. The cone is cut by a plane
T(- 10", - 120°, + 150°). Find the projections of the curve of intersection, the
true size of this section, and the development of the cone, showing the curve of
intersection.
763. [1] The line M(- 10", - 4", - 4") 0(- 12", 0", - H") is the axis of an
oblique cone, whose vertex is the point M and whose circular base of 3" diameter
is in V with its center at the point O. The cone is cut by the plane
T(- 15", + 30°, - 60°). Find the projections of the curve of intersection, the true
size of the section, and the development of the cone showing the curve of inter-
section.
764. [1] The point M(- 15^', - 2^", - 4|") is the vertex of an oblique cone,
and the axes of its elliptical base in V intersect at the point 0(- 12", 0", -1^").
These axes are respectively 3" and 2" in length, the former being parallel to the
ground Hne. The cone is cut by a plane T(- 15^", + 22^°, - 52^°). Find the
projections of the curve of intersection, the true size of this section, and the devel-
opment of this cone showing the curve of intersection.
765. [1] The point M(- 15f", -y', - 1") is the vertex of an oblique cone
whose circular base of 2|" diameter is in a horizontal plane 4^" below H with its
center at the point 0(- 11^", - 3", - 4^"). The cone is cut by a plane
T(- 7|", + 120°, - 150°). Find the projections of the curve of intersection, the
true size of this section, and the development of that part of the cone between
its base and the plane T.
766. [1] The point M(- 15f", - 2|", - ^t") is the vertex of an oblique cone,
whose base is a circle of 4" diameter located in a plane parallel to and 4|'" below
H, with its center at the point 0(- 11|", - 2^", - 4^"). The cone is cut by a
plane T(- 8", + 90°, - 135°). Find the projections of the curve of intersection,
the true size and shape of this curve, and the development of the cone between
the vertex, the plane T and the plane of the base.
9.— SPHERES CUT BY PLANES.
767. [2] A sphere of 4" diameter is located in the first quadrant with its cen-
ter at the point 0(- 5|", f 2^", + 2|"). Find the curve of intersection of this
sphere with a plane R^- 1", - 135°, + 135°).
768. [2] A sphere of 3" diameter rests upon a horizontal plane in the third
quadrant at the point M(- 3|", - 2", - 3|"). Find the intersection of the sphere
with a plane R(- 5", + 60°, -120°).
769. [2] P at - 3y\ A sphere of 2|" diameter is located in the third quadrant
with its center at the point 0(- If", - 1|", - If"). Find the curve of intersec-
tion of this sphere and a plane R( oo, - 3", - 1^").
55
10.— ELLIPSOIDS.
Cut by planes; true size of curve of intersection.
770. [3] An ellipsoid of revolution is formed in the third quadrant by the ro-
tation about its vertical major axis of an ellipse whose axes are respectively 3"
and 2''. The ellipsoid is cut by a plane R which passes through the middle point
of its vertical axis and makes an angle of 30° therewith. Find (1) the curve of
intersection between the surface and the plane R, (2) the true size of this curve.
771. [2] An ellipsoid of revolution is formed in the third quadrant by the rota-
tion about its vertical minor axis of an ellipse whose axes are 3" and If respec-
tively. The ellipsoid is cut by a plane R which intersects the vertical axis at a
point ^" below its highest point and makes an angle of 45° therewith. Find
(1) the curve of intersection between the surface and the plane R, (2) the true
size of this curve.
772. [2] An ellipsoid of revolution is formed in the third quadrant by the
rotation about its vertical major axis of an ellipse whose axes are 4'' and 2^''
respectively. The ellipsoid is cut by the plane R which intersects its vertical axis
at a point V' below its highest point and makes angles of 60° and 30° respectivelv
with H and V. Find (1) the curve of the intersection between ellipsoid and
plane R, (2) the true size of this curve.
773. [2] An ellipsoid of revolution, axes 6'' and 3'' respectively, has its center
at the point 0(- 4|", + 2", O'O, and its long axis perpendicular to H. Find
(1) the intersection of the ellipsoid with a plane R(- 2^", - 90°, + 135°), (2) the
true size of this curve of intersection.
11.— PARABOLOIDS AND HYPERBOLOIDS OF REVOLUTION.
Intersection by planes; true size of curves of intersection; lines tangent to
intersection curves.
775. [2] The line A(-51^-2r,-D B(- 1", - 2^, - 1") is the directrix
and the point F(- 3^", - 2:^", - 1") the focus of a parabola which is the genera-
trix of a Paraboloid of Revolution, formed by rotating the parabola about its axis.
Find (1) the intersection of this paraboloid with a plane R(- 5^", + 67|°, - 45°),
(2) the true size of this curve of intersection, (3) a line tangent to the curve of
intersection at its lowest point.
776. [2] Find (1) the curve of intersection of the Paraboloid in Prob. 775
with a plane R, parallel to V and f" in front of the axis of the paraboloid, (2)
the true shape of this curve.
777. [2] A hyperbola, in a plane parallel to V, has its foci at the points
F(-4i^-2i",-lf'') and G(- 4y^ - 2|",- 3i'') and its vertices at the points
V(- 4:V', - 2V', - 2'') and W(- 4V', - UJ', - 3"). This hyperbola spins about its
vertical transverse axis and generates a Hyperboloid of Revolution. Find (1) the
curve of intersection of this surface with the plane R(- 5:|'', + 90°, - 67^°), (2)
the true size and shape of this curve, (3) a line PX tangent to the curve of inter-
section at the highest point of that portion of the curve on the lower nappe of the
surface.
778. [1] The line A(- 8^, - 2'', - 4") B(- 81'', - 2", 0'') is the directrix and
the point F(- 7|", - 2", - 2'') the focus of a parabola which is the generatrix of
a Paraboloid of Revolution with AB as its axis. Find (1) the intersection of this
paraboloid with a plane R(- 7", + 105°, - 120°), (2) the true shape of that
branch of the curve nearest the H trace of R.
56
12.— TORUS.
Intersection by plane; tangent line to curve of intersection at given point.
779. [2] A Torus is formed by the revolution of a circle of 1^" diameter about
the axis A(- 4f", + 2f ', + 4'0 B(- 4-f ', + 2-f' , 0''). The circle in its initial posi-
tion is parallel to V with its center at the point 0(- 6^", + 2|", + If ). Find
(1) the intersection of the torus with the plane R(- If", - 120°, + 120°), (2) a
line tangent to the curve of intersection at that point P on the upper portion of
said surface which is 1" in front of V.
780. [2] A Torns is formed by the revolution of a circle of If diameter about
the axis A(-3^^-2f ,-3'0 B(- 31'', - 2f , 0"). The circle in its initial po-
sition is parallel to V with its center at the point 0(- 4^", - 2|'', - If"). Find
(1) the intersection of the torus with the plane R(- 7f , + 60°, - 45°), (2) a line
tangent to the curve of intersection at that point P on the upper portion of the
surface which is 1^'' behind V.
781. [1] A Torus is formed by the revolution of a circle of 2" diameter, about
the axis A(- Q>V' , - 1^", - 4'') B(- Q>^" , - 1^', 0")- The circle in its initial posi-
tion is parallel to V with its center at the point 0(- 7f'', - 1|'', - 2''). Find
(1) the intersection of the torus with a plane R(- 13^", + 17|°, - 34J°), (2) the
true size of this curve of intersection.
782. [1] (G. L. par. to short edge of sheet.) A Torus is formed by the revo-
lution of a circle of 2]''' diameter about the axis A(3f , - 5", - 4|")
B(- 3f , 0'', - 4^"). The circle in its initial position is parallel to H with its
center at the point 0(- 6-J", - 3'', - 4^''). Find (1) the intersection of the torus
with a plane R(- 10^'', -f- 45°, - 60°), (2) a line tangent to the curve of intersec-
tion at its highest point.
13.— SQUARE RING.
Cut by plane ; true size of curve of intersection.
783. [1] The axis of a Square Ring is the line M(- 13'', + 2^', + |")
N(- 13", + 2^", + 3"). The. generatrix is a 1^" square with 2 sides parallel to
H, whose center moves in a horizontal circle of 3" diameter in a plane If" above
H. Find (1) the intersection of the ring with a plane T(- 8^", - 90°, + 157^°),
(2) the true size of this intersection when swung into H.
784. [2] The axis of a Square Ring is the line M(- 5^", + 2^", + ^")
N(- 5^", + 2^", + 3"). The generatrix is a 1|" square with 2 sides parallel to
H, whose center moves in a horizontal circle of 3" diameter in a plane If" above
H. Find the intersection of the square ring and a plane T(-^", - 150°, + 135°).
785. [2] The axis of a Square Ring is the line M(- 4^", - 2f", - f )
N(- 4:J", - 2f", - 2-J-"). The generatrix is a If" square with 2 sides parallel to
H, whose center moves in a circle of 3Y' diameter in a plane parallel to H and
If" below it. Find the intersection of this square ring with the plane
T(-7", + 45°,-45°).
14._HYPERBOLIC PARABOLOIDS.
Assuming elements ; intersection of surface by planes ; true size of intersec-
tion curves.
794. [2] Thelines M(-6", + f", 0") N(- 4", + 2", + 2") and P(- 2", + 1", 0")
Q(- 1", - 1:}", + If") are the directrices of a Hyperbolic Paraboloid whose plane
directer is H. Find the intersection of this surface with the plane
T(- 6^", - 60°, f 30°) and the true size of this intersection.
57
795. [2] The lines M(- 6", + £", 0'') N(- 4", + 2", + 2") and P(- 2'', + V, 0")
0(- 1'', - 1:^'', + If ) are the directrices of a HyperboHc Paraboloid whose plane
directer is H. Find the intersection of this surface with V.
796. [1] The two right lines M(- 131", + 4|'', + 3J-'') N(- ll^'', + 1'', 0'') and
P(-8i", 0",+3'') Q(-5V', + 5J:", 0") are the directrices of a Hyperbolic Para-
boloid, whose plane directer is D(- 13|", - 150°, + 120°). Find (1) nine ele-
ments of the first generation through points on the directrix MN which divide it
into 8 equal parts, (2) the projections of the intersection of the Hyperbolic Para-
boloid as determined by these elements with the plane T(- 11", - 60°, + 120°),
(3) the true size of this curve of intersection when swung into H.
797. [1] The two right lines M(- 12", - 5^", - |") N(- 6^:", - 3", - 4^") and
P(-llJ:",-l",-4") 0(-7f",-l",-l-f' ) are the directrices of a Hyperbolic
Paraboloid. The lines MQ and PN are elements of the first generation. Find
(1) thirteen elements of the first generation through points on the directrix PQ
which divide it into 12 equal parts, (2) the projections of the intersection of the
Hyperbolic Paraboloid as determined by these elements and the plane
T(- 6", + 1574°,- 90°), (3) the true size of this curve of intersection when
swung into H.
798. [1] The two right lines M(- 12|", - 5", - 3") N(- 12J-", 0", - 3") and
P(- 8|", 0", - li") Q(- 5i", - 5", - 4^-") are the directrices of a Hyperbolic Par-
aboloid whose plane directer is V. Find (1) eleven elements of the first genera-
tion through points on the directrix MN which divide it into ten equal parts, (2)
the projections of the intersection of the Hyperbolic Paraboloid as determined by
these elements, with the plane T(- 13^", + 45°, - 45°), (3) the true size of this
section when swung into V.
799. [1] The two right lines M(- 13", 0", - 1") N(- 9", - 3^", - 3^") and
P(- 7", 0", - 2") Q( - 5^", - 3i", - 2i") are the directrices of a Hyperbolic Par-
aboloid whose plane directer is V. Find (1) nine elements of the first generation
through points on the directrix MN which divide it into 8 equal parts, (2) the
projections of the curve of intersection of the Hyperbolic Paraboloid and the plane
T(- 81", + 135°, -60°), (3) the true size of this curve of intersection.
800. [2] The two right lines A(- 6|", - I", - i") B(- 5", - 2", - 4") and
C(-3",-J:",-3y') D(-|",-4",-2") are directrices of a Hyperbolic Parabo-
loid, of which BC and AD are elements. Find 13 other elements of the same gen-
eration and the intersection of the surface with the plane T(- 4^", + 90°, - 70°).
801. [2] The two right lines A(- 61", - i", - i") D(- i", - 4", - 2") and
B(-5", -2",-4") C(-3", -Jr", -3^') are directrices of a Hyperbolic Parabo-
loid of which AB and CD are elements. Find seventeen other elements of the
same generation and the intersection of this surface with the plane
T(-6", + 90°,-45°).
802. [2] The two right lines A(- 6i", - V', - f) B(- 5", - 2", -4") and
C(-3", -J-/', -3^") D(-^", -4", -2") are directrices of a Hyperbolic Parabo-
loid of which AC and BD are elements. Find thirteen other elements of the same
generation and the intersection of the surface with a plane T(- 6", + 90°, - 60°).
803. [2] The two right lines A(- 6^', - i", - i") B(- 5", -4", - 2") and
C(-3", -3|", -i") D(-^", -2", -I") are directrices of a Hyperbolic Parabo-
loid of which AC and BD are elements. Find thirteen other elements of this
same generation, and the intersection of the surface thus determined with a plane
T(-7i", + 60°,-60°).
58
15.— HYPERBOLOIDS OF REVOLUTION OF ONE NAPPE.
Assuming elements; intersection by oblique planes.
804. [1] The line A(- 9'', - 3^'', - 2^) B(- 9'', 0'^ - 2i'0 is the axis of a Hy-
perboloid of Revolution of one Nappe, located in the third quadrant with its base
in V. The generatrix in its initial position is M(- 11", 0", - If")
N(- 9'', - If", - 1|"). Find (1) 32 elements of one generation of the surface,
which pierce V at equal intervals around the circumference of its base, (2) the
projections and true size of the curve of intersection between the surface and a
plane T(- 14^", + 25°, - 45°).
805. [1] The line A(- 11", f 2f' , + 5") B(- 11", + 2|", 0") is the axis of a
Hyperboloid of Revolution of one Nappe which stands in the first quadrant with
its base in H. The generatrix in its initial position is M(- 13^', + 2", 0")
N(- 11", + 2", + 2|"). Find (1) 32 elements of one generation of the surface
which pierce H at equal intervals around the circumference of its base, (2) the
vertical projection of the meridian section parallel to V, (3) the projections and
true size of the curve of intersecti on between the surface and a plane
T(-12|",-75°, + 45°).
806. [2] The line A(- 5", - 2", - 4") B(- 5", - 2", 0") is the axis of a Hyper-
boloid of Revolution of one Nappe located in the third quadrant with its upper
and lower bases respectively in H and in a horizontal plane 4" below H. The
generatrix of this surface is, in its initial position, the line M(- 6^", - 1^", -4")
N(- 5", - 1^", - 2"). Find (1) 32 elements of one generation of the surface,
which pierce H at equal intervals around the circumference of its base, (2) the
projections and true size of the curve of intersection of the surface and a plane
T(-10.85", f 90°,-120°).
807. [2] A Hyperboloid of Revolution of one Nappe stands in the 1st quad-
rant with its base of 4" diameter in H at the point 0(- 3^", + 2i", 0"). The
circle of the gorge of 2|" diameter is in a horizontal plane 1-J" above H, with its
center directly above O. Find 32 elements of the surface and the intersection of
the surface thus determined with the plane T(- 8", - 54°, + 23°).
808. [1] (G. L. par. to short edges of sheet.) A portion of a Hyperboloid of
Revolution of one Nappe is generated by the revolution of the line
M(-6f", hlf", + f") N(-4y',-4f", + 5i) about the line A(- 5|", + 3", + |")
B(- 5^", + 3", + 5^") as an axis. Assume 32 elements of the surface and find
the intersection of the surface thus determined with plane T(- 10|", -45°, + 45°).
809. [1] (G. L. par. to short edges of sheet.) Find the intersection of the
surface determined in the above problem with a plane R(- 10|", - 60°, + 39°).
810. [1] (G. L. par. to short edges of sheet.) Find the intersection of the sur-
face determined in the above problem with a plane S(- 10", - 52°, + 38°).
16.— HELICOIDS.
Oblique and right. Assume elements; intersections with oblique planes,
or with H or V ; true shape of curves of intersection.
811. [1] The line A(- 12", + 2", + 4") B(- 12", + 2", 0") is the axis of an
oblique Helicoid ; the generatrix in initial position is M(- 13^", + 2", 0")
N(- 12", + 2", + H") which moves in such a way that its horizontal projection
appears to rotate counter-clockwise. The pitch of the helix which is generated
by the point M is 2" . Find (1) sixteen elements of the surface generated during
one complete movement of the generatrix about the axis, (2) the intersection of
the surface thus determined with the H plane of projection, (3) the projections
\
59
and true size of the curve of intersection between the surface thus determined and
the plane T(- 9f', - 90°, + 150°).
812. [2] The line A(- 4^, - If, - 4'') B(- 4^", - If", 0") is the axis of an
oblique Helicoid, whose generatrix in initial position is M(- 5-|", - If", - 4")
N(-4^", -If , - lij")- This generatrix moves in such a way that its H projec-
tion appears to rotate clockwise and the point M generates a helix whose pitch
is 2 1". Find (1) sixteen elements of the surface generated during 1 complete
movement of the generatrix about the axis, (2) the intersection of the surface
thus determined with a horizontal plane 4" below H, (3) the projections and true
size of the curve of intersection between the surface thus determined and the
plane T(- 8f ', f 90°, - 30°).
813. [1] The line A(- lOV', ~ 4", - 2") B(- lOi", 0", - 2") is the axis of a
Right Helicoid whose generatrix in its initial position is determined by the points
M(- 9", 0'', - 2") and N(- 10^'', 0", - 2")- This generatrix moves in such a way
that its V projection appears to rotate clockwise, and the point M generates a
helix whose pitch is 2". Find (1) 32 elements of the surface, generated while
the generatrix swings about the axis twice, (2) the projections of the curve of
intersection between the surface thus determined and plane T(- 7", + 135°, - 135°).
Determine and mark the asymptotes to the projections of the intersection curve
in both H and V.
814. [1] The line A(- 9'', - 2-^, 0") B(- 9'', - 2^", - 5'') is the axis of a
Right Helicoid, whose generatrix in its initial position, is determined by the points
M (- lir', - 2|", - 5'') and N (- 9", - 2^', - 5") • This generatrix moves in such
a way that its H projection appears to rotate counter-clockwise, while the point M
generates a helix whose pitch is 2-^". Find (1) sixty-four elements of the surface,
generated while MN swings about the axis twice, (2) the curve in which the sur-
face thus determined intersects the vertical plane of projection. Determine and
mark the asymptotes to this curve.
815. [1] (G. L. par. to short edges of sheet.) An Oblique Helicoid is given
in Prob. 610. Find its intersection with H, and with a plane T(- 10",- 67^°,+ 30°).
816. [2] Find the intersection of the Oblique Helicoid of Prob. 611, (a) with
the PI plane of projection, (b) with a horizontal plane 1^^" above H.
817. [2] The cast iron marking post in Fig. 46 is to be cut off by a plane T,
which intersects the axis 1' - !{/' below the top of the post and makes an angle
of 30° with said axis. Find the projections of the curve of intersection and the
true size and shape of the section formed. Scale, 1^-" = I'-O"..
818. [2] One side of the marble pillar cap of Fig. 47 was injured and in order
to use the cap as an ornament in another place, it was cut off by the plane T,
which intersects the axis at a point 2' - 8" above the bottom plane of the cap and
makes an angle of 30° with the axis. With scale, f" = 1' - 0", find the projections
of the intersection, and the true size of the section formed.
819. [2] Assume a convenient scale for the mortar shown in Fig. 48 whose
outer surface is an Hyperboloid of revolution of one nappe and whose interior
is partly conical, partly spherical as indicated. Find the projections of the mortar,
and the true size of the intersection of the mortar with an oblique plane cutting
the axis at a convenient angle.
820. [1] G. L. par. to short edge of sheet. The cast iron snubbing post shown
in Fig. 46 is cut by a plane T(- 18'6'', + 67|°, - 60°). Find the line of intersec-
tion. The axis of the post is" the line A(- 11' 0", - 8' 6", 0)
B(- 11' 0", - 8' 6", - 16' 0") and its base stands on a plane which is 14' 4^" be-
low H. Draw to scale of 2" = 1' 0".
6o
SHORTEST DISTANCE BETWEEN POINTS ON SURFACES.
Prisms and Pyramids.
825. [1] Find the shortest distance on the surface from the point
A(- 14|'', y, - 2|'0 on the front to the point B(- 13", y, - i") on the back of the
right prism of Prob. 725.
826. [1] Find the projections of the shortest Hne that can be drawn on the
surface from the point on the back of the right prism of Prob. 726 vertically pro-
jected at A(- 14", 0'', - 3i") to the point on the front vertically projected at
.B(-12r,0",-2").
827. [1] Find the shortest distance on the surface from the point
K(-3f",-t-li", z) to the point L(- 4^", + 2", z) lying on diflferent faces of the
oblique prism of Prob. 727.
828. [1] Find the projections of the shortest line lying on the surface of the
right pvramid of Prob. 730, joining the points on its surface horizontally projected
at A(- 3|", + 2f' , 0") and B (- 4-|", + If', 0") .
829. [1] Find the shortest distance along the surface from the point
C(-lli",-f", z) to the point D(- 13^', - 1^", z) on the surface of the oblique
pyramid of Prob. 731.
830. [1] Find the projections of the shortest line that can be drawn on the
surface of the oblique pyramid of Prob. 732 from point E(- 13^", y, - 1^") to
F(- 12f", y, - If") on the front face.
Cylinders and Cones.
831. [1] Find the projections of the shortest line that can be drawn on the
surface from the point A(- 15", y,+ 2") on the front to the point B(- 13", y,+ |")
on the back of the right cylinder of Prob. 734.
832. [1] Find the shortest distance on the surface of the right cylinder of
Prob. 736 from the point C(- 4^", y, - f") on the back of the surface to the point
D(-3|", y,-H") on the front of the surface.
833. [1] P at - 10". Find the shortest distance on the surface of the right
cylinder of Prob. 738 from the point E(- 3", - 2§", z) on the bottom to point
F(- I", - li", z) on top of the cylinder.
834. [1] Find the projections of the shortest line that can be drawn on the sur-
face of the oblique cylinder of Prob. 739 from the point A(- 12", + 3", z) on the
top to the point B(-9|-", + 2f", z) on the bottom of the cylinder.
835. [1] Find the shortest distance along the surface from the point
A(-10i",y, -2^") on the front to point B(- 12f' , y, - 2") on the back of the
oblique cylinder of Prob.. 740.
836. [1] P at -8^". Find the projections of the shortest Hne that can be
drawn on the surface of the oblique cylinder of Prob. 741 from the point
C(-5i", -2|", z) on the top to point D(- 3^", - If ', z) on the under side of
the surface.
837. [1] Find the shortest distance from point A(-10f , + 1", z) on the top
to point B(-12f', + 3i", z) on the bottom of the oblique cylinder of Prob. 743.
838. [1] Find the projections of the shortest line that can be drawn on the
surface of the right cone given in Prob. 747 from C(-.10f ', + 2f", z) to
D(-12f', + 4^z).
6i
839. [1] P at -of. Find the projections of the shortest hne that can be
drawn on the surface of the right cone of Prob. 752 from point E(- 3|", - 1^", z)
to point F(-2'', -2V', z).
840. [1] P at -11". Draw the projections of the shortest hne that can be
drawn on the surface of the right cone of Prob. 756 from point A(- I" , - 2%" , z)
on the top to point B(- 3^", ~ If", z) on the under side of the surface.
841. [1] Find the shortest distance on the surface from point C(- 12^", y, - 3")
on the front to point D(- llf , y, - 2|") on the back of the obHque cone of
Prob. 758.
842. [1] Find the shortest distance on the surface of the obHque cone of Prob.
759 from the point E(- lOf , + 2", z) on the top to point F(- 13^", + 3", z) on
the bottom of the surface.
Spheres.
843. [2] Find the projections of the shortest hne that can be drawn on the
surface of the sphere of Prob. 767 from point A(- 6", y, + 1|") on the front to
point B(- 4|", y, + 31") on the back of the surface.
844. [2] Find the shortest distance on the surface of the sphere of Prob. 768
from point C(- 4", - 1^", z) on the top to point D(- 2f", - 2f', z) on the bottom
of the surface.
845. [2] P at - 3^". Find the projections of the shortest Hne that can be
drawn on the surface of the sphere of Prob. 769 between point E(- 2|", y, - If")
on the front and point F(- 1^", y, - 2^") on the back of the surface.
II. INTERSECTIONS OF TWO SURFACES.
1.— INTERSECTION OF TWO CONES.
Intersection curves ; line tangent to said curve, or development of one cone.
850. [1] (Take ground line parallel to short edge of sheet.) The axis of an
oblique cone is the line A(- 3^'', + f", + 6") B(- 6f", + 3^'', 0'')- The base of
^this cone is an ellipse in H with center at B, and axes 3^'' and 2^" respectively,
the latter being at right angles to the H projection of the cone axis. The line
C(-6f", + 2^'', + e3'^'') D(-2|", + 2f'', 0") is the axis of another oblique cone,
whose circular base of 3f diameter is in H with its center at D. Find (1) the
intersection of these 2 cones, (2) a line tangent to the intersection at some con-
venient point.
851. [1] The line A{-7V',- -y',0'') B(- IH", - 2", - 4") is the axis of the
oblique cone whose vertex is at A and whose circular base of 3" diameter is in a
horizontal plane 4" below H with its center at B. The line C(- 10", 0", - 1^-")
D(- 7|'', - 3", - 4") is the axis of another cone, whose vertex is at C and whose
circular base of 3" diameter is in the same horizontal plane, with its center at D.
Find (1) the intersection of these 2 cones, (2) a line tangent to the curve of inter-
section at a point of that element of cone AB which is parallel to V.
852. [1] A right circular cone stands in the third quadrant on a horizontal
plane 5" below H, with the center of its circular base (diameter 3") at the point
B(-ll'',-2",-5")- Its altitude is 4V'. The line C(- 12", - 1^", - 1|'0
D(- 9|", - 2^", -5") is the axis of an intersecting oblique cone, whose circular
base of 4" diameter is in the same horizontal plane, with its center at the point D.
Find (1) the intersection of these two cones, (2) the development of the
right cone showing the intersection.
853. [1] A right circular cone of 32" altitude stands on H with the center
of its circular base of 2|" diameter at the point B(- 13", + 2", 0"). It is inter-
sected by an oblique cone whose vertex is at the point C(- 15^", + 2", + 5") and
whose circular base of 3" diameter is in H, with its center at D(- 12|", + 2", 0").
Find (1) the intersection of the two cones, (2) the development of the right
cone, showing this intersection.
854. [1] P at -6". A right circular cone has its circular base of 3" diameter
in the profile plane and its axis parallel to the ground line through the vertex
0(- 5", - 2", - 2"). Another right circular cone has its base of 2V^ diameter
these two cones.
855. [1] The line A(- Hi", + 4i", + 1") B(-7i", 0", + 3i") is the axis
of a cone whose vertex is at A and whose base is a 3^' circle in V with center at B.
Line C(-lli", + li", !-4i") D(-7i", -hlf", 0") is the axis of a second cone
whose vertex is at C and base a 2V' circle in H with center at D. Find the curve
of intersection of the two cones.
63
2.— INTERSECTION OF TWO CYLINDERS.
Intersection curves, development of one cylinder or line tangent to curve of
intersection.
850. [1] One cylinder cuts clear through another. The line
A(-10^ + 3^ + 2i'O B(-7^0", + 3i") is the axis of a cylinder whose circular
base of 2^' diameter is in V with its center at the point B. The line
A M(- nf, + 1^'', 0'') is the axis of another cylinder whose axis intersects
that of the first cylinder at A and whose circular base of IJ" diameter is in H
with its center at the point M. Find (1) the intersection between these two
cyHnders, (2) the development of the larger cylinder showing the intersection.
857. [1] The line A(- 11'', - 4^', - 1^") B(- 14i", - 2-]", - 4f") is the axis
of a cylinder whose circular base of 3'' diameter is in a horizontal plane
4|'' below II with its center at the point B. A second cylinder has the line
C(-14",-5",-f') D(-ll",-lf',-4f') for its axis and its circular base of
2|:'' diameter is in the same horizontal plane as the base of the first, with its center
at the point D. Find (1) the intersection of the two cylinders, (2) the develop-
ment, showing this intersection, of the larger cylinder between the base and a
right section taken above the intersection of the two cylinders.
858. [1] A line AB passing through the point A(- 12|", + 3'', + 3") and
making angles of 45° with H and 30° with V respectively, is the axis of an
oblique cylinder whose base of 1^" diameter is in H. A line D(- 12|", + 3", + 2")
E(-9f", + 1'', 0") is the axis of another cylinder whose circular base of 3|''
diameter is in H with its center at E. Find (1) the intersection between these
cylinders, (2) the development of the smaller cylinder.
859. [1] The line A(- 8^'', - 3", - 4") B(-ll", 0", - l^") is the axis of an
oblique cylinder whose circular base of 2" diameter is in V with its center at B.
The line C(- 8;^, - 2", - 3") D(-6", 0",-l") is the axis of another cylinder
whose circular base is 3V' in diameter, and is in V with its center at D. Find
(1) the intersection of these two cylinders, (2) a line tangent to the intersection
at a point on the highest element of the smaller cylinder.
860. [l](Take the ground line parallel to short edge of sheet.) The line
A(- 5^", + }", + 5") B(- 7V', + 7", 0") is the axis of a cylinder whose circular
base of 2rr' diameter is in H with its center at B. The line C(- 7^", + l^", + 4f' )
D(- 31'', + 5", 0'') is the axis of another cylinder whose base is an ellipse in H
with its center at D. The axes of this ellipse are 4|" and 3^" respectively and the
minor axis is at right angles to the H projection of the cylinder axis. Find
(1) the intersection of the two cylinders, (2) a line tangent to this intersection
at some convenient point P.
861. [11 Partially cutting. The line A (-13",- 4|",-li") B(- 10^", 0",- l:?-")
is the axis of a cylinder of circular right section of 2" diameter whose base is in
V. The line Cf-ll", -3f", - If) D(- 14f", 0", - 2f") is the axis of another
cylinder whose circular base of 2" diameter is in V with its center at D. Find
(1) the intersection between these two cylinders, (2) the development of the first
mentioned cylinder.
862. [1] The line A(- 10", - U", 0") B(- 14^', - 1V\ - 4^') is the axis of an
oblique cvlinder, whose circular base of 2" diameter is in a horizontal plane iV'
below H with its center at the point B. The line C(- 13f", + 2^", 0")
D(- 11", - 2:1", - 4J") is the axis of another cylinder whose circular base of 4".
diameter is in the same horizontal plane with its center at D. Find (1) the inter-
section of these 2 cylinders, (2) the development of the smaller cylinder.
04
8()3. [2 I A right circular cylinder in the 3rd quadrant has its upper base of 3"
diameter in H with its center at the point A(- 6", - 1^", 0") and its axis per-
pendicular to H. Another cylinder has a circular tapper base of 3" diameter in
H and its center at the point B(- 7", - 3", 0"). Its axis is in a plane parallel
to V and the vertical projection of this axis makes an angle of 45° with that of
the axis of the first cylinder. Find (1) the intersection of these 2 cylinders,
(2) the development of the first cylinder, showing the intersection.
864. [2] A vertical cylindrical pipe, diameter 3", intersects a horizontal cylin-
drical pipe, diameter 1-J". Their axes are 1" apart. Find the projections of the
curve of intersection, and develop one cylinder, showing this intersection.
8()5. [2] A horizontal cylindrical pipe, diameter 25", is joined by vertical cyl-
indrical pipe, diameter 1 Y\ Their axes are ^" apart. Find the projections of the
intersection curve, and develop one cylinder, showing this intersection curve.
86(). [2] Draw the projections of a 135° elbow for a 2" sheet iron pipe and
develop one branch.
867. [2] Draw the projections of a Y intersection for a 5" drain pipe and
develop one branch (Scale, half size).
3._INTERSECTION OF CONE AND CYLINDER.
Curves of intersection; line tangent to said curves, or development of one
surface.
868. [1] (Take ground line parallel to short edge of sheet.) The point
A(- 8^", + ^", + 6|") is the vertex of an oblique cone whose elliptical base is in
H with its center at the point B(- 3^", + 5|", 0''). The axes of this base are 5"
and 4" respectively, the minor axis being at right angles to the H projection of
the axis AB of the cone. The line C(- 5", + i", + 5^") D(- 6^", + 6", 0") is the
axis of an oblique cylinder whose base of 2" diameter is in H with its center at D.
Find (1) the intersection between the cone and cylinder, (2) a line tangent to the
intersection curve at some convenient point.
869. [1] The point A(- 10", - 4^", - 5^") is the vertex of a cone which ex-
tends downward into the third auadrant from a circular base of 5" diameter in
H with its center at the point B(- 5^', - 2f' , 0"). The line C(- 7", - 5", - 2^")
D(- 12", 0", - 2-i") is the axis of a cylinder whose circular base of 3" diameter
is in V at the point D. Find (1) the intersection between cone and cylinder,
(2) a line tangent to the intersection curve at some convenient point.
870. [1] The point A(- 9", - 5", - 1") is the vertex of an oblique cone in the
third quadrant, whose circular base of 3" diameter is in V with its center at
B(-14", 0",-3f"). The line C(- 13^", - 5|", 0") D(- 10", 0", - 3^") is the
axis of an oblique cylinder whose circular base of 3" diameter is in V with its
center at the point D. Find (1) the intersection between cone and cylinder,
(2) the development of the cone, showing the intersection.
871. [1] An oblique cone has its vertex at the point A(- 9|", - 3|", - 5") and
its circular base of 4" diameter in V with its center at the point B(- 13^, 0, - 2^").
An intersecting oblique cylinder has the line C(- 14^", - 3", - 4|")
D(- 9^", 0", - 2") as axis, its circular base of 2" diameter being in V with its
center at the point D. Find (1) the intersection between cylinder and cone,
(2) the development of the cylinder, showing this intersection.
872. [1] The point A(- 11", - 4^", - 3^") is the vertex of an oblique cone in
•the 3rd quadrant whose circular base of 3" diameter is in V with its center at the
point B(-13^", 0",-2"). The line C(- 13^', - 3", - 4^') D(- 9|", - H", 0")
is the axis of an oblique cylinder whose circular base of 2" diameter is in H with
6s
its center at the point D. Find (1) the intersection of cone and cyhnder, (2) the
development of the cyhnder showing the intersection.
873. [11 An obhque cyhnder in the 3rd quadrant has for its axis the hne
A(-2", -V', -i'O B(-6|'',-2'', -5^') ; its circular base of 3" diameter is lo-
cated in a horizontal plane 5'' below H with its center at the point B. An inter-
secting oblique cone has its circular base of 2Y' diameter in the same horizontal
plane with center at a point D(- 3:|", - 2f", - 5'') ; its vertex is at the point
C(- 5|", - ^".. - f ). Find (1) the intersection of cone and cylinder, (2) the
development of the cylinder, showing this intersection.
874. [1] A right circular cone stands in the third quadrant on a horizontal
plane 5" below H. Its circular base of 3^" diameter has its center at the point
B(- 4'', - 2^/', - 5") its vertex being the point A(- 4", - 2|", - V')- An oblique
cylinder has the line C(- 6y',-V',- V') D(- 3f', - 3", - 5") "as axis, with its
circular base of 2" diameter in the same horizontal plane as the cone, with its
center at the point D. Find (1) the intersection of cone and cylinder, (2) the
development of the cone, showing this intersection.
875. [1] A right circular cone stands in the third quadrant on a horizontal
plane 5" below H. Its circular base of 3" diameter has its center at the point
B(- 4", - 3V', - 5"), its vertex being the point A(- 4", - 3^', - ^"). An oblique
cylinder has for its axis the line C(- 2'', - 4^", - If) DC"- 71", - H", - 5") its
circular base of 2V' diameter being in the horizontal plane above mentioned with
its center at the point D. Find (1) the intersection of concand cylinder, (2) the
development of the cone, showing this intersection.
876. [1] An oblique cone has its vertex at the point A(- 13|", + f', + of") and
its circular base of 2|" diameter in H wnth its center at point B(- 9§", + 'iV', 0").
An intersecting oblique cylinder has for its axis the line C(- 11", + :|", + 3^')
D(- 13", + 3:5",0"), with its circular base of 2-^-" diameter in H with its center
at the point D. Find (1) the intersection between the cone and cylinder, (2) the
development of the cylinder showing this intersection.
877. [1] (Take ground line parallel to shorter edge of sheet and i" above
middle of sheet.) The line A(- 6-J", + 6|", + 7") B(- 3-1", + 3,f', 0") is' the axis
of an oblique cylinder whose circular base of 2" diameter is in H with its center
at the point B. An intersecting oblique cone has its vertex at the point
C(- 2f", + 7", + 5^") and an elliptical base in H with its center at the point
D(- 7f", f 3", 0"). The axes of this ellipse are respectively 5" and 4", the major
axis being at right angles to the FI projection of the cone axis CD. Find (1)
the intersection of cone and cylinder, (2) a line tangent to the intersection at a
point P which is 3^ from H, on that part of the curve of intersection which is
visible in the V projection.
878. [1] (Ground line parallel to short edges of sheet, and one inch above the
middle of sheet.) An oblique cone has its vertex at the point A(- 8", + V\ + G-J")
and the center of its elliptical base in H at the point B(- 3", + 5^", 0"). The
axes of this ellipse are respectively 5" and 4", the minor axis being at right angles
to the H projection of the cone axis AB. The line C(- 5", + 11", + 5|")
D(- 6^", + 7", 0") is the axis of an intersecting oblique cylinder whose circular
base of 2|" diameter is in H with its center at the point D. Find (1) the inter-
section of cylinder and cone. (2) a line tangent to this intersection at some con-
venient point.
879. [1] (Ground line parallel to short edges of sheet and 1^" above middle
of sheet.) An oblique cone has its vertex at the point A(- 7^^", - 6-|", - 1") and
the center of its elliptical base in V at the point B(- 2$", 0", - 5"). The axes of
the ellipse are respectively iV' and 3f", the minor axis being at right angles
66 INTERSECTION OF CONE AND CYLINDER
to the V projection of the cone axis AB. The Hne C(- 4^'', - 5", -^")
D(- 6^'', 0'', - 7'') is the axis of an intersecting obHque cyHnder whose circular
base of 2-Y' diameter is in V with its center at the point D. Find (1) the inter-
section of the cone and cyHnder, (2) a line tangent to this intersection at some
•convenient point.
880. [1] Line A(- 14^", + 3'', + 3") B(- 14^", + 2'', 0") is the axis of a right
circular cone vertex at A and center of 3'' diameter at B. Line
C(-U-y', + 3Y', + ^¥') D(-11J", 0", + 11") is the axis of a right circular cyl-
inder of 2" diameter. Find (1) the base of cylinder in V, (2) the intersection
between the two surfaces.
88L [1] An oblique cone has its vertex at the point A(- 9V', - SV', - 5") and
its circular base of 4" diameter in V with its center at point B(- 13|'', 0", - 2^").
An intersecting oblique cylinder has the line C(- 14^'', - 3", - 4f")
D(- 9^", 0", - If") as axis, its circular base of 2" diameter being in V with its
center at the point D. Find (1) the intersection between cylinder and cone,
(2) the development of the cone, showing this intersection.
882. [1] The point A(- 11", - 4|", - 3^") is the vertex of an oblique cone in
the third quadrant whose circular base of 3" diameter is in V with its center at
point B(- 131", 0", - 2"). The line C(- 13|", - 3^", - 4|") D(- Of", - H", 0")
is the axis of an oblique cylinder whose circular base of 2" diameter is in H with
its center at the point D. Find (1) the intersection between cone and cylinder,
(2) the development of the cone showing this intersection.
883. [1] The line A(- 7-J", + 4i", + 3|") B(- 4", 0", + 2|") is the axis of an
oblique cone, whose vertex is at A and whose circular base of 3f " diameter is in
V with its center at B. An intersecting cylinder has for its axis the line
C(-5", + 4", + 4") D(-:)", ^If", 0") its circular base of 3" diameter being in
H with its center at the point D. Find (1) the intersection of cone and cylinder,
(2) a line tangent to this intersection at a convenient point.
884. [1] A right circular cone stands in the third quadrant on a horizontal
plane 5" below H, with the center of its circular base of 3|" diameter at the point
B(-12", -2^", -5"). Its altitude is 4|". An obHque intersecting cylinder has
as axis the line C(- lo|", - f", - f") D(- 11^", - 3^", - 5"), and has its circular
base of 3" diameter in the same plane as the cone base with its center at the
point D. Find (1) the curve of intersection of cone and cylinder, (2) the devel-
opment of the cone, showing this intersection.
885. [1] A right circular cylinder and a right circular cone have vertical axes,
and bases in a horizontal plane 44" below H. The cone base is of 4" diameter
with center at the point B(- 13|", - 2^", - 4-i") ; the cylinder base is of 2" diam-
eter, with center at the point D(- 14f", - 1-|", - 4^") ; the cone altitude is 4",
Find (1) the intersection of cone and cylinder, (2) the patterns (developments)
for cone and cylinder, showing the intersection in each case.
886. [1] A right circular cylinder and a right circular cone have horizontal
axes, and bases in a vertical plane 4" behind V. The cone base is of 4:|" diame-
ter, with center at the point B(- 13", - 4", - 2|") ; the cylinder base is of 1^"
diameter, with center at the point D(- 14^", - 4", - 2") ; the cone altitude is 3|".
Find (1) the intersection of cone and cylinder, (2) the patterns ( developments)
for cone and cvHnder, showing the intersection in each case.
67
4.— GENERAL INTERSECTIONS OF TWO SURFACES.
Including combinations of single-curved, warped, and double-curved sur-
faces.
887. [1] A helical convolute is formed by drawing tangents to a helix, whose
axis is the line M(- 13", + l^'', + 4") N(- 13", + l^", 0"), and which is gener-
ated by a point which starts from a point 0(- 14", + 1^", 0") in H and moves
in such a way that its horizontal projection appears to revolve counter-clockwise.
The pitch of the helix is 4". An oblique intersecting cone has its circular base of
If" diameter in V with its center at a point B(- 10", 0", + ^") and its vertex at
A(- 13", + 3f", + If"). Find (1) the intersection of cone and helical convolute,
(2) the development of the helical convolute, showing this intersection.
888. !"1] A hemisphere of 4" diameter stands on H in the first quadrant with
its center at the point 0(- 10^", + 2^, 0"). It is cut by an oblique cylinder
whose axis is the line A(- 7^', + 5^", + 3^") B(- 11", + 2", 0") and whose cir-
cular base of 2^" diameter is in H with its center at B. Find (1) the intersection
of hemisphere and cylinder, (2) a line tangent to the curve of intersection at some
convenient point.
889. [1] A hemisphere of 4-^-" diameter stands in the third quadrant on a hor-
izontal plane 5" below H, with the center of its base at the point
0(- llg^", - 2f", - 5"). It is cut by an oblique cylinder, whose axis is the line
A(- 9", - 2", - 1") B(- 13", - 2", - 5") and whose circular base of 2|" diameter
is in the above horizontal plane with its center at the point B. Find (1) the inter-
section between hemisphere and cylinder, (2) the development of cylinder, show-
ing the intersection.
890. [1] A sphere of 4" diameter is in the third quadrant with its center at
the point 0(- 9", - 2^", - 2^"). An oblique intersecting cylinder has the line
A(- 6f", - 2Y', - i") B(- 11-i", - 21", - 5") as axis, and its circular base of 2^"
diameter is in a horizontal plane 5" below H wnth its center at the point B. Find
(1) the intersection of the sphere and cylinder, (2) a line tangent to the curve
of intersection at some convenient point.
891. [1] A sphere of 4" diameter is located in the third quadrant with center
at a pomt 0(- 2f", - 2^", - 2^"). A right circular cylinder of 2" diameter has
its axis parallel to the ground line through the point A(- 3", - 2f", - 3"). Find
(1) intersection of cylinder and sphere, (2) development of cylinder showing this
intersection.
892. [1] (G. L. par. to short edge of sheet.) An ellipsoid of revolution is
generated by the revolution about its major axis of an ellipse which is in a plane
parallel to V, with its center at the point 0(- 5^", + 2", + 3") and with axes 6"
and 4" respectively, the latter being parallel to H. An oblique cone has its ver-
tex at the point A(- 3^", - 2^", + 3|") and its circular base of 4" diameter is in
H with center at B(- 7^", + 2^", 0"). Find the intersection of cone and ellipsoid
of revolution.
893. [2] A sphere of 5" diameter has its center in the third quadrant at
0(-2",'-2",-2^"). An oblique cone has the vertex A(- 2", - 3^', - 4i") and
its circular base of 2-|" diameter is in V with center at the point B(- 5^", 0", - 2").
Find the intersection of sphere and cone.
894. [2] A torus with axis perpendicular to H through the point
0(-3|", + 4|", 0") is generated by a circle of 2" diameter, which in initial posi-
tion is in a plane parallel to V with its center at the point M(- 5f", + 4|", + 1^").
A right circular cone of 4" altitude, stands in H with the center of its base of
68 . gi!;neral inteirsections oi'' two surface;s
3|" diameter at the point C(-4^", + 2|^'', 0"). Find the projection of the inter-
section between torus and cone.
895. [1] A torus stands on H in the first quadrant. Its axis is the hne
M(- 9i'', + 2^', O'O N(- 9^", + 2-y', + 3"), and its generating circle of If" diam-
eter in initial position is in a plane parallel to V with center at the point
0(- 11|'', + 2|^", + ^"). A right circular cone stands upon H, with the center of
its circular base of 4" diameter at the point B(- 8|'', + S"*, 0"). Its altitude is
3^''. Find the intersection of torus and cone.
896. [1] (G. L. par; to short edge of sheet and f above middle of sheet.)
A torus in the first quadrant has the line M(-7", f4", 0") N(- 7'', + 4", + 4")
as its axis, and its generating circle of 2|" diameter in initial position is in a plane
parallel to H with its center at the point 0(- 9^'', + 4'', + 2"). An oblique inter-
secting cone has its circular base of 5" diameter in V with center at point
B(-3'', + 6", 0''), and its vertex at a point A(- 9'', + 2'', + 7"). Find the inter-
section of cone and torus.
897. [1] A torus has the vertical axis A(- 12'', - 4", - 4") B(- 12", - 4'', 0")
and its generating circle of 3|" diameter is in initial position in a plane parallel
to V with its center at 0(- 14|", - 4", - If"). Another intersecting torus has
the horizontal axis C(- 7", - 5|", - 5^") D(- 7", 0", - oi") and its generating
circle of 4" diameter is in initial position in a horizontal plane with its center at
M(- 10", - 2^", - B-Y'). Find the intersection of these two tori.
898. [1] A triangular pyramid stands in the first quadrant with its vertex at
the point M(- 12", + 2", + 5") and the three vertices of its triangular base at the
points A(-13r, + §",0") B(-12y',-4i",0") and C(- 9f", + If", 0"). It
is intersected by a regular triangular prism whose lower base is in the plane
T(- 14^", -90°, + 120°) with its center at a point 1" from the H trace of the
plane T, and 2" from V. Each side of the base is 2" long, that side nearest V
is parallel to V, and the altitude of the prism is 5^". Find (1) the intersection
of pyramid and prism, (2) the development of each, to half scale, showing this
intersection.
899. [1] A regular square pyramid of 4^" altitude has the axis
M(-12i",-2i",-5") N(-12y',-2i",-yO, and its base is in a horizontal
plane 5" below H, with the line A(- 12|", - f", - 5") B(- lOf", - If", - 5") as
one side. A right circular cylinder of 1:|" diameter has the axis
P(-14i", -2J", -3i") Q(-10|",-2i",-lf"). Find (1) the intersection of
pyramid and cylinder, (2) the development of the pyramid, showing the inter-
section.
900. [1] The lines A(- 15", 0",- 5") B(- 12",- 5",0") and C(- 8",- 5"- 5")
D(- 3^", 0", - y') are the directrices of a hyperbolic paraboloid whose plane
directer is V. It is intersected by an oblique cone whose vertex is the pwint
M(- 8y, - 5f", 0") and whose circular base of 4" diameter is in V with its cen-
ter at the point N(- 10", 0", - 3V'). Find the intersection of cone and hyperbolic
paraboloid.
901. [1] A hyperbolic paraboloid has the lines A(- 13f", + 2^", 0")
B(-10|", + 3",-3") and C(- 13^", + 1", + 2^") D(- lOJ", + i", 0") as direc-
trices and H as plane directer. A right circular cylinder of 2" diameter lies on
H with the line M(- 14i", + y', + 1") N(- 10", + 3^", + 1") as axis. Find (1)
the intersection of the cylinder and hyperbolic paraboloid, (2) the development
of cylinder showing this intersection.
902. [1] The right line directrices of a certain hyperbolic paraboloid are
given as A(- 14^", - 5f", 0") B(- 11^", + T, + 5'0 and C(- 7^, + Sf", + 5")
D(- 2", -1- 1", 0") and H as plane directer. A hyperboloid of revolution of one
GENERAL INTERSECTIONS OF TWO SURFACES 69
nappe has as axis the line E(- 9^', + 2|", 0") F(- 9^", + 2|'', + 5") and its gen-
eratrix in initial position is the line M(- 12^'', + f', 0") N(- 9^'', + f", + 3")-
Find (1) the meridian curve of the latter which is parallel to V, (2) the curve
of intersection of the two surfaces.
903. [1] A rig-ht circular cone of 5" altitude stands on a horizontal plane 5:["
below H. with the center of its circular base of 3^" diameter at the point
Of- 13", - 2^', - 5:^"). An oblique helicoid has the same axis as the cone, its
generatrix in initial position pierces the plane of the cone base at the point
M(-U-^",-2V',-H") and it makes an angle of 30° with the axis. Pitch of
helix generated by point M = t". Find (1) the intersection of cone and helicoid,
(2) the development of the cone, showing this intersection.
904. [1] A right circular cone stands on a horizontal plane 51'' below H, with
the center of its circular base of 4" diameter at the point 0(- 12", - 2f", - 5|").
The altitude of the cone is 6". An oblique helicoid has the same axis as the cone,
its generatrix in initial position is the line M(- 12", - 2^", - 4^')
N(-14y', -2J", -o-j") and its pitch is 2". Find (1) the intersection of cone
and helicoid, carrying the generatrix of the latter surface through 2 complete
revolutions about the axis, (2) the development of the cone showing the inter-
section.
905. [1] An oblique helicoid has a vertical axis which pierces H at the point
0(- 13|", + 2", 0"). Its pitch is 4" and its generatrix in initial position is the
line A(-15", + 2",0") B(- 13^', + 2", + 1|"). The generation is such that in
H projection the line generatrix appears to rotate counter-clockwise. An oblique
cone has a circular base of 3" diameter in H with center at the point
N(-8|", + 3|", 0") and vertex at M(- 13^", -1- 2", + 2i"). Find (1) the line in
which the helicoid cuts H, (2) the intersection of helicoid and cone, (3) the de-
velopment of cone showing the intersection.
III. GENERAL INTERSECTIONS.
Including three surfaces.
90G. [1] (Ground line parallel to shorter edge of sheet.) A right circular
cylinder of 3" diameter has as axis the line A(- 7^", + 5", 0") B(- 3^", 0", 0") ;
a sphere of 4" diameter has its center in H at the point 0(- 7^", + 4", 0") ; a hy-
perbolic paraboloid whose plane directer is H, has directrices M(- 7", + 3", + 3")
N(-oi", + y', 0") and P(-3|^ + 2", 0'') Q(- Sf, 0", + 1|"). Find (1) the in-
tersection of the cylinder with the sphere and hyperbolic paraboloid, considering
only that part of each surface which is above the H plane, (2) the development
of the semi-cylinder between the intersecting surfaces.
907. [1] (Ground line parallel to shorter edge of sheet.) A hexagonal ring
stands on H in the first quadrant and has for its axis the line M(- 6f", + 3V', + 3'')
N(- 6f", -F 3|", 0"). Its generating hexagon in initial position stands in a plane
parallel to V, with its base in H and its center at the point P(- 8|", + 3^", + 1|")-
A regular hexagonal pyramid of 4|" altitude stands on H with two sides of its
base parallel to the ground line, and the center of said base at the point
0(- 8|", + 3^", 0"). Sides of base are IJ". An oblique cylinder has as axis
the hne A(- 2", + 1^", + 3^') B(- 5^", + 5", 0") with the center of its circular
base of 2^" diameter in H at the point B. Find the lines of intersection of the
hexagonal ring with the pyramid and cylinder.
908. [1] (Ground line parallel to shorter edges of sheet.) A sphere, an
oblique cylinder and an oblique cone are located in the third quadrant. The
sphere is of 5" diameter with center at the point 0(- 5|", - 3", - 3") ; the cylin-
der has the axis A(- ,5J-", - 3", - 5fO B(- 8^", - 3", 0") with its circular upper
base of 2]'' diameter in H at the point B, the cone has its vertex at the center of
the sphere, and its elliptical base, axes of 3" and 3^" respectively, is in V with
center at the point M(- 3f , 0'', - 3") and with its major axis parallel to H.
Find the intersections of the sphere with cylinder and cone.
909. [1] (Ground line parallel to shorter edges of sheet.) A torus, a cylin-
der and a cone are located in the first quadrant. The axis of the torus is a line
perpendicular to H through the point M(- 5^", + 4^", 0"). and its generating
circle, of 2^" diameter, in initial position is in a plane parallel to \" with its cen-
ter at the point X(- 7f", + 4^", -i- 3|"). The oblique cylinder has as axis the
line A(-6V', + 4V', + H") B(- 9". + 2", 0"), the center of its circular base, of
3'' diameter, being in H at the point B. The vertex of the cone is at the point
C(- 5|", + 4|", + 7") and the center of its circular base of 4'' diameter is in H
at the point D(- 2y, + 6|'', 0"). Find the intersections of the torvis with cylin-
der and with cone.
910. [1] A right circular cylinder of 2^' diameter has the axis
M(-13|", 0", + 2'') N(-6|'^ + 2'^ + 2"); a sphere of 5" radius has its center
in H at the point 0(- -J'', + 1'', 0") ; a hyperbolic paraboloid has H as plane
director and as directrices the lines A(- 16", 0'', + 4'0 B(- 13'', + 5", + 1") and
C(-6", 0", + 4'0 D(-6'', 0'', 0''). Find (1) the intersections of the cylinder
with the sphere and hyperbolic paraboloid, (2) the development of that part of
the cylinder between the other two surfaces, said development to be made on a
separate sheet.
GENERAL APPLICATIONS
BASED ON
INTERSECTIONS AND DEVELOPMENTS.
915. [2] A spiral riveted pipe is shown in Fig. 45. Design such a pipe,
making pitch of hehces twice the pipe diameter, and find the pattern lay-out for
a length of pipe equal to four times the diameter.
916. [1] Fig. 49 shows the ventilator connections to be located in the ridge of
a factory building. Assume a convenient scale and find the projections of the
pipes A, B and D, of their intersections, and the patterns (developments) for the
same. Also the pattern for a cap C, to overhang 3" all around.
917. [1] A sheet metal spire is shown in Fig. 55, located on a symmetrical
gabled- tower. Assuming reasonable dimensions, find and dimension the pattern
for the spire.
918. [1] A stone monument is shown in Fig. 59. Assuming a convenient
scale, find the development of all surfaces except the sphere and to scale construct
from a separate sheet a model for the monument.
919. [2] A sheet metal moulding is shown intersecting a spherical newel post
in Fig. GO. Find (1) the curves of intersection, (2) the pattern lay-out for the
moulding.
920. [2] A symmetrical triangular galvanized iron panel is shown in Fig. 61.
Find the projections of the panel, and the pattern lay-out for one side.
921. [1] In Fig. 62 is shown two views of a copper cornice ornament. Con-
struct the projections of the ornament, and find and dimension the complete
pattern lay-out.
922. [2] A galvanized iron pillar base is shown in Fig. 63. Find the projections
of the base, and a pattern lay-out for the intermediate section.
923. [1] In Fig. 64 a metal gable moulding is shown at its intersection with
a vertical pilaster. Assuming convenient scale and dimensions, find the pro-
jections of the intersection curves, and pattern for a short length of both moulding
and pilaster.
924. [1] Design a cofifee-pot with conical body, spout and cover, and handle
similar to that shown in Fig. 65, showing its projections including intersection
curve of spout and body, and lay out patterns for cutting the metal for all parts
of the pot.
925. [1] Scale, f" = 1' - 0''. An iron pouring-pot has a cylindrical body,
conical top and cylindrical spout as indicated in Fig. 66. Find the projections
of the pot, the curves of intersection and the developments of the parts.
926. [2] A pipe fitting is shown in Fig. 67. Find its projections, the curves
of intersection, and develop outer and inner surface of one branch of the fitting.
927. [2] Scale, half size. A copper sink drainer. Fig. 68, is made of a conical
perforated front and two triangular vertical back pieces at right angles. Con-
struct the three projections of this drainer and lay out the pattern to cut from one
piece allowing a Y^ lap, cut ofif at the correct angle at each end.
928. [1] Scale, 1|" = I'-O". Find the projections and pattern development
of the metal reducer in Fig. 69 designed to connect a square with a round pipe.
Dimension in full, so as to be clear to the one who lays out the pattern.
y2 GENERAL APPLICATIONS
929. [2] Scale, 14" = I'-O". A conical offset pipe is shown in Fig. 70, connect-
ing the two cylindrical pipes. Find its projections and lay out a pattern for the
same.
930. [2] Find the projections of the hopper of Fig. 71, and determine all the
angles which would be necessary in construction. Make a working drawing of
one side of the hopper.
931. [1] A cylindrical pipe passes through a factory roof in Fig. 72. Find the
projections and pattern for the flange as shown, assuming the flange base to be
a square, the roof to be inclined at 45°, and the center line of the pipe to pass
through the center of the square flange base.
932. [2] Scale, V = I'-O". Show three views of the bath tub of Fig. 73, and
find the pattern lay-out for the same, dimensioned in full for the tinsmith.
933. [2] A tin tea pot has a conical spout as shown in Fig. 74. Find projec-
tions and pattern for this spout.
934. [1] A tin tea pot is designed with conical body and conical spout as in
Fig. 74. Find and dimension its projections and complete patterns. Scale, full
size.
935. [2] Construct 2 views and the complete pattern for the steam exhaust
head shown in Fig. 75.
936. [2] Draw two views and lay out the pattern for a roof flange similar to
that shown in Fig. 76, assumed to be a portion of a cone of revolution.
937. [1] Scale, 1^'' = I'-O". Make the necessary drawings for a transition
connection between square and round pipes similar to that shown in Fig. 77.
Lay out and dimension the complete pattern.
938. [1] Design a metal oil can as shown in Fig. 78. Show necessary views,
and lay out and dimension the complete pattern.
939. [2] A hexagonal nut has a conical chamfer as shown in the top figure of
Fig. 79. Construct 3 views, assuming a = 2^", d = \\" , and the chamfer angle
= 45°.
940. [2] A hexagonal nut has a spherical top as shown in the lower figure of
Fig. 79. Construct 3 views, assuming a = ^\" , d = \\" , r = 1" , c = yV'.
941. [2] An upset iron rod as shown in Fig. 80, is turned down to a conical
transition, connecting the square and cylindrical portions. Construct 2 views of
such a rod, finding accurately the curves of intersection.
942. [1] Design a metal sugar scoop such as is sketched in Fig. 81, showing
at least two views and a complete pattern for the same.
943. [1] Construct two views and complete pattern lay-out for the tin meas-
ure shown in Fig. 82.
944. [4] Scale, \" = I'-O". The Fig. 83 shows a hopper used on a rotating
cylindrical dryer. The hopper top is an 18'' square, the sides of hopper being
faces of a regular pyramid whose vertex is in the center line of the cylinder as
shown, b = 2'-0''. a = 2'-6''. Show three views, find intersection of hopper
with cylinder, and develop pattern for one-half the hopper.
945. [2] Two cylindrical air shafts intersect as shown in Fig. 84. a = 3'-0''.
c = 12''. Center line of small shaft is 9" from that of large. Show 3 views and
develop smaller shaft. Scale, f " = I'-O".
946. [4] Construct to convenient scale, two views of the stove pipe elbow in
Fig. 85. Develop pattern for one half.
947. [2] To a convenient scale, construct two views of the furnace elbow
shown in Fig. 86, and lay out the patterns for the intermediate section and one
end section.
GENERAL APPLICATIONS 73
948. [4] Fig. 87 shows a square furnace pipe elbow. Construct two views,
and lay out a pattern for the complete elbow.
949. [3] Fig. 92 shows a water tank with conical boss. Assuming reasonable
dimensions and scale, show 2 views of the tank and develop the boss and the
tank.
950. [1] A locomotive smoke stack is dimensioned in Fig. 88. To scale,
I" = I'-O'', draw two views of the same, and lay out sheets for the several sec-
tions, dimensioning ready for cutting.
951. [2] A pipe connection is made up of portions of a torus and a cylinder,
flanged as shown in Fig. 89. Find two views, with the curve of intersection,
and develop the cylindrical portion. Scale, 1" = I'-O".
952. [2] Design a tin funnel similar to that shown in Fig. 90. Show 2 views
and lay out complete pattern, dimensioning in full.
953. [4] A scale scoop is formed of portions of 2 equal intersecting cylinders
as in Fig. 91. Lay out and dimension a complete pattern.
954. [2] A portion of a connecting rod is shown in Fig. 93. Assume dimen-
sions about twice the size of the blue print drawing, and find three projections of
the curve of intersection between the square shaft and the sphere as shown.
955. [1] A transition connection for rectangular to round pipe is shown in
Fig. 94, with round pipe development and transition pattern. Lay out and di-
mension the complete problem to scale, 3'' = I'-O". Develop the conical surface
BGE by Church's method, and the conical surface AGK approximately by tri-
angles.
956. [2] Find two views and pattern lay-out for the furnace partition transi-
tion pipe shown in Fig. 95. Scale, 1^" = I'-O''.
957. [1] A furnace connection is to be constructed as shown in Fig. 96. Find
pattern for all parts shown, assuming reasonable proportions and scale.
958. [2] Design a metal gable moulding and wash as shown in Fig. 97 and
find the pattern lay-out for the metal cutter.
959. [2] A locomotive slope sheet is shown in Fig. 98. Assume reasonable
dimensions and find the slope sheet pattern.
960. [1] Design two gusset plates for a boiler stack such as indicated in Fig.
99. Show three views of the plates and their intersections with stack and boiler,
then find the development of one of these gusset plates.
FIGURE INDEX.
The following is a list of blue print figures, with numbers of those problems which are
based thereon.
I-TGS. PROBS.
1-8 Introduction
9 17, 18
TO 48, 49, 92, 93> 94. 95
II 74. 75. 82, 83, 84, 85
12
13
14
15
16
17
18.... '..126 [4], 127 [4], 128
295 [4]
19 156 [4]. 157 [4I. 158
160 [4], 191 [4I. 208
210 [4]. 249 [4]. 269
401 [4]. 402 [4], 426
451 [2], 452 [2]
20 t6i [2], 211 [2], 267
403 [2], 457 [2I, 464
21 165 [8], 251
[4], 294
23 196 [2], 458
24 434 [2], 454
25-
26.
27.
.163
215
435
[2],
[2],
[2]
28 166 [4],
2r6 UL
428 [2]
29 192 [4I,
270 [2],
453 [2I.
530 [2],
.13T [2],
30.
298 [2],
31 129 [8],
32 130 [8],
433 [4]
33
34
35
36
37
38..
39
40
41
42
43
44
45
46
164 [2],
324 [2],
167 l4l.
274 [4].
212 [4L
271 [2],
455 [2],
531 [2],
132 [2],
431 [2]
168 [8],
162 [8],
•517 [i].
•527
193
325
194
323
213
276
456
532
169
273
272
S18
275
277
[I]
[2]
[2]
[4l
[4]
[4l
[2]
[2]
[2]
[4]
[8]
[8]
[I]
[2],
[2],
•534 [i]
.460 [4]
.279
.817
[2]
[2]
.85
.87
..90
..91
[4].
, 159
[4l
, 209
UJ
. 322
Ul
. 427
I2J
, 268
[2]
. 521
Ul
. 522
I2J
• 523
liJ
, 524
hJ
, 525
li]
526
[il
1, 528
III
, 214
l2|
, 404
I2J
. 195
[4]
, 405
I2J
. 217
[4l
, 429
[2\
. 529
I2J
, 197
[2]
. 430
[4l
, 432
l4j
, 5 19
[I]
296
I2J
297
L2I
533
liJ
535
III
536
[2\
537
[2\
, 542
{2\
541
\2\
538
[I]
539
llJ
. 540
[2\
915
|2|
, 820
llj
FIGS.
47-
48.
49.
50.
51-
52.
53-
54-
55-
90.
91.
92.
93-
94.
95-
96.
97-
98.
99.
. . . »I8
... 819
. . . 916
614 [8]
615 [8]
616 [8]
617 [8]
618 [8]
917
56 619 [8]
620
621
918
919
920
921
922
923
924
92s
926
927
928
929
57-
58.
59-
60.
61.
62.
63.
64.
65-
66.
67.
68.
69.
70.
71 543 [i], 930
72 931
73 932
74 933 [2], 934
75 935
76 936
77 937
78 938
79 939 [2]. 940
80.
81.
82.
83.
84.
85.
86.
87.
941
942
943
944
945
946
947
948
950
951
952
953
949
954
955
544 [i], 956
957
958
959
960
Fig. I.
Fig. 3.
Fig.S.
Fig. 2.
Fig. 4.
Fig. 6.
Fig. 7.
Fig. 8.
OF /
■>p the'
VERSJTY
OF
Fig. 12.
'4'.<"
Fig.a
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if?','V^
F/^./4.
RK3/^*KTTna.
F,-g./3.
F,g.l6.
Fig. 17.
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U 3f
-*f* 3i"
CAP FOR FA:7Vffy /E/VT/LATOR SH4FT.
F/^.2/.
Q^K MISSION LAMP SHADE.
rCEJSca/cr-S'.J
F,cf.22.
STEEL TANK FOR AN ELE)/ATOP B007.
P/afcs to be bc^n^ afK/
THE
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ng.3/.
MULE TOiV/h/6 CAhJAL BOAT
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UNIVERSITY FRESH-SOP^ SPRING (JAMES.
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UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
<^^^
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RECD i-^
NOV 5 19S7
JUN 12 1958 R
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6Noy'57WW
DEC0 3 199Z
LD 21-100m-9.'47(A57028l6)476
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YE 02300
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