CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 Given to the 
 COLLEGE OF ENGINEERING 
 
 Ronald J. Sweenqr 
 
Cornell university Library 
 QA 501. B64 1920 
 
 Elements of descrjptwegaffiil 
 
 '!J924 004 047 852 =- 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004047852 
 
BY THE SAME AUTHORS 
 
 ELEMENTS OF DRAWING 
 
 6 by 9, xviii + 193 pages, 151 figures. Cloth, 
 $2.00 net. 
 
 Contents; — Selection, Care, and Use of Drawing 
 Instruments and Materials. Letters, Numerals, 
 and Lettering. Mechanical Drawing and Drafting 
 Room Practice. Free-hand Working Sketches. 
 Isometric Drawing and Sketching. 
 
Elements of 
 
 Descriptive Geometry 
 
 BY 
 GEORGE F. BLESSING, M.E., Ph.D. 
 
 Professor of Mechanical Engineering and in charge of Engineering, Swarthmore College 
 
 Formerly Assistant Professor of Machine Design, Cornell University 
 
 Member American Society of Mechanical Engineers 
 
 And 
 LEWIS A. DARLING, E. in M.E. 
 
 Mechanical Engineer, Electric Service Supplies Co.; Formerly Assistant 'ProfessoB 
 of Machine Design, Cornell University 
 
 SECOND EDITION 
 {Corrected and partly rewritten") 
 
 TOTAL ISSUE FIVE THOUSAND 
 
 NEW YORK 
 
 JOHN WILEY & SONS, Inc. 
 
 London: CHAPMAN & HALL, Limited 
 1920 
 
Copyright, 1912, 1920 
 
 BY 
 
 GEORGE F. BLESSING and LEWIS A. DARLING 
 
 Stanbope ipress 
 
 F. H. GILSON COMPANY 7-20 
 
 BOSTON, U.S.A. 
 
PREFACE 
 
 This book, like its companion volume, " Elements of Drawing " 
 by the same authors, is based on the work of this character 
 required of all first-year students in Sibley College, Cornell 
 University. It is the outgrowth of an effort to modify and 
 shorten somewhat the scope and method of presentation of 
 this subject, from the lengthy discussion often presented, to 
 one more in keeping with the relative importance of the sub- 
 ject in an engineering curriculmn. 
 
 This work was undertaken for me some years ago by Pro- 
 fessor Blessing, and the book is the outgrowth of a series of 
 lectures and an accompanying drawing-room course first given 
 by him and later continued by Professor DarUng; both being 
 at the time members of the instructing staff of the Department of 
 Machine Design and Construction of Sibley College. The 
 work of putting this material into the form of a book was imder- 
 taken by the authors at the writer's request, the object in view 
 being twofold, namely: to obtain a book exactly suited to the 
 needs of the Department, which we had hitherto been imable 
 to do, and also to put into permanent shape the methods and 
 principles used in this work, thus forming one of a series of 
 correlated textbooks which eventually it is expected will cover 
 the entire work of the Department. 
 
 The authors brought to the task a full knowledge of the 
 more advanced work of the Department, having had expe- 
 rience in teaching the advanced subjects in design, which with 
 their experience elsewhere both in practical and teaching posi- 
 tions was of great aid in improving and simpKfjdng this more 
 elementary work. For five years this material has been used 
 in mimeographed form in this and other institutions with such 
 unqualified success as to warrant the belief that the methods 
 of presentation are sound and the scope sufficient to give the 
 
IV PREFACE 
 
 Student all the training and information necessary to pursue 
 advanced work in drawing and design, without omitting any 
 essential details necessary in training engineering students. The 
 following suggestions by the authors as to the use of the book 
 may make their point of view clearer: 
 
 "Tfe presentation oj an experiment as a means of bringing out 
 fundamental principles has been found to appeal to the student 
 and it has enabled mtoy to grasp the subject with compara- 
 tive ease who had considerable difl&culty by other methods in 
 ' visuahzing ' these simple problems. For the average student 
 the mere reading of the experiment is sufficient but others may 
 reqiiire the actual 'building up' of the experiment to fully 
 grasp the principles involved. The analysis and solution of 
 each problem should be thoroughly imderstood and, as a test of 
 the thoroughness and accuracy, the solution of the ' check ' will 
 serve. 
 
 "A somewhat unusual feature of the book is the use of the 
 first angle for the solution of many problems relating to the 
 point, line and plane, while the third angle is used exclusively 
 for problems relating to sohds. It is beheved that the funda- 
 mental principles of the subject are more easily grasped by 
 beginners if they are presented in the first angle on account 
 of the ease with which the problem can be built up in this angle 
 when necessary and if these principles are once thoroughly 
 understood the student will find no difficulty in applying them 
 to any other angle. Experience in the use of this method of 
 presentation justifies the belief that this is a good way of ap- 
 proaching the subject. The third angle is made use of only 
 after the student has gained some proficiency, and is used for 
 all work involving solids in order to conform to modern prac- 
 tice of making drawings in the third angle. 
 
 " The drawing-room course of problems in Chapter X can be 
 varied and others introduced by the instructor as occasion 
 demands. It is believed that the method of definite measure- 
 ments used in this drawing-board work is fully justified by the 
 training it gives the student in method and accuracy, although 
 admittedly a sHghtly greater number of problems can be solved 
 in a given time by placing them on the sheet in a haphazard 
 
PREFACE V 
 
 way. This systematic and definite arrangement of problems 
 
 will be found helpful to the instructor where the work of a large 
 
 number of students must be handled." 
 
 There is nothing of an experimental nature in the book, as 
 
 the material it contains has been used in Sibley College for 
 
 several years with increasing success; and it is hoped that it may 
 
 be found helpful elsewhere. 
 
 Dexter S. Kimball, 
 
 Professor of Machine Design and Construction, 
 
 Sibley College, Cornell University. 
 Ithaca, N. Y. 
 July 29, 1912. 
 
 AUTHORS' NOTE 
 
 While teaching this work and collecting data for this book 
 the authors have studied carefully all available literature on 
 the subject. When a suggestion, method of presentation or 
 problem was found especially good it was adopted. So much 
 sjmiUarity exists in these books that individual credit carmot 
 be given, but the following books were found especially useful : 
 
 Elementary Descriptive Geometry by C. H. McLeod ; De- 
 scriptive Geometry by A. E. Church; Essentials of Descriptive 
 Geometry by G. H. Follows; Notes on Descriptive Geometry 
 by C. L. Adams; Descriptive Geometry by V. ' T. Wilson; 
 Descriptive Geometry for Engineering Students by J. A. Moyer; 
 Elements of Descriptive Geometry by C. E. Ferris. 
 
 Grateful acknowledgment is also made to the following per- 
 sons for assistance and helpful criticism : 
 
 Prof. Dexter S. Kimball and Prof. John T. Williams, Sibley Col- 
 lege, Cornell University; Prof. James G. White, State University 
 of Kentucky; and Prof George W. Lewis, Swarthmore College. 
 
 PREFACE TO THE SECOND EDITION 
 All typographical and other errors discovered in the first 
 edition have been corrected. Chapter X has been rewritten and 
 an entirely new set of exercises has been inserted. The notation 
 for the exercises has been improved and new material added. 
 
 The authors wish to acknowledge valuable assistance received 
 from Prof. C. E. Townsend of Cornell University, in the form of 
 corrections, suggestions and new material. 
 
 G. F. Blessing. 
 
 Swarthmore, Pa. 
 Jan. s, 1920. 
 
CONTENTS 
 
 Page 
 
 Preface iii 
 
 Basis on which the Work is Planned. — General Purpose of the Book. — 
 "Experiments" and the "Building Up" of Data as a Feature of this 
 Work. — Unusual Features of the Book. — Text Matter Assigned as 
 Needed in Solving an Exercise. — Flexibility of the Book. — Authors' 
 Acknowledgment. 
 
 CHAPTER I 
 
 Introduction 
 Section 
 
 1. Descriptive Geometry Defined i 
 
 2. The Practical Value of Descriptive Geometry i 
 
 3. Properties of Data Dealt with i 
 
 4. " Position," Defined 2 
 
 5. Experiment I. — Explanation and Illustration of the Principles of 
 
 Orthographic Projection 2 
 
 6. Meaning of the Term " in space'' 5 
 
 7. Meaning of the Expression "projecting a point in space." The 
 
 Maximum Projection of a Line; The Minimimi Projection of a Line . . 5 
 
 CHAPTER II 
 
 Definitions, Notation and First Principles 
 Problems on the Point 
 
 8. Principal Planes of Projection 8 
 
 g. Horizontal Plane of Projection 8 
 
 10. Vertical Plane of Projection 8 
 
 11. Ground Line, Defined 8 
 
 12. Space Projector, Defined 8 
 
 13. Vertical Projection Defined and Illustrated 8 
 
 14. Horizontal Projection Defined and Illustrated 10 
 
 15. Ruled Projector Defined and Illustrated 10 
 
 16. Orthographic Projection Defined and Applied 10 
 
 1 7. Theorem I. — The Vertical and the Horizontal Projection of a Point 
 
 Determines its Position in Space with Reference to V and H 10 
 
 18. Corollary. — The Vertical and Horizontal Projection of a Point are 
 
 on One and the Same Straight Line 11 
 
 19. Corollary. — A Point Situated on Either Plane of Projection is 
 
 its Own Projection on that Plane and its Other Projection is in 
 G-L II 
 
 20. The Six Diflferent Positions a Point may Occupy with Reference to 
 
 V and H 11 
 
 vii 
 
vm CONTENTS 
 
 Section Page 
 
 11. Problem i. — Given the Position of a Point in Space to Determine 
 
 its Projections 12 
 
 22. Problem 2. — Given the Projections of a Point on V and H to De- 
 
 termine its New Vertical Projection on any New Vertical Plane ... 13 
 
 23. What the Data for Problems in Descriptive Geometry Consist of ... . IS 
 
 24. The Four Dihedral Angles Defined and Illustrated IS 
 
 25. General Notation for Points in Space; Horizontal Projections of 
 
 Points; Vertical Projections of Points; Lines in Space; The Hori- 
 zontal Projections of Lines; and The Vertical Projections- of 
 Lines 18 
 
 26. Line Conventions for the Ground Line; A Ruled Projector; A Visible 
 
 Line; Construction Lines; Invisible Lines; The Path Traced by a 
 Point; and the Method of Indicating the Direction of Sight 18 
 
 CHAPTER III 
 Point and Line Problems 
 
 27. Experiment n. — -Important First Principles as to the Projections of 
 
 Lines both Given and Illustrated 20 
 
 28. Positions which it is Possible for a Line to h^ve with Reference to 
 
 V and H 22 
 
 29. Principle. — A Line in Space is Represented by its Projections and 
 
 Two Projections are Necessary to Completely Define the Line. ... 25 
 
 30. Principle. — Two Points of a Line or a Point and the Direction Com- 
 
 pletely Determine the Line 25 
 
 31. The Projections of Two Points of a Line Determine the Projection 
 
 of the Line, also the Projection of One Point and the Direction 
 may Locate the Projection of a Line 25 
 
 32. Theorem n. — Every Point on a Line in Space has its Horizontal Pro- 
 
 jection on the Horizontal Projection of the Line, and its Vertical 
 Projection on the Vertical Projection of the Line 25 
 
 33. Principle. — The Projections of Lines Intersect on the Same Ruled 
 
 Projector if the Lines in Space Intersect 26 
 
 34. Problem 3. — To Determine the Piercing Points (or Traces) of a 
 
 Line 26 
 
 35. Theorem III. — If a Line in Space is Parallel to a Plane of Projection 
 
 its Projection on that Plane Shows the True Length of the Line. ... 28 
 
 36. Corollary. — The True Distance Between Two Points in Space is 
 
 the Length of the Straight Line Joining Them 29 
 
 37. Corollary. — -The Angle Between a True Length Projection and 
 
 G-L is Equal to the Angle the Line in Space Makes with its Other 
 Projection, Hence the Same Angle as the Line in Space Makes 
 with the Plane Containing its Other Projection 29 
 
 38. Corollary. — To Determine the True Angle Between Two Inter- 
 
 secting Lines, Find the Projection of these Lines on a Plane which 
 
 is Parallel to Both of Them , 29 
 
 39. Corollary. — To Measure off a True Distance on a Line Measure it 
 
 off on that Projection of the Line which is Parallel to a Plane of 
 Projection ., 29 
 
CONTENTS IX 
 
 Section Page 
 
 40. Theorem IV. — If Two Lines in Space are Parallel their Vertical Pro- 
 
 jections will be Parallel, also their Horizontal Projections will be 
 Parallel 29 
 
 41. Corollary. — The Projections of a Line which is Parallel to V and H 
 
 will be Parallel to Gr-L ._ 30 
 
 42. Corollary. — A Line in Space Projects in its True Length on a Pro- 
 
 jection Plane to which it is Parallel, and on the Other Projection 
 Plane it Projects Parallel to G-L 30 
 
 43. P'roblem 4. — To Pass a Line Through a Given Point and Parallel 
 
 to a Given Line 31 
 
 44. Experiment HI. — Principles of the Rotation of a Point about a Line, 
 
 . Given, Discussed, and Illustrated 32 
 
 45. Axiom. — If one of Two Lines which Intersect at Eight Angles is 
 
 Parallel to a Plane of Projection the Projection of the Right 
 Angle will be a Right Angle unless the Second Line Projects as 
 a Point 34 
 
 46. Terms Used in Connection with the Revolution of Points, Lines, 
 
 Planes, and Solid? — Defined and Illustrated 35 
 
 47. Problem 5. — To Revolve a Point through a Given Angle and around 
 
 an Axis which is Perpendicular to V 35 
 
 48. Problem 6. — To Revolve a Point in Space into H about an Axis 
 
 which lies in H and is Oblique to V 38 
 
 49. Problem 7. — To Revolve a Point in Space into V about an Axis 
 
 which lies in V and is Oblique to H 39 
 
 50. Problem 8. — -To Determine the Distance between Two Points in 
 
 Space 40 
 
 51. Problem 9. — To Determine the True Size of the Angle between Two 
 
 Intersecting Lines 41 
 
 52. Problem 10. — To Measure off a Given Distance on a Given Line. . . 44 
 
 53. Definitions of An Element of a Cone; The Axis of a Cone; A 
 
 Right Circular Cone; A Cone of Revolution; and The Base Angle 
 
 of a Right Circular Cone 45 
 
 54. Problem 11. — To Draw the Projections of a Line which makes a 
 
 Given Angle with V and H 46 
 
 CHAPTER IV 
 
 Line and Plane Problems 
 
 55. Definitions and Principles 49 
 
 56. Notation and Line Convention for Planes — • The Horizontal Trace 
 
 of a Plane; The Vertical Trace of a Plane 49 
 
 57. Positions which a Plane may have with Reference to V and H 50 
 
 58. Theorem V. — If a Line lies in a Plane- its Vertical Piercing Point (or 
 
 Vertical Trace) must lie in the Vertical Trace of the Plane; and 
 its Horizontal Piercing Point in the Horizontal Trace of the 
 
 Plane 51 
 
 59. Problem 12. — -To Assume a Line in a Given Oblique Plane 51 
 
 60. Problem 13. — To Pass a Plane through Two Intersecting Lines. . . S4 
 
 61. Problem 14. — To Pass a Plane through Three Given Points 53 
 
X CONTENTS 
 
 Section Page 
 
 62. Problem 15. — To Pass a Plane through a Point and a Line SS 
 
 63. Problem 16. — To Determine the Traces of a Plane which will Con- 
 
 tain any Given Plane Figure SS 
 
 64. Theorem VI. — If Two Planes in Space are Parallel their Vertical 
 
 Traces are Parallel, also their Horizontal Traces are Parallel .... sS 
 
 65. Principles. — Through a Given Straight Line a Plane can be Passed 
 
 Parallel to any Other Straight Line; also a Plane is Parallel to 
 a Given Straight Line when it Contains any Straight Line Parallel 
 to the Given Line S7 
 
 66. Problem 17. — To Pass a Plane through a Given Straight Line and 
 
 Parallel to Another Given Straight Line. . S7 
 
 67. Problem 18. — -To Pass a Plane through a Given Point and Parallel to 
 
 Two Given Straight Lines 57 
 
 68. Problem 19. — ^To Determine a Plane which shall be Parallel to a 
 
 Given Plane and Contain a Given Point 58 
 
 69. Theorem VH. — If a Plane in Space is Intersected by a Plane Paral- 
 
 lel to H, the Horizontal Projection of the Line of Intersection will • 
 be Parallel to the Horizontal Trace of the Plane; and the Verti- 
 cal Projection will be Parallel to G-L 58 
 
 70. Corollary. — Any Line which is Parallel to H and which Lies in a 
 
 Given Plane will be Parallel to the Horizontal Trace of that 
 Plane S9 
 
 71. Axiom. — The Line of Intersection of Two Planes is a Straight Line 
 
 Common to Both Planes S9 
 
 72. Problem20. — To Determine the Line of Intersection of Two Planes. 60 
 
 73. Principle. — Parallel Planes Intersect a Third Plane in Parallel 
 
 Lines 61 
 
 74. Problem 21. — To Determine the Line of Intersection of Two Planes 
 
 when the Horizontal Traces do not Intersect within the Limits 
 
 of the Drawing 62 
 
 75. Problem 22. — To Determine the Line of Intersection of two Planes 
 
 when neither the Horizontal nor the Vertical Traces Intersect 
 within the Limits of the Drawing 64 
 
 76. Problem 23. — To Determine the Point in which a Given Straight 
 
 Line Pierces a Given Plane 64 
 
 77. Definition of a Dihedral Angle 65 
 
 78. Principle. — If Each of Two Intersecting Planes are Perpendicular to 
 
 a Third Plane their Line of Intersection is Perpendicular to that 
 Plane 65 
 
 79. Theorem VIII. — If a Line is Perpendicular to a Plane the Vertical 
 
 Projection of the Line is Perpendicular to the Vertical Trace of 
 the Plane; and the Horizontal Projection of the Line is Perpen- 
 dicular to the Horizontal Trace of the Plane 65 
 
 80. Problem 24. — To Draw through a Given Point a Straight Line 
 
 Perpendicular to a Given Plane, and to Determine the Distance 
 from the Point to the Plane 68 
 
 81. Notation for the Projections of a Point on a Plane Other than V 
 
 or H 69 
 
 82. Problem 25. — To Project a Given Straight Line on a Given Oblique 
 
 Plane 69 
 
CONTENTS xi 
 
 Section Page 
 
 83. Problem 26. — To Determine a Plane ivhich Shall Pass through a 
 
 Given Point and be Perpendicular to a Given Straight Line 69 
 
 84. P*rinciple. — How to Measure the Angle between Two Planes 72 
 
 85. Problem 27. — To Determine the Angle that a Given Plane Makes 
 
 with V and H 72 
 
 86. Problem 28. — To Determine the True Angle between the Traces of 
 
 a Plane 73 
 
 87. Problem 29. — To Determine the Angle between Two Intersecting 
 
 Planes 75 
 
 88. Principle. — If a Line in Space is Perpendicular to a Plane in Space 
 
 the Line and the Plane make Complemental Angles with the 
 Plane of Projection 77 
 
 89. Problem 30. — To Determine the Traces of the Plane that Contains 
 
 a Given Point and Makes Given Angles with V and H 77 
 
 90. Principle. — The Angle which a Straight Line Makes with a Plane 
 
 is the Angle which the Line Makes with its Projection on the 
 Plane 78 
 
 91. Problem 31. — To Determine the Angle which a Given Straight Line 
 
 Makes with a Given Plane 78 
 
 92. Problem 32. — To Draw a Straight Line through a Given Point so 
 
 as to Intersect a Given Line at a Given Angle 80 
 
 93. Problem 33. — To Determine the True Form of any Plane Figure 
 
 Given by its Projections 81 
 
 94. Axiom. — • The True Angle between Two Intersecting Lines can be 
 
 Determined by Revolving the Plane of the Lines into a Plane 
 
 of Projection 83 
 
 95. Problem 34. — To Determine the Third Angle Plan and Elevation 
 
 of any Plane Figure when One Edge of the Figure is Inclined 
 at any Given Angle to V and the Plane of the Figure is Perpen- 
 dicular to V and Inclined at any Angle to H. Also to Deter- 
 mine the Angle the Given Edge Makes with H 83 
 
 96. Note. ^ How to Determine the Angle a Line Makes with a Plane 
 
 of Projection 85 
 
 97. Principle. — The Projection of a Circle is Determined by Finding 
 
 the Projections of a Sufficient Number of Points of the Circle 
 and Connecting the Proper Projections by a Curved Line 85 
 
 98. Problem 35. — To Draw the Third Angle Projections of a Circle 
 
 Lying in a Given Plane 85 
 
 99. Problem 36. — To Determine the Third Angle Plan and Elevation 
 
 of a Plane Figure Inclined to both V and H 88 
 
 CHAPTER V 
 Solids 
 
 100. Definition of a Solid; Method of Projecting a Solid 90 
 
 101. Problem 37. — To Draw the Third Angle Plan and Elevation of a 
 
 Cube when one Face is in H and a Vertical Face Makes a Given 
 Angle with V 90 
 
XU CONTENTS 
 
 Section Page 
 
 102. Problem 38. — To Draw the Third Angle Plan and Elevation of a 
 
 Cube when One Edge Lies in H and Makes 30° or any Given 
 Angle with V, and Two Parallel Faces Make 60° or any Given 
 Angle with H 91 
 
 103. Problem 39. — To Draw the Third Angle Plan and Elevation of a 
 
 Cube when One Edge is Inclined at 45° or any Given Angle to 
 H and 30° or any Angle to V, and the Diagonal of pne Face is 
 Perpendicular to V 92 
 
 104. Problem 40. — To Draw the Third Angle Plan and Elevation of a 
 
 Cube when One Face is Inclined to H at a Given Angle and an 
 Edge of that Face also Makes a Given Angle with H 94 
 
 105. Definitions. — Pyramid; Apex of a Pyramid; Altitude of a Pyra- 
 
 mid; Axis of a Pyramid; Right Pyramid; Oblique Pyramid .. ^ . 95 
 
 106. Problem 41. — To Draw the Third Angle Plan and Elevation of a 
 
 Right Hexagonal Pyramid when its Axis is Inclined at any 
 Angle j8 to H and a to V 95 
 
 107. Problem 42. — To Draw the Third Angle Plan and Elevation of a 
 
 Right Pyramid of Given Base and Altitude and with its Base in 
 
 a Given Oblique Plane 97 
 
 108. Problem 43. — To Draw the Third Angle Plan and Elevation of an 
 
 Oblique Hexagonal Pyramid when its Axis is Inclined to Both 
 
 V and H , 98 
 
 109. Problem 44. — To Draw the Third Angle Projection of a Right Penta- 
 
 gonal (Five-angled) Pyramid when One Slant Face C-Q-D is Paral- 
 lel to H and the Horizontal Edge C-D of the Base is Inclined at any 
 
 Angle to G-L 98 
 
 no. Definitions. — Conical Surface; Element of a Cone; A Cone; Base 
 of a Cone; Altitude of a Cone; Circular Cone; Axis of a Cone; 
 Right Circular Cone; Oblique Circular Cone 100 
 
 111. Problem 45. — To Draw the Projections of a Cone in any Position 
 
 Similar to the Pyramids of Problems 41, 42, 43 and 44 100 
 
 112. Definitions. — ■ A Prism; Altitude of a Prism; .Right Prism; Oblique 
 
 Prism 100 
 
 113. Problem 46. — To Draw the Third Angle Plan and Elevation of an 
 
 Hexagonal Prism when the Axis is Parallel to V but Inclined to H 
 at any Given Angle and an Edge of One End is Parallel to H 
 and Perpendicular to V loi 
 
 114. Problem 47. — To Draw the Third Angle Plan and Elevation of an 
 
 Hexagonal Prism when One Edge of an End is in H and Inclined 
 at any Given Angle to V and the Axis is Inclined at any Given 
 Angle to H 102 
 
 115. Problem 48. — To Draw the Third Angle Plan and Elevation of a 
 
 Right Cylinder when its Axis is Parallel to V but Inclined to H 
 
 at any Given Angle 103 
 
 116. Problem 49. — To Draw the Third Angle Plan and Elevation of a 
 
 Right Cylinder when its Axis is Inclined to both V and H 104 
 
 117. Definitions. — Helix; Pitch of a Helix; Helicoid 104 
 
 118. Problem 50. — ■ To Project a V Screw-Thread 106 
 
CONTENTS Xlll 
 
 Section Page 
 
 CHAPTER VI 
 
 Tangent Planes and Double-Curved Surfaces of Revolution 
 
 119. Definitions and General Considerations. — Tangent to a Curve; 
 
 Point of Tangency; Tangent to a Surface; A Tangent Plane; 
 Element of Tangency; A Normal to a Surface 107 
 
 120. Problem 51. — To Pass a Plane Tangent to a Cone through a Given 
 
 Point on a Surface 108 
 
 121. Problem 52. — To Pass a Plajie Tangent to a Cone through a Point 
 
 without the Surface 109 
 
 122. Problem 53. — To Determine the Traces of a Plane which shall Con- 
 
 tain a Given Line and Make a Given Angle with H . . no 
 
 123. Problem 541 — To Pass a Plane Tangent to a Cone and Parallel 
 
 to a Given Straight Line iir 
 
 124. Problem 55. — To Pass a Plane Tangent to a Cylinder through 
 
 a Given Point on the Cylinder 112 
 
 125. Problem 56. — To Pass a Plane Tangent to a Cylinder and through 
 
 a Given Point without the Surface 112 
 
 126. Problem 57. — To Pass a Plane Tangent to a Sphere at a Given 
 
 Point on the Surface 114 
 
 127. Problem 58. — To Pass a Plane Tangent to a Sphere and through 
 
 a Given Straight Line without the Sphere 115 
 
 128. Definitions. — Hyperboloid of Revolution; Hyperboloid of Revo- 
 
 lution of two Nappes; Torus 116 
 
 129. P'roblem 59. — To Draw the Third Angle Plan and Elevation of an 
 
 Hyperboloid of Revolution with its Axis Perpendicular to H 
 and to Assume a Point on its Surface 116 
 
 130. Problem 60. — To Draw the Third Angle Plan and Elevation of a 
 
 Torus which is Inclined to both V and H 117 
 
 CHAPTER VII 
 
 Sections 
 
 131. Definitions and General Considerations. — A Cutting Plane; 
 
 A Section; The Line of Intersection; Longitudinal Section; Trans- 
 verse Section; Oblique Section; The Five Sections which may be 
 cut from a Cone, viz: A Circle, A Triangle, An Ellipse, An 
 Hyperbola, A Parabola; The Section which can be Cut from a 
 Sphere 119 
 
 132. Problem6i.^ — ToDetermineany Section of a Right Square Prism. 120 
 
 133. Problem 62. — To Determine any Section of any Right Prism 122 
 
 T34. Problem 63. — To Determine the Section Cut from a Right Cylin- 
 der by a Plane Making a Given Angle with its Axis 122 
 
 135. Problem 64. — To Determine the Section of a Pyramid when Cut 
 
 by a Plane Making a Given Angle with its Axis 124 
 
 136. Problem 65. — To Determine the Section of a Right Cone when 
 
 Cut by a Plane Making a Given Angle with its Axis 124 
 
 137. Problem 66. — To Determine the Section of an Annular Torus 
 
 when Cut by a Plane Making a Given Angle with H 127 
 
XIV CONTENTS 
 
 Section Pagb 
 
 CHAPTER VIII 
 
 Intersections and Developments 
 
 138. Definitions and General Considerations. — The Line of Intersec- 
 
 tion; The Development of a Surface; General Method of Deter- 
 mining the Line of Intersection; Development of a Solid; 
 Double-curved Surfaces cannot be Developed; Development 
 of a Cube, Prism, Pyramid, Cone, Cylinder 129 
 
 139. Problem 67. — To Determine the True Form of the Line of Inter- 
 
 section of an Hexagonal Prism when Cut by a Plane Meeting its 
 Axis at any Given Angle; and to Develop the Surface of the 
 Prism ■ 130 
 
 140. Problem 68. — ^To Determine the True Form of the Line of Intersection 
 
 of a Right Cylinder Cut by a Plane Meeting Its Axis at a Given 
 Angle and to Develop the Surface of the Cylinder 131 
 
 141. Problem 69. — To Determine the True Form of the Line of Inter- 
 
 section of a Pyramid Cut by a Plane which Meets the Axis of the 
 Psnramid at Any Given Angle, and to Develop the Surface of the 
 Pyramid 132 
 
 142. Problem 70. — To Determine the True Form of the Line of Inter- 
 
 section of a Right Cone which is Cut by a Plane Meeting the Cone 
 Axis at Any Given Angle, and to Develop the Surface of the Cone. . 134 
 
 143. Problem 71. — To Determine the Line of Intersection of a Prism 
 
 when Cut by a Plane Inclined to V and H. Also to Develop the 
 Surface and to Determine the True Form of the Line of Intersection 134 
 
 144. Problem 72. — To Determine the Line of Intersection of a Cylinder 
 
 Cut by a Plane Inclined to V and H. Also to Develop the Surface 
 and Determine the True Form of Intersection 136 
 
 145. Problem 73. — To Determine the Line of Intersection of a Pyramid 
 
 Cut by a Plane which is Inclined to V and H. Also to Develop the 
 Surface and Determine the True form of the Line of Intersection . . . 137 
 
 146. Problem 74. — To Determine the Line of Intersection of a Cone Cut 
 
 by a PIcme Inclined to V and H. Also to Develop the Surface of 
 
 the Cone and Determine the True Form of the Line of Intersection. 138 
 
 147. Problem 75. — To Determine the True Form of the Line of Intersection 
 
 of Two Square Prisms Meeting at Any Angle and to Develop the 
 Surfaces of the Prisms 138 
 
 148. Problem 76. — To Draw the Plan and Elevation of Two Right Cylin- 
 
 ders the Axes of which meet at any Given Angle; to Develop the 
 Surfaces and to Determine the True Form of the Line of Inter- 
 section 142 
 
 I4Q. Problem 77. — To Draw the Plan and Elevation of a Pyramid and 
 Prism which Intersect; to Develop the Surfaces and Determine the 
 True Form of the Line of Intersection 144 
 
 1 50. Problem 78. — To Draw the Plan and Elevation of a Cone and Cylinder 
 
 which Intersect and to Develop the Surfaces and Determine the 
 True Form of the Curve of Intersection 148 
 
 151. Problem 79. — To Determine the Curve of Intersection of Two Double 
 
 Curved Surfaces of Revolution Whose Axes Intersect 151 
 
CONTENTS XV 
 
 Section Page 
 
 CHAPTER IX 
 
 IsoMETKic Projections 
 
 152. Introductory. — Isometric Drawings Explained; Method of Making 
 
 Isometric Drawings — Coordinate Axes; Coordinate Planes; Isometric 
 Origin and Isometric Axes, Defined. — Isometric Scale Explained. ... 133 
 
 153. Fundamental Principles on which Isometric Drawing is Based 155 
 
 IS4-, Problem 80. — To Determine the Isometric Projection of a Point 
 
 Which Lies in one Coordinate Plane and at a Given Distance from 
 
 the Other Two Coordinate Planes 155 
 
 155. Problem 81. — To Determine the Isometric Projection of Any Point in 
 
 Space when Its Position Relative to the Coordinate Planes is Known 156 
 
 156. Problem 82. — To Determine the Isometric Projection of Any Straight 
 
 Line when the Position of Two of Its Points is Known with Reference 
 
 to the Coordinate Planes „ 156 
 
 157. Problem 83. — To Draw the Isometric Projection of a Square Figure of 
 
 Given Dimensions which Lies in one Coordinate Plane and at 
 Known Distances from the Other Two 157 
 
 158. Problem 84. — ^To Draw the Isometric Projection of a Cube 157 
 
 159. Problem 85. — To Draw the Isometric Projection of an Hexagonal 
 
 Pyramid 158 
 
 160. Problem 86. — To Make an. Isometric Drawing of a Circle 159 
 
 161. Problem 87. — To Make an Isometric Drawing of Any Plane Figure. . i'S9 
 
 162. Problem 88. — To Make an Isometric Drawing of a Mortise and 
 
 Tenon 159 
 
 163. Problem 89. — To Make an Isometric Drawing of a Piping System . . . 159 
 
 164. Cavalier or Cabinet Projection, Explained and Illustrated 160 
 
 165. Pseudoperspective, Explained and Illustrated 160 
 
 CHAPTER X 
 
 Set op Drawing Exercises in Descriptive Geometry 
 
 166. Introductory — Instructions on the Solution of Exercises 162 
 
 167. Plate I. Exercises 1 to 23. — Fundamentals of Point, Line and 
 
 Plane Figure Projections. Change of Position of Points and Use 
 
 of Profile Plane 165 
 
 168. Platen. Exercises 26 to 32. — Elementary Problems of the Line 169 
 
 169. Plate in. Exercises 33 to 42. — Problems of Revolution 172 
 
 170. Plate IV. Exercises 43 to 49. — Line Problems Illustrated by 
 
 Practical Applications i75 
 
 171. Plate V. Exercises 50 to 59. — Elementary Problems of the Plane 178 
 
 172. Plate VI. Exercises 60 to 69. — Continuation of Plane Problems. 180 
 
 173. Plate Vn. Exercises 70 to 76. — Problems of the Plane and Cone 
 
 with its Base in a Given Plane 182 
 
 174. Plate Vm. Exercises 77 to 81. — Problems of the Helix 185 
 
 174(a). Substitute for Plate vm. Exercises 77 to 81 185 
 
 175. Plate IX. Exercises 82 to 88. — Problems of the Tangent Plane 
 
 and Double Curved Surfaces 186 
 
XVI CONTENTS 
 
 Section page 
 
 175 (a). Substitute for Plate IX. Exercises 82 to 88. 190 
 
 176. Plate X. Exercises 89 to 90. — Intersections and Developments 
 
 of Pyramids 192 
 
 177. Plate XI. Exercises 91 to 93. Intersections and Developments 
 
 of Prisms 195 
 
 177 (a). Substitute for Plate XI. Exercises 91 to 93. — Intersection and 
 
 Development of 90 Degree Cylindrical Elbow 195 
 
 178. Plate Xn. Exercise 94. — Intersection and Development of Coni- 
 
 cal Connection for Gutter 198 
 
 179. Plate Xni. Exercise 95. — Intersection and Development of In- 
 
 tersecting Pipes 198 
 
 180. Plate XIV. Exercises 96 to 97. Drawings of Back Outlet Flanged 
 
 Elbow; and End of Connecting Rod 199 
 
 181. Plate XV. Exercises 98 to 100. — Isometric Drawing of a Cube, 
 
 Chipping Exercise and Eccentric 200 
 
 181 (a). Substitute for Plate XV. Exercises 98 to 100. — Isometric 
 Drawing and Cabinet Drawing of Cube and of Tool Rest Support 
 Slide. Isometric of Tail Stock Clamp 202 
 
 182. PlateXVI. Roof and Stack Problem 203 
 
 183. Plate XVn. Belt Problenj 203 
 
 184. Plate XVin. Transition Piece. — From Rectangular Opening to 
 
 Round : 207 
 
 185. Plate XIX. Locomotive Slope Sheet. — Development of Warped 
 
 Surface 207 
 
 186. Plate XX. Development of the Sphere. — Zone Method and Gore 
 
 Method 209 
 
 187. Material for Additional Exercises — Fig. 177, Layout for Plate XX. 
 
 Fig. 178, Conical Helix and Development. Fig. 179, Screw 
 Conveyor. Fig. 180, Funnel. Fig. 181, Locomotive Smoke 
 Stack. Fig. 182, Hopper. Fig. 183, Hood. Fig. 184, Scale 
 Scoop. Fig. 185, Boat Ventilator. Fig. 186, Exhaust Head. 
 Fig. 187, Breeching 211 
 
ELEMENTS OF 
 DESCRIPTIVE GEOMETRY 
 
 CHAPTER I 
 INTRODUCTION 
 
 1. Descriptive Geometry is that branch of geometry which 
 deals with the solution of problems involving points, lines, planes, 
 and solids; also the methods of representing these problems and 
 their solutions upon a sheet of drawing paper in such a way that 
 the position of points, the position and length of lines, the posi- 
 tion, length, and breadth of planes, and the position, length, 
 breadth, and thickness of solids can be determined with unfailing 
 accuracy. 
 
 2. The practical value of Descriptive Geometry lies in the 
 knowledge gained in solving graphical problems which arise in 
 engineering and architecture, and in making and reading work- 
 ing drawings; also, in the sense of accuracy acquired, and in 
 the training of the mind to analyze, and the imagination to vis- 
 ualize, graphical problems. The power to visualize a problem 
 consists in being able to call up a mental picture that will enable 
 the draftsman to see all the parts of the problem in their true 
 geometrical relations without the aid of a model. This power 
 is characteristic of all successful inventors, designers, and con- 
 structive engineers, and the student should endeavor to develop 
 along this Hne by visualizing each problem presented. 
 
 3. The latter part of the definition of Descriptive Geometry 
 (see § i) is very important, since it indicates the properties of the 
 data dealt with in the subject: that is, the only geometric prop- 
 erty of a point is its position, and therefore it is entirely 
 defined when its position in space is known; a line has not only 
 
ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 position, but also length; a plane has position, length, and 
 breadth; while a solid has position, length, breadth, and thickness. 
 
 4. Position indicates the exact location of an object in space, 
 and the simplest way to give the position of an object is to state 
 the distance and direction it is from a second object, whose posi- 
 tion is always known. 
 
 5. EXPERIMENT I. Cut and letter a piece of drawing 
 paper as shown in Fig. i. 
 
 Crease the paper along the lines G-L and Gi-Li and glue the 
 sides with gummed stickers to form the " solid angle " shown in 
 
 Fig. 2. Secure a fine plumb 
 
 <- 
 
 FRONT WALL V 
 
 o 
 
 FLOOR H 
 
 Fig. 1. — Paper Cut and Marked for Experiment i. 
 
 line with a small "bob"; tie 
 a knot D in the line so that 
 it will be exactly three inches 
 from the point of the bob, 
 after the knot E has been 
 tied. Other knots are then 
 tied to correspond to A, B, 
 and C. 
 
 Hold the soHd angle formed 
 by the paper so that the 
 " front waU " and " side wall " 
 are vertical and the " floor " 
 is horizontal, or in the position of an ordinary floor. Then, facing 
 the front wall (this places the side wall to the left), hold the plxunb 
 Hne so that it is located according to the dimensions shown in 
 Fig. 2. If instructions have been accurately followed the plumb 
 line will be perpendicular to and the point of the bob wiU just 
 touch the " floor." Mark this point d where the bob touches 
 the " floor " and note that it marks the foot of a perpendicular 
 (the plumb line) to a horizontal plane (the " floor ") and through 
 the point D in space. The point d on the " floor " is called the 
 horizontal projection of the point D in space. 
 
 Next, with the plumb line still in position, pass the wire No. i 
 through the point D (the knot) in such a manner that it will 
 just touch the "front-wall" and be perpendicular to it. This 
 
INTRODUCTION 
 
 point where the wire touches the ""front-wall " is marked d', and 
 it is termed the vertical projection of the point D in space. In 
 a similar manner, with wire No. 2, locate and mark the point 
 di on the side wall. The 
 point di is also termed a 
 vertical projection, because 
 it, too, marks the foot of a 
 perpendicular to a vertical 
 plane and through the point 
 D in space. At this stage 
 of the experiment note that 
 the number of vertical pro- 
 jections a point in space may 
 have depends only upon the 
 number of vertical planes con- 
 sidered. If only the front 
 plane is considered, the point 
 being projected has one ver- 
 tical and one horizontal pro- 
 jection. When the " side 
 wall" is also considered the 
 point has two vertical pro- 
 jections {d' and di) but only 
 one horizontal (</) projection. 
 A third vertical wall might 
 have existed on the right, in which case a third vertical projec- 
 tion coxild have been located on this wall and so on for any 
 number of vertical planes. Note especially that the vertical 
 projection shows how far the point in space is above the hori- 
 zontal plane (the floor). Hence, all vertical projections of the 
 'same point on different vertical planes are the same distance 
 from their respective G-L lines, because these G-L lines lie in 
 both the vertical and horizontal planes. 
 
 With wires No. i and No. 2 stiU in position (that is, through 
 the point D in space and perpendicular respectively to the front 
 and side walls) observe that three measurements must be made 
 to definitely locate the point D in the paper angle. 
 
 For example, suppose the point D to be an electric light bulb 
 
 Fig. 2. — To Illustrate the Principles of Orthograpliic 
 Projection. 
 
4 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 which is to occupy some exact position over a desk in a room, it 
 wUl be necessary to instruct the workman exactly how far the 
 light is to be from the side wall, the front wall, and the floor. 
 These measurements are always understood to be the shortest 
 distances, i.e., the perpendicular distances, and all three must 
 be known in order to fix the point where the light is to hang. 
 To illustrate, unless the distance D-d is fixed, the bulb could 
 occupy any position on the plximb line such as ^, B, C, or E, and 
 its distance from the floor would not be fixed, although' it would 
 be the proper distance from both vertical walls. Similarly, unless 
 the distance of the bulb from the front wall is fixed it might be 
 the proper distance from the side wall and floor, but could occupy 
 any position whatever along the wire No. i. Similar reasoning' 
 applies to wire No. 2. 
 
 Next observe that the following principle is true, namely the 
 horizontal and one vertical projection of the point D would fix 
 definitely its position in the paper angle ; this is due to the fact 
 that the horizontal projection shows how far the point is from 
 both vertical planes (the front and side walls) and either, one of 
 the vertical projections shows how far the point D is from the 
 horizontal plane. 
 
 Again referring to Fig. 2, place a triangle against the front 
 wall as shown and draw a dashed line d'-x from the point d' 
 to the floor; then rotate the triangle to the " floor " so it lies in 
 the dotted position shown and draw th* dashed Hne d-x from 
 the point d to the front wall. If this has been properly done the 
 dashed lines d'-x and d-x meet at the point x on G-L and each 
 is perpendicular to G-L. Also the dashed line d'-x drawn on 
 the front wall is exactly equal and parallel to the line D-d in 
 space, and the hne d-x drawn on the floor is exactly equal and 
 parallel to the part D-d' of the wire No. i. The dashed lines 
 di'-Xi and d-Xi are similarly drawn and are equal and parallel 
 to their corresponding line in space. 
 
 Next remove the plumb Hne and wires, open the paper angle, 
 spreading it out flat on the drawing board as it was originally 
 in Fig. I. The result is shown in Fig. 3 which is a graphic record 
 or Descriptive Geometry drawing of all that has taken place. If 
 the drawing as shown in Fig. 3 had been made first, the position 
 
INTRODUCTION 5 
 
 of the point D in space could have been fixed, by reversing the 
 process. The operation by which the vertical and horizontal 
 projections of the point D were found and the drawmg produced 
 is termed Orthographic (or 
 perpendicular) projection, and 
 is the method employed by 
 the draftsman in making me- 
 chanical drawings. 
 
 In making a Descriptive 
 Geometry drawing of this data 
 in Fig. 2 the rectangle to the 
 left of Gr-Li made to repre- 
 sent the side wall, in Fig. 3, 
 would be omitted. The left- 
 hand border line of the draw- 
 ing paper would serve as the 
 line Gi-Li for measuring off the distance 2 inches shown in Fig. 2 
 for the vertical projection di gives no information not con- 
 tained on the front wall. That is, both vertical projections 
 simply tell that the point in space is 3 inches above the floor. 
 
 
 
 
 FRONT WALL V 
 d' 
 
 
 
 / 
 
 ^ 
 
 r- 
 
 X . 
 
 0. 
 
 Ul 
 Q 
 
 V M 
 
 _J 
 
 4 ' L 
 
 
 . ^" ^ 
 
 3 
 
 d 
 
 FLOOR H 
 
 
 
 (. 
 
 Fig. 3. - 
 
 -Descriptive Geometry Drawing of Data 
 of Fig. 2. 
 
 6. The expression " in space " is constantly used in Descrip- 
 tive Geometry and in general means that the object (spoken of 
 as being " in space ") does not lie on or coincide with the planes 
 of projection (i.e., the " walls " or " floor "). Thus, the point D 
 in Fig. 2 is " in space " but the projections d' and d are in the 
 vertical and horizontal planes of projection. Any material object 
 such as the pliunb line and the wire projectors used in the experi- 
 ment are considered as existing " in space," but the dashed lines 
 d~x, d'-x, etc;, ruled on the paper, are spoken of as being in the 
 planes of projection. 
 
 7. The expression " projecting a point in space " means that 
 a shadow of the point has been thrown forward upon the plane 
 of projection by a sjngle ray of hght from the eye, somewhat 
 after the manner that a lantern projects the picture from a slide 
 upon a screen. Note especially that points, not objects, are pro- 
 jected by a single ray of Ught and that this ray must always be 
 
ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 perpendicular to the plane upon which the point is to be projected. 
 Figure 4 illustrates how a line A-B might be projected from the 
 slide to a screen by a lantern, and Fig. 5 (a) illustrates how it 
 would be orthographically projected by 
 the eye, upon the vertical plane of 
 projection. In Fig. 4 the rays of light 
 diverge and form a triangular sheet of 
 hght with its base on the screen, but in 
 Fig. 5 (a) a single ray of light first pro- 
 jects the point A to a' and the positi(Mi 
 of the eye must then be changed so as 
 to project B to b' and the straight line connecting a' and h' is 
 the projection of the line A~-B on the vertical plane. 
 
 Assume that in Fig. 5 the points in space {A and B) are the 
 same distance from the vertical plane V, then the line A-B in 
 space is parallel to V, the ray of Hght A-a' is equal to B-b', and 
 the two rays are also parallel because both are perpendicular to V. 
 Hence a'-b' is equal to A-B or this truth has been established: 
 namely, if a line in space is parallel to a plane of projection its 
 projection on that plane is equal in length to the line itself. 
 
 Fig. 4. — Projecting a Line by 
 Divergent Rays. 
 
 (a) Projecting a Line by Par- 
 allel Kays. 
 
 (b) How the Projection of a Line 
 is affected by Revolution. 
 
 Fig. 5. 
 
 In Fig. 5 (b) assmne that A-B is the radius of a circle, B being 
 the center. Revolve A about B and in the plane A-a'-b'-B 
 xmtil the radius A-B occupies the position Ar-B, that is, until 
 it is perpendicular to the V plane, and note the effect. When 
 the point A in space assimies such a position as Ai the eye would 
 be in the second position shown and the vertical projection of 
 the line Ai-B woiild be ai'-b'. When the point A reached A2 
 both A2 and B would be projected by the same ray of hght, 
 and the line A2-B would be projected as a point (marked ai'-V 
 
INTRODUCTION 7 
 
 for convenience) on the V plane. This serves to illustrate the 
 principle that the maximum projection of a line in space is its 
 own length, and occurs when the line is parallel to the plane of 
 projection. The minimum projection that a line in space can 
 have is a point, and occurs when the line is perpendicular to the 
 plane on which it is projected. 
 
CHAPTER II 
 
 DEFINITIONS, NOTATION, AND FIRST PRINCIPLES, 
 PROBLEMS ON THE POINT 
 
 8. Principal Planes of Projection. As stated in § 4, page 2, 
 the position of any object can be fixed by giving the distance 
 and direction it is from some other object the position of which 
 is always known. 
 
 The standard objects from which positions (that is, distances 
 and directions) are given in Descriptive Geometry, consist of 
 two planes at right angles to each other. These are termed the 
 planes of projection and they always occupy a fixed position in 
 space. 
 
 9. One of these planes is termed the horizontal plane of pro- 
 jection, because its position in space is always horizontal and 
 for brevity it is designated -as H. 
 
 ID. The other plane is at right angles to H and is termed the 
 vertical plane of projection because its position in space (being 
 at right angles to H) js always vertical. This plane is desig- 
 nated as V. 
 
 11. The line of intersection of these two planes (V and H) is 
 termed the ground line and is designated as G-L. 
 
 12. A line [real, as the wire in Fig. 2; or imaginary, as the 
 ray of light in Fig. 5 (a)] passing through a point in space' and 
 perpendicular to a plane of projection is termed a space projector. 
 
 13. The point where a space projector pierces V is termed the 
 vertical projection of all points in space, through which the space 
 projector passes. The vertical projection shows how far the 
 point in space is from H [see Fig. 6 (a)]. Thus the point a', 
 where the ray of light from the eye (perpendicular to V) pierces 
 V, is the vertical projection of the point A which is in space. 
 
DEFINITIONS, NOTATION, ETC. 
 
 
 (a) To Project a Point which is in Space. 
 
 (b) The Projection of the Point recorded on Paper. 
 Fig. 6. — Method of Projecting a Point. 
 
lO ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 14. The point where a space projector pierces H is termed the 
 horizontal projection of all points in space through which the 
 space projector passes. The horizontal projection shows how far 
 the point in space is from V [see Fig. 6 (a)]. 
 
 The point a, where a space projector (perpendicular to H) 
 pierces H, is the horizontal projection of A . 
 
 15. A straight dashed line ruled on the drawing paper per- 
 pendicular to G-L and containing the vertical and the horizontal 
 projections of a point is termed a ruled projector. If the paper 
 is creased along G-L and held in the position of V and H [see 
 Fig. 6 (a)], each ruled projector is equal and parallel to a space 
 projector which is directly opposite and for this reason can repre- 
 sent space projectors on the drawing. 
 
 16. Orthographic projection is a method of projecting points 
 in space upon planes of projection (V and H), by lines (space 
 projectors) perpendicular to these planes. By projecting a suf- 
 ficient number of points of an object an accurate drawing or 
 projection of the object can be made. Projecting by the use o^ 
 space projectors is usually an imaginary operation [see Fig. 6 
 (a)], and the real process [see Fig. 6 (b)] consists in dividing the 
 drawing paper into two parts, by a straight line to represent 
 G-L; all the area of the paper on one side of G-L being consid- 
 ered as if in V and all on the other side as if in H, and the V and 
 H projections of the points are located in these areas respectively 
 and on ruled projectors drawn perpendicular to G-L. 
 
 ' 17. THEOREM I. The vertical and the horizontal projec- 
 tion of a point determines its position in space with reference 
 to V and H. 
 
 Proof. — If a perpendicular to V is passed through the vertical 
 projection of the point it will pass through the point in space. 
 If a perpendicular to H is passed through the horizontal pro- 
 jection of the point, it, also, will pass through the point in space. 
 
 Hence, as the point in space is common to both perpendiculars 
 it must be at their intersection. These perpendiculars coincide 
 with the space projectors and the point is fully determined. 
 
DEFINITIOlsrS, NOTATION, ETC. 
 
 II 
 
 i8. Corollary I. The vertical and horizontal projections of a 
 point are on one and the same straight line (the ruled pro- 
 jector), perpendicular to the ground line (see Fig. 6 (b), page 
 9), because the ruled projector from the vertical projection is 
 perpendicular to H and therefore to G-L; the ruled projector 
 from the horizontal projection is perpendicular to V and there- 
 fore to G-L. The ruled projectors are parallel and equal to the 
 space projectors, and these two sets of linestform a rectangle, 
 and hence the ruled projections must meet at a point. The only 
 point possible is on G-L and when V and H are folded into 
 one plane [see Fig. 6 (b)], the ruled projectors drawn on V and 
 H fall 180 degrees apart, hence lie in the same straight hne. 
 
 19. Corollary II. A point situated upon either plane of pro- 
 jection is its own projection on that plane, and its other projection 
 is in G-L [see Fig. 7, IV, V, and VI, this page]. 
 
 20. A point in space may occupy any one of six different posi- 
 tions with reference to V and H [see Fig. 7 (a)]. 
 
 (a) The Point Itself. 
 
 I 
 
 II 
 
 in 
 
 IV 
 
 V VI 
 
 a' 
 
 W 
 
 
 Bd' 
 
 V 
 
 » 
 
 t 
 
 0' 
 
 T 
 1 
 
 T 
 1 
 1 
 1 
 
 e' f' 
 I Pi L 
 
 
 * 
 
 1 
 
 d 
 
 T ^f 
 
 i 
 
 b 
 
 i 
 
 
 1 
 
 a 
 
 
 c 
 
 
 Ee 
 
 H 
 
 Fig.'7. 
 
 (b) Projections. 
 -Positions wliich a Point may Occupy. 
 
 (i) It may be the same distance from both V and H [see 
 
 Fig. 7 (a)-I]. 
 
 (2) It may be nearer to V than H [see Fig. 7 (a)-II]. 
 
 (3) It may be nearer to H than V [see Fig. 7 (a)-III]. 
 
 (4) It may lie in V [see Fig. 7 (a)-IV]. 
 
 (5) It may lie in H [see Fig. 7 (a)-V|. 
 
 (6) It may lie in V and H (therefore in G-L) [see Fig. 
 VI.] 
 
 7(a)- 
 
12 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Figure 7 (b) illustrates the method of representing each of the 
 six cases on the drawing paper. 
 
 21. PROBLEM I. Given the position of a point in space to 
 determine its projections. 
 
 Let the point be | inch from H and i inch from V [see Fig. 
 8 (b)]. 
 
 Analysis. The vertical projection shows the perpendicular 
 distance that the point in space is from H; the horizontal pro- 
 
 (b) The Point Itself. 
 
 (a) Projections. 
 
 Fig. 8. — To determine a' Point by its Projections. 
 
 jection shows the perpendicular distance it is from V, and both 
 projections must he on the same straight Une perpendicular to 
 G-L. 
 
 Construction. See Fig. 8 (a). Draw an indefinite horizontal 
 line to represent G-L; draw an indefinite line perpendicular to 
 G-L (ordinate) and lay off a point § inch above G-L for the 
 vertical projection and a point i inch below G-L for the hori- 
 zontal projection. 
 
 Line Conventions. The ground line {G-L) is a light unbroken 
 hne. The ruled projector connecting the vertical and horizontal 
 projections is composed of a series of dashes y\- inch long, sepa- 
 rated by -^ inch spaces and is hght " weight." 
 
 Notation. The ground Une has G written at one extremity 
 and L at the other. 
 
DEFINITIONS, . NOTATION, ETC. 
 
 13 
 
 The point in space is designated by a capital letter (A in this 
 case) ; its vertical projection by a' and its horizontal projection 
 by a. 
 
 22. 
 
 PROBLEM 2. Given the projections of a point on V and 
 H to find its new vertical projection on any new vertical plane. 
 
 Analysis. The same horizontal projection serves for any 
 nrnnber of new vertical planes (Experiment I, page 2). By § 18, 
 page II, the horizontal projection and the new vertical pro- 
 jection must lie on the same straight line, perpendicular to the 
 new ground Hne Gi-Li. 'By Experiment I, also Fig. 9 (b) the 
 
 (b) Space Analysis. 
 
 (a) Projections. 
 Fig. 9. — To determine New Vertical Projections on a New Vertical Plane. 
 
 new vertical projection on any new vertical plane, such as Vi or 
 V2, will be the same distance from the new ground lines Gi-Li 
 and G2-L2 as the original vertical projection was from the 
 original G-L. 
 
 Construction. See Fig. 9 (a). Draw the indefinite straight 
 line G-L to divide the drawing paper into two parts, the area 
 above G-L to represent V revolved back into the plane of the 
 drawing board, and the area below G-L to represent H. Draw 
 an indefinite straight Hne perpendicular to G-L to serve as the 
 ruled projector for the projections of the point on V and H. 
 Assume that the point A in space is \^ inch from H and | inch 
 from V. The vertical projection is then found by measuring 
 U inch above G-L on the ruled projector. The horizontal pro- 
 
14 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 jection is found by measuring | inch below G-L on the same 
 ruled projector. 
 
 To draw Gi-Li first determine the point Zi (this would be 
 given in the problem), where the new vertical plane Vi meets V 
 and then draw Gi-Li so that it makes the same angle (a) with 
 G-L that the plane Vi makes with V. From a, the horizontal 
 projection of A, draw a-line of indefinite length and perpendicular 
 to Gi-Li. This is the ruled projector containing the vertical 
 projection of A on Vi. Measure off ^^ inch (the distance the 
 point A is from H) from Gi-Li on this projector and ai is the 
 new vertical projection sought. The vertical projection on the 
 plane V2 is found in a similar manner. 
 
 NOTE. Observe that the distance (-^^ inch) which the verti- 
 cal projections are above their respective groimd lines is the same 
 in each case, because the same H serves for V, Vi, and V2. The 
 distances, however, from the horizontal projection a to G-L, 
 Gi-Li, and G2-L2 are different because the perpendicular distance 
 from the point A in space to the planes V, Vi, and V2 is differ- 
 ent. Let no confusion arise from the fact that Oi and 02' are 
 located on a portion of the paper that was originally set aside 
 as H. When the plane Vi was folded away from the point A 
 in space to bring Vi into the plane of the paper, ai fell at the 
 place indicated, then all the surface of the paper on the left of 
 Gi-Li became vertical plane Vi area when working with Gi-Li; 
 but it represents H when working with G-L. When V2 was 
 folded away from the point A in space into the plane, of the 
 paper, it also fell on the original H and the surface below G2-L2 
 became vertical plane area when working with V2 and G2-L2. 
 Do not pass over this note without understanding it thoroughly, as 
 the principles involved are of constant recurrence. 
 
 Notation. The original or principal vertical plane is desig- 
 nated as V, the first new vertical plane as Vi, the second V2, etc. 
 The lines marking the intersection of these new planes with H are 
 notated as Gi-Li, Gi-Li, etc., and the new vertical projections 
 on these planes are primed as on V and the sub-nmnber of the 
 plane (i, 2, etc.) is added. Thus a/ is the vertical projection of 
 A on Vi and a^' is the vertical projection of the same point on 
 Va, etc. Any number of new vertical planes can thus be used. 
 
DEFINITIONS, NOTATION, ETC. 
 
 IS 
 
 23. The data for problems in Descriptive Geometry consist in 
 the location of isolated points in space and of points which limit 
 the dimensions of lines, planes, and solids. As a point has 
 no dimensions it is fully specified by its position. A line is 
 fully specified by its position and length; a plane by its posi- 
 tion, length, and width; and a solid by its position, length, 
 width, and height. 
 
 Notice very carefully that the position and length of a straight 
 Hne are determined by the position of its two ends; that the 
 position of a plane is determined by a straight Hne and a point 
 {or by three points only) which He in the plane, and its length and 
 width are determined by several Hmiting Hnes; that the position 
 and dimensions of soHds are determined by the position and 
 dimensions of the planes outHning their form. Hence it is £een 
 that all the data used in Descriptive Geometry may be stated in 
 terms of the position of points, and whether thus stated or not 
 it will be finaUy used in this form and every principle involved 
 in the Descriptive Geometry of the point must be understood 
 thoroughly. 
 
 24. The Four Dihedral Angles. Thus far all points and Hnes 
 in space have been made to occupy a position in front of V and 
 
 Fig. 10. — The Four Dihedral Angles. 
 
 above H, and no mention has been made of points lying behind 
 V and below H. In order to have the methods of Descriptive 
 
i6 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Geometry general, they must deal with data which may involve 
 any position whatever in space, and this is accomplished by con- 
 sidering V to extend b^low H, and H to extend beyond V, thus 
 forming four equal dihedral angles (see Fig. lo, page 15) and 
 numbered ist, 2d, 3d, 4th, in the order shown. 
 
 The draftsman always views an object from the first angle 
 (that is, the eye is always above H and in front of V) regardless 
 of the angle in which that object is located. 
 
 The 1st angle is above the horizontal, and in front of the vertical 
 plane. 
 
 The 2d angle is above the horizontal, and behind the vertical 
 plane. 
 
 The 3d angle is below the horizontal, and behind the vertical 
 plane. 
 
 The 4th angle is below the horizontal, and in front of the vertical 
 plane. 
 
 Figure 11 shows dimensioned planes which can be cut out of 
 cardboard, or very thin celluloid, and used as a model in studying 
 
 
 'U 
 
 rSf, 
 
 STAND SND 
 ANai.E AREA 
 
 en 
 
 1-* ^^ i 
 
 v| 
 
 -12 
 
 Hi' 
 
 ^ 
 
 s 
 n 
 
 3RD AND iTH 
 ANCLE AREA 
 
 S ■— 
 
 Fig. II. — Layout for " Planes of Reference." 
 
 problems invol-ving several angles. Figure 12 shows this assem- 
 bled model viewed edgewise with points A, B, C, D, located in 
 the ist, 2d,. 3d, and 4th angles respectively. As stated above 
 for all angles the position of the eye (as indicated by the arrows) 
 is always in front of V in determining vertical projections, and 
 above H in determining horizontal projections. Therefore for each 
 point projected the ray of light (or space projector) passes from 
 
DEFINITIONS, NOTATION, ETC. 
 
 17 
 
 3RD ANGLE 
 
 the eye through V for vertical projections in the second angle, 
 
 through H for horizontal projections in the fourth angle, and 
 
 through both V and H for 
 
 the projections in the third 
 
 angle. In order to represent 
 
 the vertical and horizontal 
 
 projections on the same 
 
 sheet of paper the planes 
 
 that stand at right angles 
 
 to one another in space must 
 
 be brought into a single 
 
 plane. This is accomplished 
 
 by opening out the first an- 
 gle as shown in Fig. 6 (b), 
 
 page 9, and this would open '''«■ "■ -«^™8 the Point. 
 
 the third angle and close the second and fourth, so that in the 
 
 second angle V and H both fall above G-L and in the fourth 
 
 both fall below G-L. The plane 
 H falls above G-L in the third 
 angle while V falls below G-L. 
 
 The positions of the vertical and 
 horizontal projections relative to 
 G-L for 
 the differ- 
 ent angles 
 (a). Also 
 
 J ST SND 3RD iTH 
 
 ANGLE ANCLE ANCLE ANCLE 
 V VH H 
 ! T T 
 
 I I I 
 
 J ! ^T 
 
 I 
 I 
 
 i 
 
 Fig. 12(a). - 
 
 i 
 HV 
 
 ■ V and H Relative to G-L. 
 
 Fig. 12 
 
 C, and D of Fig. 
 Fig. 
 
 are as shown in 
 the points A, B. 
 12 are shown in projection in 
 
 13- 
 
 With the single exception that the ver- 
 tical and horizontal projections may be 
 located on different sides of G-L for 
 the various angles, all principles estab- 
 lished for the first angle apply to all other Fig. 13.— Projection of a Point 
 , which lies in Each Angle. 
 
 angles. 
 
 25. General Notation. The following notation is general and 
 special notation will be given as required. 
 
1 8 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Points in space are designated by capital letters, beginning with 
 A and continuing through the first part of the alphabet, thus 
 A, B,C, D, etc. 
 
 Horizontal projections of points are designated by the small 
 letters corresponding to the capital letters used to designate the 
 points being projected, thus, a, b, c, d, etc. 
 
 Vertical projections of points are designated in the same man- 
 ner as horizontal projections ea;ce^< the letters are primed, thus, 
 a', b', c', d', etc. 
 
 Lines in space are designated, in general, by the two points 
 which mark their extremities and the two letters designat- 
 ing the Une are separated by a dash, thus, A-B, B-C, C-A, 
 etc. 
 
 The horizontal projections of lines are designated by the 
 small letters corresponding to the capital letters designating 
 the lines in space and are separated by a dash thus, a-b, b-c, 
 c-a, etc. 
 
 The vertical projections of lines are designated in the same 
 manner as horizontal projections except the letters are pruned, 
 thus, a'-b', b'-c', c'-a', etc. 
 
 26. Line Conventions. The following line conventions cover 
 the ordinary cases and special conventions will be given as 
 required. 
 
 (a) The ground line is an unbroken line of light weight with 
 the capital G printed at one extremity and the capital L at 
 the other, thus: G '■ l 
 
 (b) * A ruled projector is a broken line of light weight and is 
 made up of a series of dashes ^ inch long separated by spaces 
 ^inch long, thus: 
 
 (c) A visible line, given or required, is an unbroken line of 
 medium weight, thus: 
 
 (d) * Construction lines used in the solution of a problem are 
 drawn broken and of light weight and are of a long and short 
 dash construction, the long dash being \ inch, the short dash 
 being ^ inch, and aU spaces being ^ inch, thus: 
 
 * Note : Some Instructors prefer these lines made very light solid lines as such 
 lines take less time. 
 
DEFINITIONS, NOTATION, ETC. 19 
 
 (e) All invisible lines are broken and are of medium weight. 
 They are made up of 3 inch dashes separated by ^ inch, thus: 
 
 (f) The path traced by a point in changing its position is indi- 
 cated by a broken line of light weight. The line is composed of 
 dashes \ inch long separated by spaces ^ inch long, and a half 
 arrow shows the direction of motion, thus: 
 
 (g) The direction of sight in the perspective drawings is mdi- 
 cated by a/«// arrow thus: 
 
 (h) When a point changes its position, the second position 
 is indicated as subscript i ; the third position as sub. 2, and so 
 forth. Thus, if the point A changes its position, it becomes 
 Ai and the horizontal projection of Ai is Oi and the vertical 
 projection ai'. 
 
CHAPTER m 
 POINT AND LINE PROBLEMS 
 
 27. EXPERIMENT H. Crease, letter, and fold a piece of 
 drawing paper to represent V and H in space. Thrust wire No. i 
 through V and H so that it Ues across the angle as shown in 
 Fig. 14 (a), call the point where the wire pierces H the hori- 
 zontal piercing point of the wire and mark it Aa. The vertical 
 projection a' of this horizontal piercing point ^o is necessarily 
 in G-L. Call the point where the wire pierces V the vertical 
 piercing poiat and mark it BV. The horizontal projection h of 
 this vertical piercing point Bb' is in G-L. The line coimecting 
 Aa with h is the horizontal projection of the wire and the line 
 connecting a' and BV is the vertical projection of the wire. 
 
 Next thrust wire No. 2 through H at about Jj and sHding it 
 along wire No. i so that the two wires intersect (or touch) at 
 about E thrust the wire to pierce V, and mark this piercing 
 point Ff. Draw, as in the case of wire No. i, the vertical and 
 horizontal projections of wire No. 2. Determine and mark the 
 projections of the point of intersection E of the two wires also, 
 determine the projections of any other two points, as C and D, 
 on wire No. i. 
 
 Remove the wires, fold V backward into the plane of the 
 drawing board (that is, into the plane of H) and if instructions 
 have been accurately followed the result will be as shown in Fig. 
 14 (b). By referring to Figs. 14 (a) and (b), the following im- 
 portant principles are evident. 
 
 1. Lines in space can be represented on a drawing by their 
 
 projections. 
 
 2. A single line in space such as C-D reqmres two pro- 
 
 jections, c'-d' and c-i (the vertical and horizontal), 
 to define its position. 
 
 3. Since a straight line in space is determined by two of its 
 
 points, so the projections of such a line are deter- 
 mined by the projections of two points of the line. 
 
POINT AND LINE PROBLEMS 
 
 21 
 
 Thus the line C-D in space is fxilly determined by 
 the points C and D and the projections of the line 
 are determined by the projections c, c' and d, d' of 
 these points. 
 
 (b) Construction. 
 Fig. 14. — Illustrating Important First Principles. 
 
 4. Since two points determine a straight line in space, a 
 line may be determined by its intersections with the 
 planes of projection, that is, by determining its hori- 
 
22 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 zontal and vertical piercing points. A line can have 
 only two piercing points [see the line A-B, Fig. 14 (a)] ; 
 but if parallel to one plane of projection it will have 
 but one piercing point, and if parallel to G-L will 
 have none. 
 
 5. If a point such as C or Z> is on a line such as A-B the 
 
 projections c and c' or d and d! will be on the projec- 
 tions Aa-b and a'-Bb' of the line. 
 
 6. If any two lines, as A-B and J-P, intersect at some point 
 
 E in space, their vertical projections will intersect at 
 e', and their horizontal projections at e, as these are 
 the vertical and horizontal projections of the same 
 point E in space, and the ruled projector joining 
 e' and e must be perpendicular to G-L. 
 
 7. The horizontal piercing point Aa, a' of any line A-B is 
 
 found by producing its vertical projection Bb'-a' 
 imtil it meets G-L. At this jimction a' draw a per- 
 pendicular a'-aA (to G-L) which wiU cut the , hori- 
 zontal projection b-Aa of the line at the horizontal 
 piercing point. 
 
 8. The vertical piercing point Bb', b of any line A-B is 
 
 found by producing its horizontal projection Aa-b 
 until it cuts G-L at b. From, b draw a perpendicular 
 b-Bb' (to G-L) which will cut the vertical projection 
 a'-Bb' of the line at the vertical piercing point. 
 
 9. The point of intersection E; of two lines A-B and F-J 
 
 in space, has its projections e and e' at the inter- 
 section of the projections of the hne. 
 
 10. The piercing point Aa, a' or Bb', b (or the trace) of a line 
 
 is the point of intersection of the line A-B in space 
 
 with a plane such as V or H. (See problem 3, page 
 
 26, for fxurther discussion of piercing point.) 
 
 28. Possible Positions which a Line may have with reference 
 
 to V and H. A straight line may occupy any one of the following 
 
 general positions in space [see Fig. 15]. 
 
 (i) Parallel to both V and H. [Figs. 15 (a) and (b)-I.] 
 (2) Perpendicular to V (hence parallel to H). [Figs. 15 (a) 
 and (b)-II.] 
 
POINT AND LINE PROBLEMS 
 
 23 
 
 (3) Perpendicular to H (hence parallel to V). [Figs. 15 (a) 
 and (b)-III.] 
 
 (4) ParaUel to H and inclined to V. [Figs. 15 (a) and (b)-IV.] 
 (s) Parallel to V and inclined to H. [Figs. 15 (a) and (b)-V.] 
 (6) Inclined to V and H. [Figs. 15 (a) and (b)-VI.] 
 
 Fig. IS. 
 
 (b) Projections of the Line. 
 -Positions which a Line in Space ma; Occupy. 
 
 (7) In a plane perpendicular to V and H (that is, in a plane 
 perpendicular to G-L). [Figs. 15 (a) and (b)-VII.] 
 
 The lines in space aj^e shown in Fig. 15 (a), and Fig. 15 (b) 
 illustrates the method of representing, on a drawing, the line in 
 each of the above positions. 
 
 If a line does not occupy one of the above seven general posi- 
 tions in space, it may lie in a plane of projection as follows: 
 
 (i) It may lie in V and be perpendicular to H. [Figs. 16 (a) 
 and (b)-I.] 
 
24 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (2) It may lie in V and be parallel to H. [Figs. 16 (a) and 
 (b)-II.] 
 
 (3) It may lie in V and be inclined to H (either direction). 
 [Figs. 16 (a) and (b)-III.] 
 
 (4) It may lie in V and in H, tHat is, in G-L. [Figs. 16 (a) and 
 (b)-IV.] 
 
 11 
 
 (a) The Line Itself. 
 Ill IV V VT 
 
 VH 
 
 Bb 
 
 Aa' 
 
 Aa' 
 
 'ab a 
 
 I I TAaf Bb' afV of 1/ a' V a' b' 
 
 2> a ba b a 
 
 Bb I 
 
 I I 
 
 Aa 
 
 Aa Bb 
 
 Bb 
 
 BS 
 
 (b) Projections of the Line. 
 Fig. i6. — Positions which a Line which lies on a Projection Plane may Occupy. 
 
 (5) It may lie in H and be perpendicular to V. [Figs. i6 (a) 
 and (b)-V.] 
 
 (6) It may lie in H and be parallel to V. [Figs. i6 (a) and 
 (b)-VI.] 
 
 (7) It may lie in H and be inclined to V (either direction). 
 [Figs. 16 (a) and (b)-VII.] 
 
 The lines in the planes of projection are shown in Fig. i6 (a). 
 In Fig. 16 (b) is illustrated the method of representing, on a 
 drawing, the line in each of the above positions. 
 
POINT AND LINE PROBLEMS 
 
 25 
 
 29. Principle. Lines in space are represented on a drawing 
 by their projections. As' both the vertical and horizontal pro- 
 jections of a point represent but a single point in space, so a 
 single line in space reqviires two projections to define its position. 
 
 30. Principle. Two points are always necessary to fully deter- 
 mine a straight line (that is, define its length as well as its 
 directions), but one point and its direction may locate the line. 
 
 31. Principle. Since a straight Une in space is determined by 
 two of its points, similarly the projections of a line are fully 
 
 , determined by the projection^ of two of its points, but the pro- 
 jection of one point and the direction may locate the projection 
 of a line. 
 
 32. THEOREM II. Every point on a line in space has its 
 horizontal projection on the horizontal projection of the line, 
 and its vertical projection on the vertical projection of the line. 
 
 (a) Perspective. 
 
 Cb) Construction. 
 
 Fig. 17. — Projection of the Points of a Line. 
 
 Proof. — Let ^ and B [Figs. 17 (a) and (b)] determine any 
 Une in space. Pass the plane M through A-B and perpendicular 
 to H. Drop the space projectors from A and 5 to H and deter- 
 mine the horizontal projections a and b. Connect a and b with a 
 straight Une. 
 
 The space projector A-a (also B-b) must lie in M because 
 the plane M is perpendicular to H and contains every per- 
 
26 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 pendicxilar that can be drawn from the line A-B to H. The pro- 
 jections a and b therefore he in both M and H as also does the line 
 a-b joining them; hence a-b is the line of intersection of M and 
 H, and must contain every point common to these planes, there- 
 fore it contains the foot of every perpendicular line from A-B to 
 H and is the horizontal projection of A-B. In the same manner 
 a'-b' can be shown to contain the ruled projector from every 
 point of the line A-B. 
 
 33. Principle. If two lines intersect in space, their projections 
 must also intersect, and the ruled projector joining the points 
 in which the two projections intersect must be perpendicular tb 
 G-L; for the intersection of two lines must be a point (common 
 to both lines) whose projections must be on the horizontal and 
 vertical projections of each of the lines, hence at their intersec- 
 tions respectively. See Figs. 18 (a) and (b), also 19 (a) and (b). 
 
 34. PROBLEM 3. To find the piercing points (or traces) of 
 a line. 
 
 Fundamentals. The trace of a line is the point in which the 
 line pierces V or H. 
 
 Since a trace lies in one of the planes of projection it must 
 have one projection in G-L [see Figs. 14 (a) and (b), page 21]. 
 
 Since^the projections of the traces must be on the projections 
 of the line (see Theorem II, page 25), one projection of the 
 trace must lie where a projection of the line cuts G-L, and the 
 other projection, at the intersection of the ruled projector and 
 the other projection of the line. 
 
 (i) To determine where a line pierces V. 
 
 Analysis. Move the point of a pencil along the line from A 
 toward B, [see Fig. 20 (a)], then the horizontal projection of the 
 pencil point follows along a-b; also, the perpendicular distance 
 from any point on the projection a-b to G-L would measure the 
 distance the pencil point on A-B (in space) is from V when at 
 that position. Hence when the horizontal projection of the 
 point reaches G-L the point in space (the pencil point) is on V. 
 And since the vertical and horizontal projection of the point 
 must be on the same ruled projector, the point Cc' where the 
 
POINT AND LINE PROBLEMS 
 
 27 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 18. — Projection of a Point Common to Two Lines. 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 19. — Intersecting Lines have One Point Projected in Common. 
 
 VIST AND 2ND HaNDANDSRU 
 
 Q- 
 
 H1BTAND4TH V4THAND 3RD 
 
 (a) Perspective. 
 
 Fig. 20. — Tlie Vertical Trace of a Line. 
 
 (b) Construction. 
 
28 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 perpendicular from this projection c cuts the vertical projection 
 a'-V of the line (produced if necessary) determines the vertical 
 trace of the line. 
 
 Construction. Extend a-h [see Fig. 20 (b)] until it cuts G-L 
 at c and draw an indefinite perpendicular from c upward. Ex- 
 tend a'-V to intersect this perpendicular at Cc' , the required 
 point. 
 
 (2) To determine where a line pierces H. 
 
 Analysis. See Fig. 2 1 (a) and (b) . The vertical projection a'-V 
 of the line must cut G-L in order that the line shall touch H. 
 
 (a) Perspective. 
 
 (b) Construction. 
 
 Fig.[2i. — The Horizontal Trace ofja Line. 
 
 Draw an indefinite perpendicular from this point c' and extend 
 the horizontal projection a-h of the line until it intersects this 
 ruled projector. This point of intersection Cc establishes the 
 horizontal trace of the Une. 
 
 35. THEOREM ni. If a line in space is parallel to a plane 
 of projection its projection on that plane shows the true length 
 of the line. 
 
 Proof. — Let A-B [Fig. 22 (a) and (b)-I] be parallel to V, then 
 every point on A-B is the same distance from V; hence A-a' is 
 equal to B-V. Also as A-a' and B-V are both space projectors 
 to V, they must both be perpendicular to V; therefore A-a'-V-B 
 is a rectangle and a'-V is equal to A-B. . 
 
 Similarly c-d can be proven .equal to C-D. [Figs. 22 (a) and 
 (b)-II.] 
 
POINT AND LINE PROBLEMS 
 
 I" 
 
 29 
 
 n 
 
 V 
 a! 
 
 
 "^ 
 
 
 1 \ 
 
 V 
 
 1 
 
 c' 
 
 d' 
 
 
 \ L 
 
 
 1 
 1 
 
 \^ 
 
 ^a 
 
 a 
 
 '& 
 
 
 
 
 
 
 H 
 
 
 
 
 (a) Perspective. (b) Constructioii. 
 
 Fig. 22. — A Line Parallel to a Plane of Projection. 
 
 36. Corollary I. The true distance between two points in 
 space can be found by determining the length of the line Joining 
 them. 
 
 37. Corollary II. The angle a true length projection makes 
 with G-L is the same as the angle that the line in space makes 
 with its other projection; hence the same angle as the line in 
 space makes with the plane containing the other projection. 
 
 38. Corollary III. The true angle between two intersecting 
 lines can be found by bringing both lines parallel to the same 
 plane of projection and determining the projections of the Unes. 
 
 39. Corollary IV. To measure off a true distance on a line 
 in space, measure it off on a projection which is parallel to the 
 line. 
 
 40. THEOREM IV. If two lines in space are parallel, their 
 vertical projections will be parallel, also their horizontal pro- 
 jections will be parallel. 
 
 Proof. — Let A-B and C-T) [Figs. 23 (a) and (b)] be any two 
 lines in space which are parallel. 
 
 Determine their piercing points on V and H. 
 
 Then, by hypothesis Aa'-Bh [Fig. 23 (a)] is parallel to Cc'-Dd 
 and by construction Bb-b' is parallel to Dd-d'; hence the angle a 
 is equal to the angle /3. By construction Bb-b'-Aa' = 90° = 
 Dd-d'-€c'. 
 
3° 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Therefore the triangle Bh-b'-Aa' is similar to the triangle 
 Dd-^d'-Cc' and the side Aa'-V is parallel to Cc'-d'. Similarly 
 the horizontal projection a-Bb can be proven parallel to the 
 ^i^Vc - " horizontal projection c~Dd. 
 
 ■ %w 
 
 (a) Perspective. 
 
 (b) Construction. 
 
 Fig. 23. — Projection of Parallel Lines. 
 
 41. Corollary I. If aline A-B [Fig. 24 (a)] in space is parallel 
 to both V and H, then both of its projections [Fig. 24 (b)] 
 will be parallel to G-L. 
 
 (a) Perspective. 
 
 V 
 
 a' b' 
 
 1 1 
 
 
 a b 
 
 H 
 
 (b) Construction. 
 Fig. a4. — A Line Parallel to Both V and H. 
 
 42. Corollary II. If a line in space is parallel to a plane of 
 projection, its projection on that plane is the true length of the 
 line in space, and its projection on the other plane will show 
 parallel to G-L [see Figs. 22 (a) and (b), page 29]. 
 
POINT AND LINE PROBLEMS 
 
 31 
 
 NOTE. The converse of Theorem IV, namely, two lines in 
 space are parallel if both the vertical and horizontal projections are 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 25. —Projection of a Line lying in a Profile.Plane. 
 
 parallel, is true provided the lines do not lie in a profile plane 
 [see Figs. 25(a) and (b)], that is a plane perpendicular • to 
 G-L. 
 
 43. PROBLEM 4. To pass a line through a given point and 
 parallel to a given line. ^ 
 
 Analysis. The vertical projection of the line must pass 
 through the vertical projection of the point, and the horizontal 
 projection of the line through the horizontal projection of the 
 point. The vertical projections of the given and required lines 
 must be parallel, also their horizontal projections must be 
 parallel. 
 
 Construction i. When both projections are inclined to V and H. 
 Let e-e' (Fig. 26) be the projections of the point in space and a-b, 
 a'-V be the projections of the given line. Through e draw c-d, 
 parallel to a-h. Through e' draw c'^d' parallel to a'~b'. c-d and 
 c'-d' are the projections of the required line. 
 
 Construction 2. When the line lies in a plane perpendicular to 
 G-L. Assume a new V (shown at Gi-Li, Fig. 27), and determine 
 
32 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 the new vertical projections e/ of the given point E and ai'-bi 
 of the given line A-B. Through Ci draw ci'-di and determine 
 the position of e' on c'-d'. 
 
 Fig. 26. — A Line passed 
 througli a Given Point 
 and Parallel to a Given 
 Line. 
 
 Fig. 27. — A Line passed through 
 any Given Point and Parallel to 
 a Given Line in a Profile Plane. 
 
 44. EXPERIMENT HI. Crease, letter, and fold a piece of 
 drawing paper to represent V and H in space. Cut out a circular 
 disc of cardboard (about 2 inches in diameter) and mark a point A 
 on its circumference. Cut a slit in H, parallel to G-L, and 2 
 inches long. Place the disc in the slit and with the wire through 
 its center and perpendicular to the plane of the disc, as shown 
 in Fig. 28 (a). The wire must now lie on H, and the disc must 
 be parallel to V. 
 
 Place A in about the position shown and mark its vertical 
 projection a' on V and its horizontal projection a on H; also 
 mark the vertical and horizontal projections c' and Cc of the 
 center point of the disc. Keeping the disc in a plane per- 
 pendicular to the wire (that is, parallel to V), revolve the disc 
 about the wire as an axis imtil A coincides with H at Aai] 
 during this operation the vertical projection of A moves along 
 the arc of a circle a'-ai to the ground hue. The horizontal 
 projection moved along a straight line (the slit) — that is, per- 
 pendicular to the axis (the wire) — and to the new position Aau 
 
 Note that the point A is always & fixed distance Cc-A from the 
 wire during its revolution, and that in every position for A this 
 radius is always the hypothenuse of a right triangle the two sides 
 (Cc-a and A-a) of which are constantly changing. Hence, to 
 
POINT AND LINE PROBLEMS 
 
 33 
 
 revolve any point in space about an axis which lies in or parallel 
 to H until the point also lies in or is parallel to this plane, it is 
 only necessary to draw the projections of the axis [see Fig. 28 (b)], 
 
 a' 
 
 o'l 
 
 PLAKE OF 3 
 
 ROTATION 
 
 \x \a'i 
 
 \ 
 
 I 
 
 Cc « Aa-i 
 
 Fig. 28. 
 
 ^°^^tJo"jJ' 0>) Construction, 
 
 (a) Perspective. 
 — Rotation of a Point about a Line in H.and Perpendicular to V. 
 
 and through the horizontal projection a of the point draw a line 
 a-Aai perpendicular to the horizontal projection of the axis; 
 this determines the center Cc about which A is to revolve and 
 the horizontal projection Aai (the revolved position of ^) must 
 
 (a) Perspective. (c) 
 
 (b) Construction. 
 Fig. 29. — Rotation of a Point about a Line in H and Inclined to V. 
 
 be on this line; the distance from Cc to Aai will be the hypoth- 
 enuse of the right triangle A-Cc-a (in space). 
 
 When the axis of rotation is perpendicular to V, as shown in 
 Figs. 28 (a) and (b), this hypothenuse is projected in its true 
 
34 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 length on V; and hence it is only necessary to swing a' to a\ about 
 c' and draw its ruled projector to Aax as shown in Fig. 28 (b). 
 When the axis Ues in H but is not perpendicular to V as shown 
 in Fig. 29 (a), the hypothenuse Cc-A is not parallel to V and 
 consequently is not shown in true length in c'-a'. It therefore 
 becomes necessary to construct a right triangle [Fig. 29 (c)] with 
 Cc-a as the base, and with an altitude equal to x-a'. In Fig. 29 
 (b), an equivalent of this construction has been carried out by 
 revolving Cc-a parallel to G-L; but this triangle might have 
 been constructed at any convenient position as its only use is to 
 determine the radius of rotation which is then marked off from 
 Cc to determine Aax. 
 
 Figure 30 shows the method of revolving the point when the 
 axis is parallel to but not in the plane H. Note especially that 
 
 (a) Perspective. 
 
 (c) Construction. 
 
 Fig. 30. - 
 
 Rotation of a Point about a I.ine in Space which is Perpendicular to One of the 
 Planes of Projection. 
 
 in this case the altittide A-E of the triangle C-A-E becomes the 
 difference between the original distance A-a which the point A 
 in space is from H, and the distance Ay-ai after the radius has 
 been brought parallel to H. 
 
 45. Axiom. By referring to Experiment III, page 32, it is seen 
 that the radius C-A (which is perpendicular to the wire) projects 
 
POINT AND LINE PROBLEMS 35 
 
 on H as a straight line perpendicular to the projection of the 
 wire, in all positions except when it is perpendicular to this 
 plane H. Hence the important truth is evident that if one of 
 two lines forming a right angle is parallel to a plane of projection, 
 the projection of the right angle will he a right angle unless the 
 second line projects as a point. 
 
 46. The Revolution of Points, Lines, Planes, and Solids. It 
 
 frequently happens that the solution of a problem can be simpli- 
 fied by the revolution of some point, line, plane, or solid involved 
 in the data. 
 
 The method of revolving a point has been illustrated in Ex- 
 periment III, page 32, and the revolution of a line, plane, or 
 solid consists in revolving a sufficient number of points to estab- 
 lish the new position and dimensions of such data. 
 
 Referring to Experiment III, it will be seen that "the revolution 
 of a point involves the following essentials : 
 
 (i) An axis (the wire) of revolution, around which the point 
 revolves. 
 
 (2) A radius of revolution, which is the unchanging distance that 
 the point in space is from the axis during the entire revolution. 
 
 (3) A center of revolution, which is the point about which the 
 revolving point is to move; it marks the intersection of the axis 
 and radius. 
 
 (4) An arc or circle of revolution which represents the path of 
 the revolving point and measures the angle through which it 
 moves. ' 
 
 (5) The plane -of revolution which is always perpendicular to 
 the axis and contains the arc or circle of revolution. 
 
 47. PROBLEM 5. To revolve a point through a given angle 
 and around an axis which is perpendicular to V. 
 
 Analysis. Let B~D [see Fig. 31 (a)] be the axis, A the point, 
 and a the given angle. The point A will move in the arc A-A\ 
 of a circle about the center C and with a radius C-A. The path 
 of the point during revolution will be in the plane of revolution 
 (which is perpendicular to B-D), and therefore parallel to V. 
 
36 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Hence the radius, the path, and the angle moved through by the 
 point must be shown in their true size on V. 
 
 (a) Perspective. 
 
 (c) Construction with Axis in V. 
 
 (b) Construction witli Axis in H. 
 
 Fig. 31.— Revolution of a Point througli a Given Angle around an Axis in One Plane and Per* 
 ' pendicular to the Other Plane of Projection. 
 
 Constructioii. Draw the projections a, a' of the point A, and 
 Bb-Dd, h'-d' of the axis B-D. Figure 31 (b) shows the axis lying 
 inH. 
 
 Through a draw the line a-Cc perpendicular to the horizontal 
 
POINT AND LINE PROBLEMS 
 
 37 
 
 projection Bb-Dd of the axis. Draw the vertical projection c'-a' 
 of the radius of revolution and swing a' through the angle a to 
 ffli'. Draw the ruled projector from ai to meet a-Cc produced. 
 
 (a) Perspective. 
 
 (b) Construction with Axis about H. 
 
 9Lr.^< 
 
 
 »^. 
 
 (c) Construction with Axis in front of V. 
 
 Pig, 32. Revolution of a Point through a Given Angle about a Line Perpendicular to a Plane 
 
 of Projection. 
 
 Then ai is- the horizontal projection and a/ the vertical projec- 
 tion of the point A in its revolved position. 
 
 Figure 31 (c) shows a solution when the axis lies in V and is 
 perpendicular to H. 
 
38 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Figures 32 (a) and (b) show the solution when the axis is per- 
 pendicular to V but above H. 
 
 Figure 32 (c) shows the solution when the axis is perpendicular 
 to H and in front of V. 
 
 The student should be reqmred to analyze, construct, and 
 visualize problems similar to these in the third angle. 
 
 (aj Perspective. 
 
 (c) Construction n. 
 
 (b) Construction I. 
 Fig. 33. — Revolution of a Point about Any Line lying in a Plane of Projection. 
 
 48. PROBLEM 6. To revolve a point in space into H about 
 an axis which lies in H and is oblique to V. 
 
POINT AND LINE PROBLEMS 
 
 39 
 
 Analysis. Let B-D be the axis and A the point [see Fig. 33 (a)]. 
 The point A will move in the arc A-Aa^ of a circle, with center at 
 Cc and having a radius of A-Cc. This radius is the hyppth- 
 enuse of the triangle A-a-Cc which has a right angle at a. 
 The altitude of this triangle is the distance from the point A to 
 H, and the base of the triangle is the perpendi'ctdar distance 
 ,from the axis to the projection a. As the radius A-Cc is always 
 at right angles to the axis B-D, and the center of revolution, 
 Cc does not change and the point A must faU upon a line in H 
 drawn through Cc and perpendicular to B-D. If A is moved 
 toward the left it falls at the point Aa^ so that the distance 
 Cc-Aa2 is equal to the radius of revolution Cc-A. 
 
 Construction. In Fig. 33 (b) let a, a' be the projections of the 
 given point A, and Bb-Dd be the horizontal projection of the 
 axis. From a draw a perpendicular a-Aa2 to the axis Bb-Dd 
 and the intersection Cc of these Hnes is the center of revo- 
 lution. Then a-Cc is the base of the triangle and x-a' is the 
 altitude. The hypothenuse (the radius of rotation) can now be 
 foimd by constructing the triangle. For convenience draw from 
 a the perpendicular a-Aai and mark off the distance Aai-a 
 equal to x-a'; with Cc as a center describe an arc with radius 
 Cc-Aai to cut a-Cc (extended) at Aa^ which is the revolved 
 position of the point A. Figure 33 (c) shows the solution when 
 A is revolved toward V. 
 
 NOTE. If the point A is directly 
 above the axis, the base of the triangle 
 becomes zero and the radius of revolu- 
 tion is equal to the distance of the 
 point above H. 
 
 49. PROBLEM 7. To revolve a point 
 in space into V about an axis which lies 
 in V and is oblique to H. 
 
 Analysis. In this case the altitude of 
 the right triangle whose hypothenuse is Fig. 34.— Revolution of a Point 
 the radius of revolution becomes the dis- '"o"' " """'"^ *^= '"" ^• 
 tance of the point A from V; the base of the triangle is the 
 perpendicular distance of a' from the axis; and the construction 
 is made in V (see Fig. 34). 
 
40 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 50. PROBLEM 8. To determine the distance between two 
 points in space. 
 
 Fundamentals. The distance between two points will be 
 measured by the straight line joining them. This line will be 
 
 (a) Perspective. 
 
 (b) Construction. (c) Consttjiction. 
 
 Fig. 35. — To determine the Distance between Any Two Points in Space. 
 
 projected in its true length on that plane of projection to which 
 it is parallel, or, on which it lies. 
 
POINT AND LINE PROBLEMS 41 
 
 Analysis i. See Fig. 35 (a). Revolve the line A-B joining 
 the points, about the space projector (which can be used as an 
 axis) of any one of its points, say A, until the line is parallel 
 toV. 
 
 Its new projection on V (the plane to which it is parallel) is 
 its true length a'-bi'. Similarly it could have been revolved 
 parallel to H. 
 
 Construction i. See Fig. 35 (b). Let the projections of the 
 given points be a, a' and b, h' . Revolve the projection a-h 
 about a until it is parallel to G-L. Since the point B in space 
 must not change its distance from H, the new vertical projec- 
 tion hi will take a position along the line b'-bi, which is parallel 
 to G-L. The length of the horizontal projection a-b does not 
 change, therefore b moves in the arc of a circle to bi until hi 
 and a are the same distance from G-L. The projection bi 
 must lie on the ruled projector from &i, hence at bi'. Similarly 
 the line could have been revolved parallel to H [see Fig. 35 (c)]. 
 
 Analysis 2. See Fig. 36 (a). If a new vertical plane Vi 
 is assiuned parallel to the Hne A-B in space, the new vertical 
 projection ai'-bi of the line on this plane will be the true length 
 of the Une. 
 
 Construction 2. See Fig. 36 (b). The new Gy-Li must be 
 parallel to the horizontal projection a-b of the line. Ruled 
 projectors are drawn from a and b perpendicular to Gy-Li and 
 fli' and 61' are the same distance from Gi-Lx that a' and b' are 
 from G-L. Join ai and J/ with a straight line and mark it 
 T. L. (true length). 
 
 NOTE. In this method the new V is often made to pass 
 through the Kne in space, hence Gi-Li wiU coincide with a-b, 
 and the distances on the ruled projector from b and a to Gi-Li 
 become zero. The distance from Gi-Li to 61' and a/ remains 
 unchanged [see Fig. 36 (c)]. 
 
 Notation. When the solution of a problem requires that the 
 true length of a line be found, the Kne is marked T. L. as in 
 Fig. 36 and the projection is an unbroken line of medium weight. 
 
 51. PROBLEM 9, To determine the true size of the angle 
 between two intersecting lines. 
 
42 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (a) Perspective. 
 
 (b) Construction. (c) Construction. 
 
 Fig. 36. — To determine the Distance between Two Points in Space. 
 
 Analysis. The two lines must be brought either into or 
 parallel to a plane of projection without changing the angle 
 between them. When in this position the projections of the 
 lines on the plane to which they are parallel represent the true 
 
POINT AND LINE PROBLEMS 
 
 43 
 
 (a) Perspective. 
 
 length of the lines m space and 
 the angle between these projec- 
 tions is the same as the angle 
 between the lines in space. 
 
 If A-B and B-C [Fig. 37 (a)] 
 are the two lines in space and a 
 is the angle, the probleni consists 
 in determining a point (.4 and C) 
 on the lines respectively which 
 is the same distance a-A = C-c 
 from H, and. then revolving the 
 vertex B of the angle about the 
 axis A-C and in the plane of the 
 sector B-D-B2 untU the distances 
 52-62 = A-a = C-c, that is B is 
 brought to the same distance as 
 is A and C from H. The lines 
 A-B and B-C are then parallel 
 to H and a is shown on H as ai 
 in its true size. 
 
 A simple method, however, of 
 performing this operation on the drawing consists in keeping A 
 stationary and revolving B about a vertical axis O-o and in the 
 
 (b) Construction. 
 
 Fig. 37. — To determine the True Angle 
 between Two Intersecting Lines. 
 
44 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 plane of the sector B-O-Bi (which is parallel to H) until A-Bi 
 is parallel to V [see Fig. 37 (a)]. The true distance from A to 
 B is then shown by the vertical projection a'-bi marked TD. 
 
 The true distance from 5 to C is found in a similar manner 
 [not shown in Fig. 37 (a)] and the triangle a-bi-c is constructed 
 on the true distance a-c and this determines the required 
 angle ai. 
 
 Construction. Let a-b, a'-V and h-c, b'-c' [Fig. 37 (b)], repre- 
 sent the projections of the two intersecting Unes A-B and B-C. 
 
 To determine the points A and C the line a'-c' is drawn parallel 
 to G-L and to intersect the vertical projections of the lines. 
 Since a'-c' is parallel to G-L, the horizontal projection shows 
 the true distance between the points A and C in space. Then 
 a and c are determined by ruled projectors from a' and c'. 
 A-B and B-C wiU be shown in their true length by the pro- 
 jections a-b2 and &2-C when these lines in space are brought 
 parallel to H. Hence find the true length of A-B and B-C (by 
 revolving each of them parallel to V) and with the true lengths 
 as radii and a and c as centers describe arcs of circles inter- 
 secting at 62. Then a-b^-c is the required angle. 
 
 Check. The perpendicular b2-d to the line a-c and through b^ 
 must pass through b. 
 
 52. PROBLEM 10. To measure off a given distance on a 
 given line. 
 
 Analysis. The line must be parallel, or brought parallel, to 
 a plane of projection and the given distance measured on the 
 projection which shows the true length of the line. 
 
 The projections of the point limiting the distance can then 
 be found on the original projections of the given line by coimter 
 revolution, that is, by revolving the line back to its original 
 position. 
 
 Construction i. Let A-B represent the given line and A-C 
 the given distance which is to be measured from A. Swing 
 the line A-B about the space projector (from A to a) until it 
 is parallel to V [see Fig. 38 (a)]. The new projections of the 
 line are a-bi and a'-bi'. From a' mark off the length a'-c' 
 equal to A-C, the given distance. By drawing the ruled pro- 
 
POINT AND LINE PROBLEMS 
 
 45 
 
 jector, determine c. When the line is counter revolved c swings 
 to ci and Ci' is the corresponding vertical projection and ci and Ci 
 are the projections of the required point in its required position. 
 Construction 2. Similarly, swing the line A-B about the 
 space projector A-a' until the line is parallel to H [see Fig. 
 
 38 (b)]. 
 
 Construction 3. Assume a new Vi shown by Gi-Li [Fig. 
 38 (c)] at any convenient point and parallel to the Hne A-B. 
 
 b' 
 
 bi 
 
 
 
 a c 
 
 61 
 
 (a) Construction I. 
 
 Fig. 38.- 
 
 ^6' 
 ax 1 1 
 
 
 (b) Construction n. • (c) Construction m. 
 
 -To measure a Given Length on a Given Line. 
 
 Find the new vertical projection (true length) ai'-hi and measure 
 off the reqmred distance A-C from Ui to c'. Determine the 
 horizontal projection c and the required vertical projection Ci' 
 by ruled projectors. 
 
 NOTE. The plane could have been passed through the line 
 perpendicular to H. 
 
 53. Definitions. An element of a cone- is any straight Hne 
 drawn on its surface from the vertex to the base. 
 
 If the base of a cone is a circle the cone is said to be circular, 
 and a line passing through the vertex and the center of the base 
 is called the axis of the cone. If this axis is perpendicular to 
 the base, the cone is called a right circular cone. 
 
 A right circular cone is also called a cone of revolution because 
 it can be generated by the revolution of a right-angled triangle 
 about one of its shorter sides. 
 
46 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 The base angle of a right circular cone is the angle made by 
 an element and a line drawn from the center of the cone base 
 to the point of intersectionj of the element with the base [see 
 Fig- 39 (a)]. 
 
 54. PROBLEM II. To draw the projections of a line which 
 makes a given angle with V and H. 
 
 Analysis. If a line A-B in space [see Fig. 39 (a)] is revolved 
 about a second Hne A-C, which intersects it and is perpendicular 
 to V, a cone of revolution is generated. This cone projects as 
 
 (a) Perspective. 
 
 Fig. 39. — To determine the Projection of a Line which makes a Given Angle with a Plane of 
 
 Projection. 
 
 a triangle on H [see Fig. 39 (b)]. The altitude a-c of the tri- 
 angle will be the same as the altitude A-C [see Fig. 39 (a)] of 
 the cone; the base d-b of the triangle will be the same as the 
 diameter D-B of the cone, and each slant side of the triangle 
 will be the same as the line A-B of the cone. The cone pro- 
 jects on V as a circle which is equal to the base of the cone 
 since it is parallel to V. As the Une A-B revolves to generate 
 the cone, its projection on H changes from its true length (when 
 parallel to H) to a length equal to the altitude of the cone when 
 it is directly above the axis. The vertical projection of A-B, 
 however, does not change its length in changing its position. 
 See a-b, a-bi and a'-b', a'-bi, Fig. 39 (a), for the illustration 
 of these facts. 
 
POINT AND LINE PROBLEMS 
 
 47 
 
 Construction. Revolve the line A-B until it is parallel to H 
 [see a'-bi, a-bi, Figs. 40 (a) and (b)] and to the required angle 
 (say 30 degrees) with V. The vertical projection a'-bi of this 
 line gives the radius of the circle which represents the vertical 
 projection of the cone base. The horizontal projection a-bi 
 gives the slant height of the triangle that represents the horizon- 
 tal projection of the cone. Since the base of the cone is projected 
 on H as a straight line parallel to G-L, the complete projection 
 of the cone can be drawn [see Fig. 40 (c), page 48]. 
 
 a'^ 
 
 1 1 1 
 
 1 1 
 1 1 ' 
 1 1 1 
 1 1 1 
 1 1 
 
 1 I 
 1 ' 
 
 ' I i 
 
 a^ 
 
 ■ 1 1 
 1 ■ 1 
 1 1 1 
 
 1 1 1 
 
 /^ \| 
 
 ^- — ^ K 
 
 Fig. 40. 
 
 fa) Perspective. 
 -To determine the Projections of a Line that 
 
 (b) Construction, 
 a Given Angle with a Plane. 
 
 Place a line A-B^ of the same length as A-B (with one end 
 at A) and parallel to V [see a-bi„ a'-bi , Figs. 40 (a) and (b)] and 
 to the required angle (say 45 degrees) with H. The horizontal 
 projection a-b^ of this line gives the radius of the circle that 
 represents the horizontal projection of a second cone base. The 
 triangle representing the vertical projection is found by having 
 the slant height a -62' and the base a-bi. 
 
48 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 The vertical projections of the t7(ro cones intersect at ¥, and 
 a' is common to both. Therefore a'-V and a-h represent the 
 projections of an element common to both cones. It therefore 
 makes the required angles with both V and H. 
 
 Check. The ruled projector h'-b should come perpendicular 
 to G^Z,. 
 
 Fig. 40. — (c) Layout. 
 
 A second line A-D in the opposite direction [see Fig. 40 (c)] 
 would fulfill the requirements of this problem. 
 
 NOTE. If the cones are tangent, the line in space is in a 
 plane perpendicular to G-L and the sum of the two angles made 
 with V and H is 90 degrees. 
 
CHAPTER IV 
 LINE AND PLANE PROBLEMS 
 
 55. Definitions and Principles. The position of any plane 
 with reference to the co-ordinate planes (that is V and H) is 
 determined by the lines of intersection of the plane with V 
 and H. 
 
 These lines of intersection are termed the traces of the plane. 
 If the trace lies in V it is termed the vertical trace ; if the trace 
 lies in H it is termed the horizontal trace. 
 
 The vertical and horizontal traces of a plane must intersect 
 on G-L, because a plane can intersect a line only in one point. 
 
 Since the traces of a plane are lines in either V or H they are 
 their own projections on the plane containing them and their 
 other projection lies in G-L. 
 
 56. Notation and Line Convention. Planes are designated 
 
 by capital letters, the last letters of the alphabet being used, 
 thus, P, Q, R, S, T, U. 
 
 The horizontal trace of a plane is designated by the capital 
 letter of the plane to which the capital H has been prefixed, thus: 
 HP, HQ, HR, etc. 
 
 The vertical trace of a plane is designated by the capital letter 
 of the plane to which the capital V has been prefixed, thus: 
 VP, VQ, VR, etc. 
 
 Traces of planes unless used in construction are unbroken lines 
 of medium weight and the designating letters are printed close 
 to the line. See Fig. 41, page 50. 
 
 Traces of planes used in construction or the solution of a 
 problem are broken lines of light weight and are drawn the sahie 
 as construction lines (see page 18, § 26) and in addition the 
 traces are designated by letter in the same manner as the traces 
 of given or required planes. 
 
 49 
 
so 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 57. Positions which a plane may hav6 with reference to V and 
 H. A plane may have any one of the following positions with 
 reference to V and H. 
 
 (i) Perpendicular to both V and H, hence perpendicular to 
 G-L; the traces of such a plane are represented by a straight 
 line perpendicular to G-L [see Figs. 41 (a) and (b)-I]. 
 
 (a) Perspective. 
 
 n 
 
 m 
 
 IV 
 
 VI 
 
 V 
 
 
 / 
 
 
 7 
 
 VT 
 
 
 
 H 
 
 ? 
 
 
 
 \ 
 
 
 INDETERMINATE-l-i 
 
 HR 
 
 \ 
 
 HT 
 
 (b) Construction. 
 Fig. 41. — Positions a Plane may occupy. 
 
 (2) Perpendicular to one plane (either V or H) and inclined 
 to the other. One trace of this plane will be perpendicular to 
 G-L and the other wiU be inclined to it [see Figs. 41 (a) and 
 (b)-II]. 
 
 (3) Perpendicular to one plane (either V or H) and parallel 
 to the other. This plane is represented by a single trace, which 
 is parallel to G-L and the trace is located on the co-ordinate 
 plane to which the plane is not parallel [see Figs. 41 (a) and 
 (b)-III]. 
 
 (4) Inclined to both V and H hence to G-L. Both traces 
 will incline to G-L [see Figs. 41 (a) and (b)-IV]. 
 
 (5) Inclined to both V and H but paralled to G-L. Both 
 traces are parallel to G-L [see Figs. 41 (a) and (b)-V]. 
 
LINE AND PLANE PROBLEMS 
 
 SI 
 
 (6) Passed through G-L giving no trace either on V or H 
 [see Figs. 41 (a) and (b)-VI]. 
 
 58. THEOREM V. If a line lies in a plane its vertical piercing 
 point (or vertical trace) must lie in the vertical trace of the plane, 
 and its horizontal piercing point in the horizontal trace of the 
 plane. 
 
 Proof. If a line A-B [Fig. 42 (a) and 42 (b)] lies in a plane 
 R, every point of the line lies in the plane, and the line intersects 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 42. — The Traces of a Line lie in the Traces of a Plane containing it. 
 
 every other line in the plane to which it is not parallel. Extend 
 the line A-B until it pierces V. The vertical piercing point Cc' 
 of A-B is a point common to both the given plane R and to V. 
 Also, the vertical trace VR of the plane contains every point 
 common to the given plane R and V, hence the vertical piercing 
 point Cc' of the Une A-B must he in the vertical trace VR of 
 the plane. 
 
 Likewise the horizontal piercing point Dd of the Une must 
 lie in the horizontal trace HR of the plane. 
 
 59. PROBLEM 12. To assume a line in a given oblique 
 plane. 
 
 Analysis i. Let R in Fig. 43 (a) be the given plane. From 
 Theorem V, this page, any point ^a' in the vertical trace VR 
 may be assimied to be the vertical piercing point of some line 
 
52 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 in R. Also, any point Bb in the horizontal trace HR may be 
 assumed to be the horizontal piercing point of some line in R. 
 Therefore, since two points determine a straight line, A-B is 
 the required line. 
 
 Construction i. Draw VR and HR [Fig. 43(b)] to repre- 
 sent the vertical and horizontal traces respectively of the plane 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 43. — To determine the Projections of Any Line assumed in Any Given Plane. 
 
 R. Assume any point Aa' on VR. Since Aa' is on V it will 
 be horizontally projected in the ground line at a. Similarly, 
 the second assumed point Bb on HR is vertically projected at b'. 
 By joining the vertical projections Aa' and b' and next the 
 horizontal projections a and Bb, the vertical and horizontal 
 projections of the line are determined. 
 
 Analysis 2. Let R in Fig. 44 (a) be the given plane. Draw 
 a line A-B in the plane R and parallel to H. This line will be 
 parallel to the horizontal trace HR of the plane, and have its 
 vertical piercing point Bb' in the vertical trace VR of the plane. 
 
 Construction 2. Draw VR and HR [Fig. 44 (b)] to repre- 
 sent the traces of the given plane. As the line A-B is parallel 
 to HR the horizontal projection a-b must be parallel to HR, and 
 the vertical projection a'-Bb' must be parallel to G-L (since the 
 vertical projection of HR is in G-L). The point b where the 
 horizontal projection a-b cuts G-L marks the horizontal pro- 
 
LINE AND PLANE PROBLEMS 
 
 S3 
 
 jection of the point where the line A-B pierces V, hence draw 
 the ruled projector from b to VR to determine the vertical pro- 
 jection Bb'. The vertical projection of the required line is 
 Bb'-a' and is parallel to G-L. 
 
 (a) Perspective. 
 
 (b) Construction. 
 
 Fig. 44. — To determine tlie Projections of a Line which is Parallel to a Plane of Projection 
 and lies in Any Given Plane. 
 
 A line in the plane R and parallel to the vertical trace VR 
 could be similarly assumed. 
 
 NOTE. To assume a point in an oblique plane it is only 
 necessary to assume one projection of the point, pass any line of 
 
 Fig. 4S. — DetemuniJig a 
 Point in Any Plane by 
 Means of a Horizontal 
 Line of the Plane 
 passed through the 
 Point. 
 
 Fig. 46. — Determining a 
 Point in Any Plane by 
 Means of Any Line of 
 the Plane passed 
 through the Point. 
 
 the plane through this projection and the other projection of 
 the point will be found in the other projection of the line (see 
 Figs. 45 and 46). ^ 
 
54 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 60. PROBLEM 13, To pass a plane through two intersecting 
 lines. 
 
 Analysis. Let A-B and C-D intersecting at [see Fig. 47 (a)] 
 be the given lines, then since the lines lie in the required plane 
 
 (a) Perspective. 
 
 R, they must pierce H and V 
 in the traces of this plane, and 
 since two points are sufficient 
 to determine a straight line, 
 the vertical trace of the plane 
 is determined by the vertical 
 piercing points E and M of the 
 two Hnes. The horizontal trace 
 is determined by the horizontal 
 piercing points F and K of the 
 lines. 
 
 Construction. See Fig. 47 (b) . 
 Extend, the horizontal projec- 
 tion or-b of ' A-B and c-d of 
 C-D to intersect G-L and find 
 the vertical projections Ee' and Mm' of the vertical piercing 
 points (see page 51, § 58). Through these projections Ee' and 
 Mm' draw the vertical trace VR of the plane R. Similarly, 
 find the horizontal projections of the horizontal piercing points, 
 and through these projections Ff and Kk draw the horizontal 
 trace HR of the required plane R. 
 
 Fig. 47. 
 
 (b) Construction. 
 
 -To pass a Plane through Any Two 
 Lines which intersect. 
 
LINE AND PLANE PROBLEMS 
 
 5S 
 
 Check. If the traces of the plane R intersect G-L at the same 
 point, the work is correct. 
 
 NOTE. If one trace of a plane is given, the other trace can 
 be determined if a single point lying in it is known, for this trace 
 must intersect G-L at the same point as the given trace or 
 it must be parallel to G-L. 
 
 6i. PROBLEM 14. To pass a plane through three given 
 points. 
 
 It is only necessary to draw lines through the points and 
 proceed as in Problem 13, page 54. 
 
 62. PROBLEM 15. To pass a plane through a point and a 
 line. 
 
 Draw a line through the point intersecting the given line, 
 and proceed as in Problem 13, page 54. 
 
 63. PROBLEM 16. To determine the traces of a plane to 
 contain any given plane figure. 
 
 It is only necessary to extend two sides of the figure imtU 
 they pierce V. and H and draw the traces through these piercing 
 points [see Figs. 48 (a) and 48 (b)]. See Problem 13, page 54. 
 
 (a) Perspective. 
 
 (b) Construction. 
 
 Fig. 48. — To pass a Plane through Any Given Plane Figure. 
 
 64. THEOREM VI. If two planes in space are parallel, their 
 vertical traces are parallel, also their horizontal traces are 
 parallel. 
 
S6 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY" 
 
 (b) Construction. 
 
 Fig. 49. — To pass a Plane tlirough a Given Straight'Line and Parallel to Another 
 
 Given Straight Line. 
 
 As the vertical traces of the planes are their intersections with 
 V and their horizontal traces are their intersections with H, the 
 proof of this theorem follows directly from a theorem in SoHd 
 Geometry which establishes the truth that, if two parallel planes 
 are cut by a third plane, the intersections are parallel. 
 
LINE AND PLANE PROBLEMS 
 
 57 
 
 65. Principles. A theorem in Solid Geometry proves that 
 through any given straight hne, a plane can be passed parallel 
 to any other straight line. Also, a plane is parallel to a given 
 straight line when it contains any straight line parallel to the 
 given line. 
 
 66. PROBLEM 17. To pass a plane through a given straight 
 line and parallel to another given straight line. See § 65, this page. 
 
 Analysis. In Fig. 49 (a) let it be required to pass a plane R 
 through the Une C-D and parallel to the line A-B. Through 
 any convenient point on the given line draw a straight Une E-F 
 parallel to A-B. The plane containing these two intersecting 
 lines is the required plane. 
 
 Construction. In Fig. 49 (b) draw the projections Ee'-f and 
 e-Ff of the line E-F through the projections 0' and 0, and parallel 
 to the projections a'-h' and a-b. Find the piercing points of 
 the lines E-F and C-D on V and H. Draw VR and HR through 
 the vertical and horizontal piercing points respectively. 
 
 Check. VR and HR must meet on G-L, and any other straight 
 line drawn through any other point on C-D and parallel to A-B 
 must pierce V and H in VR and HR. 
 
 67. PROBLEM 18. To pass a plane through a given point 
 and parallel to two given straight lines. 
 
 Analysis. Let the two 
 lines A-B and C-D and the 
 point be given by their 
 projections. If two lines are 
 drawn through the point 
 parallel to the given lines 
 ^ A-B and C-D they will lie 
 in the plane R and the traces 
 of R will pass through the 
 piercing points of these lines. 
 
 Construction. Draw the 
 projections of the construc- 
 tion lines (see page 18, § 26) 
 E-F and K-M through the projections of the point 0, and par- 
 allel respectively to the projections of the given lines A-B and 
 
 Fig. 50. — A Plane passed Parallel to Two Given 
 Straight Lines and through a Given Point. 
 
58 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 C-D (see Fig. 50). Determine the piercing points of these lines 
 E-F and K-M on V and H and draw the traces VR and ER of 
 the required plane containing them. 
 
 Check. The traces of the plane must meet on G-L and a 
 straight line drawn through the point parallel to VR must 
 pierce H in HR or if drawn through parallel to HR must 
 pierce V in VR. 
 
 68. PROBLEM 19. To determine a plane which shall be 
 parallel to a given plane and contain a given point. 
 
 Analysis. Since the planes are parallel, their corresponding 
 traces will be parallel and one point on each trace, will determine 
 the reqmred plane. A straight Hne through the given point and 
 parallel to either trace of the given plaiie will pierce V or H in 
 the corresponding trace of the required plane. The inter- 
 section of this trace with G-L is a point on the other trace and 
 its direction is known. Hence the trace can be drawn. 
 
 Construction. (See Fig. 51.) Through 
 a draw a-b parallel to HT. Through 
 a' draw a'-Bh' parallel to G-L. The 
 point Bb' is on the vertical trace of the 
 <^^*\^b\,^^N^ [ ^ required plane, hence through BV draw 
 VR parallel to VT. The trace HR must 
 intersect VR on G-L ; therefore HR is 
 
 Fig. SI. -To determine a Plane draWU thrOUgh parallel tO HT. 
 to contain a Given Point and to 
 
 be ParaUei to a Given Plane. Check. See Problem i8, page .57. 
 
 69. THEOREM Vn. If a plane in space is intersected by a 
 plane parallel to H, the horizontal projection of the line of inter- 
 section will be parallel to the horizontal trace of the plane and 
 the v&rtical projection will be parallel to G-L. 
 
 Proof. See Figs. 52 (a) and (b). Let R be the given plane 
 cut by Hi parallel to H. The line of intersection A-B is 
 parallel to HR because HR is the line of intersection of R with 
 a plane H parallel to Hi (see Analysis 2, page 52). . li A-B and 
 HR are parallel their projections must be parallel. Hence, as HR 
 is its own horizontal projection, it is parallel to a-b, and as the 
 vertical projection of HR is in G-L, the vertical projection 
 a'-Bb' must be parallel to G-L. 
 
LINE AND PLANE PROBLEMS 
 
 59 
 
 V 
 
 \ 
 
 \ 
 
 
 VHi Of 
 
 \m! 
 
 ri 
 
 
 I5\ T 
 
 
 / 
 
 // 
 
 Ik 
 
 H 
 
 i' 
 
 
 (a) Perspective. 
 
 (b) Constiuctian. 
 Fig. S2. — A Plane in Space intersected by a Plane Parallel to H. 
 
 70. Corollary. Any line which Kes in a given plane and 
 which is parallel to H, will be parallel to the horizontal trace 
 of that plane. The vertical projection of this line will be parallel 
 to G-L, and the horizontal projection will be parallel to the 
 horizontal trace of the plane. 
 
 See Figs. 53 (a) and (b) for similar proof when the assumed 
 plane is parallel to V 
 
 (a) Perspective, 
 
 (b) Construction. 
 Fig. 53. — A Given Line in a Given Plane which is Parallel to V. 
 
 71. Axiom. The line of intersection of two planes is a straight 
 line common to both planes. Hence, to locate this line of inter- 
 section the projections of two points common to both planes 
 must be determined and the like projections of the points con- 
 nected for the projections of the line. 
 
6o 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 72. PROBLEM 20. To determine the line of intersection of 
 two planes. 
 
 Analysis. The vertical traces VT and VR .[see Fig. 54 (a)] 
 intersect at Aa' and thus determine the point A which lies on 
 V and is common to both the planes T and R. Since A is on V, 
 its horizontal projection a must be on G-L. The intersection 
 of the horizontal traces HT and HR determines a second point 
 B that is common to both planes. Since B is on H, its vertical 
 projection b' must be on G-L. 
 
 A line drawn to join the vertical projections Aa' and b' of the 
 points A and B determines the vertical projection of the hne 
 
 (a) Perspective. 
 
 (b) Constiuction. 
 Fig. 54. — To determine the Line of Intersection between Two Planes. 
 
 , of intersection of the two planes. The horizontal projection of 
 the Hne is determined by joining a and Bb, the horizontal pro- 
 jections of these points. 
 
 Construction. See Fig. 54 (b). Let VT, HT and VR, HR 
 represent the two given planes. The vertical traces intersect at 
 Aa', horizontally projected at a. The horizontal traces inters^ect 
 at Bb, vertically projected at b'. Join Aa' and b' for the vertical 
 projection of the Une of intersection and a and Bb for the hori- 
 zontal projection. 
 
 Check. Draw any line to cut the line of intersection A-B 
 and the vertical trace of the plane T; this line hes in T and 
 therefore when extended must pierce H on HT. Apply the 
 same test to R. 
 
 Special Cases. Solutions for the special cases where the traces 
 intersect within the limits of the drawing are given in Figs. 55 
 to 61 inclusive. 
 
LINE AND PLANE PROBLEMS 
 
 6i 
 
 
 
 
 -^ 
 
 Fig. ss. — Traces intersect on G-L (Construction I). 
 
 Li each gase the analysis given for Problem 20 (page 60) 
 applies, but the solutions depend upon the relative positions of 
 the traces. The student is 
 expected to analyze, "build 2::Lcc' 
 
 up," and explain in detail 
 each of the special cases 
 shown in Figs. 55 to 61 
 inclusive. 
 
 Fig. 55 represents the 
 solution when the traces 
 meet on G-L. Fig. 56 rep- 
 resents the solution when 
 all the traces are parallel 
 to G-L. Both of these so- 
 lutions require the use of 
 a profile plane. Fig. 57 
 represents a solution by the use of a plane parallel to H when 
 the traces meet on G-L, as in the case of Fig. 55. Fig. 58 
 
 represents the solution 
 when one plane is per- 
 pendicular to H and 
 inclined to V and the 
 other is perpendicular 
 to V and inclined to 
 H. Fig. 59 represents 
 the solution when one 
 plane is inclined to V 
 and H while the other 
 plane is perpendicular 
 to H but inchned to 
 V. Fig. 60 represents 
 the solution when one 
 plane is inclined to V and H while the other plane is perpendicular 
 to V but inclined to H. Fig. 61 represents the solution when 
 the line of intersection lies in a plane perpendicular to G-L. 
 
 73. Principle. In plane geometry the principle is established 
 that parallel planes intersect a third plane in parallel lines. 
 
 VB 
 
 ^CC 
 
 T 
 
 ITT 
 
 X 
 
 h'^ 
 
 -4- 
 
 N 
 
 &►- 
 
 
 i-^ 
 
 SR 
 
 HT 
 
 kd/'i 
 
 /'V 
 
 w 
 
 I 
 
 Fig. s6. — All Traces Parallel to G-L. 
 
62 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. S7. —Traces Meet on G-L. 
 (Constiuctioii n.) 
 
 Fig. 59. — One Plane Perpendicular to 
 H and Inclined to V, the other In- 
 clined to H and V. 
 
 Fig. 58. — Planes Perpendicular to V 
 and H respectively but Inclined to H 
 and V respectively. 
 
 Pig. 60. — One Plane Perpendicular 
 to V and Inclined to H, the other 
 Inclined to V and H. 
 
 < 
 
 7 
 
 \ \ 
 
 \ \ 
 
 h'. \ I ' 
 
 
 ^¥v 
 
 
 Fig. 61. —Both Planes Inclined to V and H. 
 
 74. PROBLEM 21. To determine the line of intersection of 
 two planes when the horizontal traces do not intersect within 
 the limits of the drawing. 
 
 , Analysis. Let T and R [Fig. 62 (a)], be the given planes. 
 Since the vertical traces intersect at Cc' , horizontally projected 
 at c, the point Cc' is one point on the required line of intersection. 
 
LINE AND PLANE PROBLEMS 
 
 63 
 
 If an auxiliary plane S, parallel to R, is passed, to intersect T at 
 such a position that VS intersects VT and HS intersects HT, 
 the line of intersection of the plane S with T can he. found. 
 Then, as the plane S is parallel to the plane R, the hne of inter- 
 section of S with T must be parallel to the line of intersection 
 
 (b) Constfuctioii. 
 
 Fig. 62. —Intersection of Two Planes when only One Set of Traces Intersect within the 
 Limits of the Drawing. 
 
 of R with T. Since one point and the direction determines the 
 position of a straight line, the line Cc'-d' parallel to Aa'-b' is 
 the vertical projection, and c-f parallel to a-Bb is the horizontal 
 projection of the required line of intersection. 
 
 Construction. See Fig. 62 (6). Let ET and HR, which do 
 not intersect, represent the horizontal traces of the planes T and 
 R; and VT and VR represent the vertical traces which do inter- 
 sect at the point Cc'. Assume VS and HS parallel respectively 
 to VR and HR and find the projections of the line of intersection 
 of S with T. Next draw Cc'-d' parallel to Aa'-h' and c-f parallel 
 to a-Bb. Then Cc'-d' is the vertical projection, and c-f is the 
 horizontal projection of the . line C-D in space which is the 
 required line of intersection of the plane T with R. 
 
 Check. Assume a new horizontal plane (i.e. parallel to H) 
 and if the Hnes cut from T and R intersect C-D the construction 
 is accurate. 
 
 NOTE. As the horizontal traces HT and HR do not intersect 
 within the limits of the drawing, d cannot be determined and F 
 is the last point on the Hne of intersection that can be shown in 
 
64 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 horizontal projection. A similar solution applies if the hori- 
 zontal traces of the plane intersect within the limits of the 
 drawing and the vertical traces do not. 
 
 7S. PROBLEM 22. To determine the line of intersection of 
 two planes when neither horizontal nor vertical traces intersect 
 within the limits of the drawing. 
 
 Analysis. Cut the two given planes T and R by a third 
 plane S parallel to H. The plane S will cut from T and R 
 straight lines (which will be parallel to H) and which will inter- 
 sect in a point common to both T and R, hence, in their line of 
 intersection. Having determined one point on the line of inter- 
 section, its direction can be found, and the projections drawn 
 as in Problem 21, page 62. 
 
 Construction. See Fig. 63. Let VT, HT and VR, HR repre- 
 sent the traces of the given planes. Draw VS the vertical trace 
 
 of the horizontal plane S. 
 This plane S cuts the plane 
 T along the line E-C which is 
 vertically projected at Ee'-c' 
 and horizontally projected at 
 e-c; it cuts the plane R along 
 the line C-F which is verti- 
 cally projected at c'-Ff and 
 horizontally projected at c-f. 
 Hence the point C is common 
 
 Fig. 63. — To Determine Line of Intersection tO both planCS and IS therC- 
 when the Traces do not Intersect within Draw- foj-g Q^e poiut OU their line of 
 ing Limits. , 
 
 intersection. Next determine 
 the line of intersection A-B of the planes T and the auxiliary 
 plane Q (which is passed parallel to R). Through c' draw c'-d' 
 parallel to Aa'-h' and through c draw c-k parallel-to Bh-a. The 
 hne C-D in space, represented by these projections, is the re- 
 quired line of intersection of the two given planes. 
 Check. Same as for Problem 21, page 63. 
 
 76. PROBLEM 23. To determine the point in which a given 
 straight line pierces a given plane. 
 
 i i 
 
LINE AND PLANE PROBLEMS 65 
 
 Analysis. In Fig. 64 (a) A-B is the given line and T the given 
 plane. If a plane R is passed through the Une A-B so as to 
 intersect the plane T, the line of intersection Cc-Dd' of these 
 planes contains all the points common to both R and T. As 
 the piercing point must be in the line A-B and on Cc-Dd' it 
 must be at their point of intersection. 
 
 Construction. Any number of planes could be passed through 
 A-B, but the simplest one to use in the solution of the problem 
 is a plane perpendicular to H [see Fig. 64 (b)]. The trace HR 
 must therefore pass through the horizontal projection a-b of the 
 given Hne, and VR must be perpendicular to G-L. Cc-d is the 
 horizontal projection, and c'-Dd' the vertical projection of C-D, 
 the line of intersection of the planes. The horizontal projections 
 of A-B and C-D fall along the same straight line, but their ver- 
 tical projections intersect at 0'. Hence the line A-B pierces the 
 plane T at the point in space. 
 
 Check. Pass a plane through A-B and perpendicular to V, 
 and find the piercing point 0. 
 
 77. Definitions. A dihedral angle is " the angular space in- 
 cluded between two planes which intersect, and its measure is 
 the angle formed by drawing a line on each plane from the same 
 point on their Hne of intersection and perpendicular to it (see 
 Fig. 69, page 72). 
 
 If one plane meets another in such a way as to make the 
 adjacent dihedral angles equal, the planes are said to be per- 
 pendicxilar to each other. 
 
 78. Principle. A theorem in Solid Geometry proves that if 
 each of two intersecting planes are perpendicular to a third 
 plane, their line of intersection is perpendicular to that plane. 
 Also a scholium in Solid Geometry states that a perpendicular 
 to a plane is perpendicular to every straight Hne drawn in the 
 plane through the foot of this perpendicular. 
 
 79. THEOREM Vni. If a line is perpendicular to a plane, 
 the vertical projection of the line is perpendicular to the vertical 
 trace of the plane, and the horizontal projection of the line is 
 perpendicular to the horizontal trace of the plane. 
 
66 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 64. — To determine the Piercing Point of any Line on any Plane. 
 
LINE AND PLANE PROBLEMS 
 
 67 
 
 (a) PerspectiTe. 
 
 V 
 
 a' 
 
 \ y^ 
 
 X 
 
 a y 
 
 
 
 
 \ 
 
 
 /6 
 
 
 
 i/ 
 
 / ^N^^^ 
 
 
 
 a 
 
 
 \ 
 
 H 
 
 
 
 
 (b) Construction. 
 
 Fig. 65. — The Projections of a Line Perpendicular to a Plane are respecttvely 
 
 Perpendicular to the Traces of that Plane. 
 
68 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Proof. (See Fig. 65.) If an auxiliary plane R is passed 
 through the ]in.e -A-B and perpendicular to H, the horizontal 
 trace HR of this plane must coincide with the horizontal pro- 
 jection of the line. The plane R will also be perpendicular to 
 the plane S since it was passed through the line A-B which is 
 perpendicular to S. The traces HR and HS are perpendicular 
 to each other, being the traces of two intersecting planes 
 which are at right angles, and which two planes are cut by a 
 third plane H. Hence the horizontal trace of the given plane 
 S is perpendicular to the horizontal projection a-b of the 
 given line. Similarly, it can be proven that the vertical projec- 
 tion of the line is perpendicular to the vertical trace of the 
 plane. 
 
 80. PROBLEM 24. To draw through a given point a straight 
 
 line perpendicular to a given plane, and to determine the distance 
 
 from the point to the plane. 
 
 Analysis. Through the given point pass a perpendicular to 
 
 the given plane. Find the point in which the perpendicular 
 
 pierces the plane. Determine the true distance between this 
 
 piercing point and the given point. 
 
 Construction. Let VT and HT (Fig. 66) represent the traces 
 
 of the plane T and a' and a the projections of the point. Through 
 
 a' draw a'-b' perpendicular to VT, 
 and through a draw a-b perpendicular 
 to HT. To find where A-B pierces 
 T pass the plane R through A-B and 
 perpendicular to H. Find the line of 
 intersection C-D of R and T, and the 
 point B where A-B crosses this line 
 of intersection C-D is a point common 
 to the line A-B and the plane T, 
 hence is the piercing point of the 
 / line on the plane. Revolve the line 
 
 Fig. 66. —To Draw through a Point A-B SO t"hat it is parallel to V and the 
 
 a Line Perpendicular to any Plane. ^^^^ distance from the poiut tO the 
 
 plane is determined since the line then projects on F in its true 
 length. 
 
LINE AND PLANE PROBLEMS 69 
 
 Check. Any line in the plane T drawn through B must be 
 perpendicular to A-B. Also to test the distance of the point 
 from the plane, revolve the line A-B into H, 
 
 81. Notation. A projected point on a plane other than V or 
 H takes the subscript of the plane, thus the point A projected on 
 the plane T is marked A-r and the projection of ^t on V is a'x 
 and on H is ot- 
 
 82. PROBLEM 23. To project a given straight line on a 
 given oblique plane. 
 
 Analysis. If the Une is noi perpendicular to the plane its 
 projection on the plane will be a straight line. 
 
 Let A-B [Fig. 67 (a)] be the given hne in space, and its pro- 
 jections on V and H are a'-b' and a-b. The projection of A-B 
 on the given plane T, however, is At-Bt, and the required 
 projections (i.e. on V and H) are a'T-b'x and ot-St- 
 
 Through the points A and B of the line in space, pass per- 
 pendiculars to the given plane T. Through these perpendiculars 
 to T, pass the planes R and S perpendicular to H, and determine 
 the point in which the perpendiculars pierce T. The straight 
 line joining the two vertical projections (on V) of these piercing 
 points on T is the vertical projection of the required Hne, and 
 the straight line joining the horizontal projections of the two 
 piercing points of the perpendiculars to T is the horizontal pro- 
 jection of the required hne. 
 
 Construction. Let a'-b' and a-b represent the given line and 
 VT, HT [Fig. 67 (b)] the given plane. From A and B draw 
 perpendiculars and determine the projections a'r, ar and 6't, 6t, 
 of the points where these perpendiculars pierce the plane T. 
 The straight line a'r-b'T is the vertical projection and ar-bT is 
 the horizontal projection of the required projection Aj^Bt. 
 
 Check. Any point on the given line other than A or B when 
 projected on T must have its projections on a'T-b'x and Ot-St. 
 
 83. PROBLEM 26. To determine a plane which shall pass 
 through a given point and be perpendicular to a given straight 
 line. 
 
70 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 67. — To Project a Given Straight Line in a Given Oblique Plane. 
 
LINE AND PLANE PROBLEMS 
 
 71 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 68. —To Pass a Plane through a Given Point and Perpendicular to a Given Straight Line. 
 
72 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Analysis. See Fig. 68 (a). Since the plane T is to be per- 
 pendicular to the line B-D, its vertical trace VT must be per- 
 pendicular to the vertical projection b'-d' of the line, and its 
 horizontal trace fl^r must be perpendicular to the horizontal projec- 
 tion i-d of the line. If, therefore, a line A-C be drawn through 
 the point E and perpendicular to the line B-D, also parallel to 
 the horizontal trace of this plane, it will be a line of the reqmred 
 plane. The vertical piercing poiat Cc' of the line A-C is a 
 point on the vertical trace VT of the plane^ and since the direc- 
 tion of the trace is known, one point is sufficient to determme it. 
 
 Construction. Let h'-d' and i-d in Fig. 68 (b) be the vertical 
 and horizontal projections of the given line. Let e' and e be 
 the projections of the given point. Through e draw e-c per- 
 pendicular to h-d and determine the piercing point Cc'. Through 
 Cc' draw VT perpendicular to b'-d'. Draw HT parallel to a-c 
 and to meet VT on G-L. 
 
 Check. Pass a plane through B-D and perpendicular to H. 
 Revolve the line of intersection of this plane with the required 
 plane into V, carrying the line B-D also into V. If the line B-D 
 and the line of intersection of the planes are perpendicular when 
 revolved into V the construction is correct. 
 
 84. Principle. The angle a between any two planes such as 
 R and T (see Fig. 69) must be measured in a plane P, perpen- 
 dicular to both R and T, hence 
 it is the angle made by the lines 
 of intersection of P with R and 
 T. 
 
 85. PROBLEM 27. To deter- 
 mine the angle that a given plane 
 makes with V and H. 
 
 Analysis. A plane S [Fig. 70 
 (a)] passed perpendicular to the 
 
 Fig. 69. -The An^^e^^etoeen Two Intersect horizontal traCe HR of the givCU 
 
 plane R will cut a line from R 
 and also one from H. The angle between these two intersecting 
 hues is the measure of the angle the plane R makes with H. 
 
LINE AND PLANE PROBLEMS 
 
 73 
 
 Construction. Let the given plane be represented by VR and 
 HR [Fig. 70 (b)]. Through any convenient point Aa on ER 
 draw HS, the horizontal trace of a plane perpendicular to HR; 
 the vertical trace of the plane S must be perpendicular to G-L 
 at c. The plane S cuts from R the line A-C given by its pro- 
 
 (b) Construction. 
 Fig. 70. — To Determine the Angle a Given Plane makes with V or H. 
 
 jections a'-Cc' and Aa-c. Revolve A-C into V and ai is the 
 required angle that R makes with H. 
 
 In a similar way by taking an auxiliary plane perpendicular 
 to the vertical trace of the given plane, the angle jS that the given 
 plane makes with V is determined. See Fig. 70 (b). 
 
 Check. Revdlve the plane S about HS into H and determine 
 the angle a. 
 
 86. PROBLEM 28. To determine the true angle between 
 the traces of a plane. >^.,^^ 
 
 Analysis. This is not the angle the traces make with G-L nor 
 the angle between these traces when V and H are folded into the 
 plane of the paper, but it is the angle a [see Fig. 71 (a)] between 
 the traces measured on the plane itself. To find this angle, a 
 
74 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (b) Construction. 
 Fig. 71. — To Find the True Angle between the Traces of a Plane. 
 
 plane S perpendicular to H and the trace HT is passed to inter- 
 sect the given plane T, and the line of intersection Aa-Cc' of 
 these two planes and the trace VS are revolved about HS into H. 
 Construction. The given plane is represented by VT and HT 
 in Fig. 71 (b). Pass the plane S to intersect T also perpendicular 
 
LINE AND PLANE PROBLEMS 75 
 
 to H and to T; find the projections of the line of intersection 
 A-C. When C is revolved about HT into H it will fall along 
 the line HS and at a distance from Aa equal to its true length. 
 Swing VS into H as shown at VSi, to find the true length of 
 A-C. Swing Cci to its proper position at Cc^. Then the angle 
 between HT and VTi is the one required. 
 
 Check. Since the point Cc' must fall on HS when revolved 
 into H, and as 0-Cc' is a true length (being on V) the arc from 
 with the radius O-Cc' must cut HS at Cc^. 
 
 87. PROBLEM 29. To determine the angle between two 
 intersecting i)lanes. 
 
 Analysis. See Fig. 72 (a). Pass a plane S perpendicular to 
 the line of intersection A-B of the two planes, and it will cut 
 from the plane R the straight line C-E and from the plane T 
 the straight line E-D. Each of these lines is perpendicular to 
 the Hne of intersection A-B at a common point E. The angle 
 between the lines C-E and E-D will be the measure of the 
 required angle, and may be found by revolving the plane S con- 
 taining the desired angle into H. 
 
 Construction. Draw the traces HR, VR, and HT, VT of the 
 planes R and T [see Fig. 72 (b)] and determine the Une of inter- 
 section A-B. The horizontal trace of the plane S is perpendicular 
 to the horizontal projection of the line A-B. The line of inter- 
 section of S with R and T is foimd by revolving the Kne A-B into 
 H at Aai-Bb carrying with it the point E (which is the vertex 
 of the required angle) to Ei. If, however, the plane S is revolved 
 into H about HS, the point E would fall in the perpendicular 
 a-Bb to HS and at a distance from Ff equal to Ff-Ei, hence 
 at E2. As Cc and Dd are both on HS their positions remain 
 fixed when the plane S is revolved into H and Cc-E^-Dd is the 
 required angle. The vertical projection of the angle has not 
 been shown, as it is not necessary to the solution of the problem. 
 
 Check. Take another plane perpendicular to A-B. The 
 intersection of this plane with R, T, and H forms a triangle 
 whose base lies in H; find the true length of the other two sides 
 and on this base construct the triangle in H. The angle at the 
 vertex of the triangle so found is the required angle. 
 
76 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 72. To Determine the Angle between Two Intersecting Planes. 
 
LINE AND PLANE PROBLEMS 
 
 77 
 
 88. Principle. If a line in space is perpendicular to a plane 
 in space, the line and the plane make complemental angles with 
 the plane of projection. 
 That is, if as shown in 
 Fig. 73 the plane T makes 
 a degrees with H then 0-B 
 which is perpendicular to 
 T must make go-a degrees, 
 in order that the sum of 
 the angles of the triangle 
 C-O-B equal i8o degrees. 
 
 Fig- 73- — The Angle a Line Perpendicular to One of 
 Two Intersecting Planes makes with the Other 
 Plane is the Complement of the Angle between 
 the Two Planes. 
 
 89. PROBLEM 30. To 
 determine the traces of the 
 plane that contains a given point and makes given angles with 
 V and H. 
 
 Analysis. If in Fig. 74 (a) the line A-B in space is brought to 
 a position where it makes the complemental angles with V and H 
 that the plane is required to make, then the plane T perpendicular 
 to A-B and containing the given point D fulfills the require- 
 ments of the problem. 
 
 Construction. Assume that the required plane is to make 
 60 degrees with H and 45 degrees with V. Draw a construc- 
 tion line, A-Bhi , [see Fig. 74 (b)] of any convenient length, and 
 in V so that it makes 90° — 60° = 30° with G-L. A-b there- 
 fore represents the length of the horizontal projection of a line, 
 whose real length in space is A-Bbi , when it makes 30 degrees 
 with H. Let the point A remain stationary and swing B out 
 from V. If the angle Bbi'-A-h is kept constant b must move 
 along the arc b-hz and Bbi moves along the straight line Bbi'-b's 
 parallel to G-L. Similarly, draw a line of the same length as 
 A-B in H so that it makes 90°— 45" = 45° with G-L. This 
 detei«Hunes the length of the vertical projection of the original 
 line when it makes the proper angle (9o°-45°) with V. Hence the 
 intersection of an arc of a radius A-b^ (having center at A) with 
 the straight line Bbi'-bs' determines bs'. Similarly, 63 is deter- 
 mined and the check on the accuracy of the work is that bz' and 
 ba wilt fall on the same line perpendicular to G-L. Draw a line 
 
78 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 D-C through the given point D so that D-C is parallel to H 
 and its horizontal projection d-c is perpendicular to A-hz. This 
 
 (a) Perspective. 
 
 line will be a line in the 
 required plane and parallel 
 to its horizontal trace. De- 
 termine the piercing point 
 Cc' of this line. Draw VT 
 through Cc' perpendicular to 
 A-W; draw HT through 
 and parallel to d-c. 
 
 Check. See Problem 27, 
 page 72. 
 
 90. Principle. The angle 
 
 lb) Construction. 
 
 Fig. 74- —To Determine the Plane that will con- 
 tain a Given Point and make Given Angles with which a straight Hne makes 
 VandH. . , , . , , 
 
 with a plane is the angle 
 which the line makes with its projection on the plane. Also, 
 this angle is the complement of the angle formed by the line in 
 space, and a perpendicular to the plane from some point on the 
 line. Therefore, if this last angle is determined, and subtracted 
 from 90 degrees, the result will be the angle required. 
 
 91. PROBLEM 31. To determine the angle which a given 
 straight line makes with a given plane. 
 
LINE AND PLANE PROBLEMS 
 
 79 
 
 (a) Perspective. 
 
 (b) Construction. 
 Fig. 75. — To Determine the Angle between a Given Line and a Given Plane 
 
8o ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Analysis. See Fig. 7S,(a). From any convenient point Don 
 the given line A-B draw a perpendicular D-C to the given plane 
 R. The projections of D-C will pass through the projections 
 of the assumed point and be respectively perpendicular to the 
 traces of the plane. Find where the line A-B and the per- 
 pendicular D~C each pierce H and a line Ff-Ee drawn through 
 these points will be the horizontal trace of a plane S containing 
 the line A-B and the perpendicular D-C. Revolve the plane S 
 into H about its horizontal trace and the true measure of the 
 complementary angle is found; subtracting this angle from 90 
 degrees gives the required angle. 
 
 Construction. In Fig. 75 (b) draw HR and VR to represent 
 the plane R, also a'-b' and a-b to represent the line A-B. From 
 d and d' draw the projections d-f and d'-f perpendicular respec- 
 tively to HR and VR. The line D-F pierces H at Ff and A-B 
 pierces H at Ee. Through Ff and Ee draw HS the horizontal 
 trace of the plane containing the complementary angle. Re- 
 volve S into H. The points Ff and Ee remain fixed and D falls 
 at Ddi. Ff-Ddy-Ee is the angle included between the lines. 
 From Ff draw a perpendicular to Ff-Ddi to meet Ddi-Ee and a 
 is the angle the line A-B makes with R. 
 
 Check. Project the line A-B upon the plane R and this pro- 
 jection will be the base of a right angle triangle, in which the 
 line itself is the h3^otenuse, and the perpendicular to the plane 
 the altitude. Revolve this triangle into V or H. 
 
 92. PROBLEM 32. To draw a straight line through a given 
 point to intersect a given line at a given angle. 
 
 Analysis. Pass a plane through the given point and given 
 line; revolve this plane parallel to V or H, and when in this posi- 
 tion draw the required line from the point to intersect the given 
 line at the given angle. Coimter-rotate the plane and find the 
 new projections of the required Hne. 
 
 Construction. Through the given point D (Fig. 76) draw a 
 line D-F to intersect the given line A-B at any convenient point 
 F. Determine the traces of the plane T which contains the 
 lines A-B and D-F. Revolve the plane T about HT as an axis 
 into H. From Ddi the revolved position of the point D, draw 
 
LINE AND PLANE PROBLEMS 
 
 8l 
 
 the line Di-Ci to intersect A-Bi at the required angle a. Coun- 
 ter-rotate Ci, and 
 Z>-Ci shown by its 
 projections d-c and 
 d'-c' is the required 
 line. 
 
 93. PROBLEM 
 33. To determine 
 the true form of any 
 plane figure given 
 by its projections. 
 
 Analysis i. Let 
 A-B-C (Fig. 77) 
 be the figure whose 
 trueformisrequired. 
 
 If the true length of ^* ^^' ~ *■ straight Line drawn through a Given Point to In- 
 , . r tersect a Given Line at a Given Angle. 
 
 each side is found 
 
 by revolving each of these sides in turn parallel to H or V, the 
 triangle can be constructed. If the figure is other than a 
 triangle it can be divided into several triangles 
 by diagonal lines. 
 
 Construction i. Swing each line of the fig- 
 ure parallel to H and thus determine the true 
 lengths in H. On the most convenient true- 
 length, a-c, construct the triangle. 
 Analysis 2. Determine the traces of the 
 ■'^2 plane which contains the figure and revolve 
 each vertex of the figure into V or H and 
 about one of the traces. 
 
 Construction 2. See Fig. 78. Extend the 
 sides of the figure imtil they pierce V and H. 
 Two such piercing points Ee', Ff and Kk, Mm 
 Fig. 77. -To Determine iu each plane of projectiou determines the ver- 
 ae True Form of a ticar trace VR and the horizontal traCe HR of 
 
 Plane Figure. (Con- 
 struction I.) the plane containing the figure. The radius 
 
 of rotation for the point A will be the true distance from this 
 
 point in space to the point y on HR. This may be found by 
 
82 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 constructing the right triangle a.-A-y at any convenient place. 
 
 A-a. is equal to- the perpendicular distance from the point A in 
 
 space to a, that is, the same as from A to H, hence is equal to 
 
 a'-x, and a-y is equal to 
 the perpendicular distance 
 from a to HR, hence is 
 equal to a-y. 
 
 Swing the figure about 
 HR into H. The hori- 
 zontal projection of the 
 path traced by each point 
 as it swings about HR is a 
 straight line perpendicular 
 to HR. Hence through a 
 draw a perpendicular to 
 HR, and lay off the dis- 
 tance y-Aai equal to y-A. 
 Similarly, construct the 
 other triangles and deter- 
 mine the revolved position 
 of B and C. The figure 
 Aai-Bbi-Cci is the one 
 required. 
 Analysis 3. Determine as before the traces of the plane 
 
 which contains the figure, and revolve this plane into H or V 
 
 about its horizontal or vertical trace. 
 
 Construction 3. (See Fig. 79.) When the plane R is revolved 
 
 about HR into H, the distance O-Ff must equal 0-FJx because 
 
 both are true lengths. Also Ff when revolved must fall on the 
 
 perpendicular to HR through /, hence at F/i and as Dd is on 
 
 HR it does not change position and the revolved position of 
 
 F-D is determined. 
 
 Similarly, determine the revolved position of E~M. The point 
 
 Aai falls on the intersection of the line Ffi-Dd and the per- 
 
 pendiculdlr to HR through a. 
 The points B and C in their revolved position are similarly 
 
 determined. 
 Check. As a check that the plane contains the figure, draw 
 
 MmS- 
 
 Fig. 78. — To Determine the True Form of a Plane 
 Figure. (Construction II.) 
 
LINE AND PLANE PROBLEMS 
 
 83 
 
 two intersecting'lines through the figure and if these lines pierce 
 V and H in the traces of the plane the work is accurate. Any 
 one of the above methods may be used as a check on another 
 to determine the true 
 shape of the figure. 
 
 94. Axiom. The 
 above method can 
 be used for finding 
 the angle between 
 two intersecting lines 
 such as A-B and 
 B-C. When the 
 angle is shown in 
 its true size it can 
 be bisected or other- 
 wise divided and the 
 dividing line can then 
 be revolved back to 
 the original position 
 with the plane R, 
 and its projections 
 found. 
 
 -To Determine the True Form of a Plane Figure. 
 (Construction ni.) 
 
 9S. PROBLEM 
 34. To determine 
 the third angle plan and elevation* of any plane figure when one 
 edge of the figiure is inclined at any given angle to V and the 
 plane of the figure is perpendicular to V and inclined at any 
 given angle to H. Also to determine the angle the given edge 
 makes with H. 
 
 Analysis. Fold the paper to represent the third angle [see 
 Fig. 80 (a)]. Draw HR and VR to represent the traces of the 
 plane that will contain the required figure. HR will be per- 
 pendicular to G-L and VR will make the angle a with G-L 
 that the plane of the figure A-B-C makes with H. Draw EiDd 
 
 * The vertical projection is technically known as, an elevation and the horizontal 
 projection as a plan view. 
 
84 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (a) Perspective. 
 
 (b) Construction. 
 
 Fig. 80. — To Determine the Third Angle Projection of a Plane Figure to Lie in a Definite 
 
 Position. 
 
LINE AND PLANE PROBLEMS 85 
 
 on H SO that it makes with G-L the given angle (3 that the edge 
 of the figure is to make with V. Construct the required figure 
 Ai-Br-Ci on the line Ei-Dd. Cut H along the double line as 
 shown in the figure and swing it about HR until the edge cut 
 along the G-L coincides with VR. The figure Ai-Bi-Ci then 
 occupies the required position shown by A-B-C, and its plan 
 on H would be represented by a-b-c found by the space projectors 
 shown. The elevation would necessarily be in VR and at a'-b'-c'. 
 
 Construction. [See Fig. 80 (b)]. Draw HR perpendicular to 
 G-L and VR at the reqmred angle a that the plane of the figure 
 must make with H. Draw Ei-Dd to make the angle /3 with 
 G-L that the edge of the figure is required to make with V. On 
 Ei-Dd construct the required figure Aai-Bbi-Cci. Swing 
 Aai-Bbi-Cci around HR into the plane R. Thus ax moves in 
 the arc of the circle to a' and Aai moves along the perpendicular 
 Aai-x to the horizontal projection a, determined by the ruled 
 projector from a'. 
 
 Check. Keeping Ei against V, swing the line Ei-Dd into the 
 plane R; e-Dd must then pass through a-c. 
 
 96. NOTE. To determine the angle that the edge A-C 
 makes with H, swing the line E-D about E into V. If the 
 edge A-C is required to make the angle d with H and the angle 
 a is the angle the plane R makes with H and which is to be 
 determined, assume the point Ee' and draw Ee'-Di making the 
 angle d with H. Swing Di to Dd and then swing E-D into H. 
 On Ei-Dd construct the figure and swing it into the plane R. 
 
 97. Principle. The projection of a circle is determined by 
 finding the projections of a number of points in its circumference 
 and drawing an ellipse through the points thus projected. In 
 general the projections of circles are ^ellipses, but if the plane of 
 the circle is perpendicular to the plane of projection the projection 
 of the circle becomes a straight line, having a length equal to 
 the diameter of the circle. 
 
 98. PROBLEM 3S. To draw the third angle projections of a 
 ciicle in a given plane. 
 
86 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Analysis. Let it be required to find the plan and elevation 
 of a circle which lies in the plane R, represented by its traces 
 HR and VR (see Fig. 8i). 
 
 The plane R which contains the circle is revolved about its 
 vertical trace VR into V and the circle is drawn in its true size. 
 
 —z- 
 
 i V/ \ A /X\i 
 
 
 ^^^---^ x~ 
 
 r, 
 
 n ^^/ 1/ 
 
 
 >-^.v 
 
 / 
 
 S\ / 
 / 
 / 
 / 
 / 
 / 
 
 M 
 
 /^ 
 
 ,/ps; 
 
 Fig. Si. — Third Angle Projection of a Circle in a Given Plane. 
 
 The circle is divided into eight (or any number) equal parts 
 by the points B-M-F-N , etc., marked on its circumference. 
 By counter-revolution thqse points are brought into their true 
 position (in the plane R) and the ellipses drawn through the 
 
LINE AND PLANE PROBLEMS 87 
 
 projections of these points give the required plan and elevation 
 of the circle. 
 
 Construction. Determine the traces VR and HR of the plane 
 R so that this plane contains the point C and makes the required 
 angles with V and H (see Problem 30, page 77). Revolve the 
 plane R and with it the point C about VR into V; to do this, 
 pass the plane T through the point C and perpendicular to VR. 
 The revolved position of Ss must be on the trace VT and at a 
 distance 0-5 from 0, hence it falls at the position Ssi. The 
 true distance of Ci from Oo' is Ooi-Cci, and is found by revolv- 
 ing the line of intersection OS of the planes R and T into H. 
 On Ci as a center describe the circle of the given size. Divide 
 the circumference into eight equal parts as shown. As the points 
 Qq' and Kk' are on VR which is the axis of rotation, and the 
 projections of C are already determined, the projections of the 
 lines K-C-J and Q-Y-C can be drawn through the respective 
 projections of the point C and these lines will then he in the 
 plane R. 
 
 Since the points X, Y, and M, N must move (in projection) 
 during coimter-revolution along perpendiculars to VR, the pro- 
 jections x', y', m', and n' are determined by the intersections of 
 these perpendiculars and the vertical projections Kk'-j' and 
 Qq'-m' of the lines; also y, x, n, and m must be on the hori- 
 zontal projections of the lines K-J and Q-M on ysrhich the points 
 N, X, Y, and M are located, and on the ruled projectors from 
 ^', y\ w'l 3.nd m', hence at the points shown. The line D-C-F 
 is drawn parallel to VR, hence when counter-revolved it will 
 pass through c' and fall parallel to VR, therefore its projections 
 are d'-c'-f and Dd-c-f and the projections of E and F fall as 
 shown. 
 
 To determine A and B in projection lay off the diameter of 
 the circle on Ooi-Ss and revolve the points to o-Ss as shown. 
 Then a' and b' are at the intersection of the ruled projectors 
 from a and b and the vertical projection of 0-S. 
 
 Check. Pass any Une through the circle parallel to HR, and 
 see if its projections cut the projections of the circle on the same 
 ruled projectors or, in other words, pass horizontals or verticals 
 through the circle. 
 
88 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 99. PROBLEM 36. To determine the third angle plan and 
 elevation of a plane figure inclined to both V and H. 
 
 Analysis. Construct the figure Aa'-Bh'-Cc'-Dd' in V, draw- 
 ing it in its true size and shape; revolve it about an axis VT 
 lying in V and perpendicular to H, xuitil its horizontal projection 
 makes the given angle, say 45 degrees, with G-L; revolve the 
 vertical projection about an axis through A and perpendicular 
 to H, until an edge of the figure makes the given angle, say 60 
 degrees, with G-L. 
 
 Construction. If Aa'-Bb'-Cc'-Dd' (Fig. 82) represents the 
 true shape and size of the plane figure, its horizontal projection 
 
 a-b-c-d must be in G-L. 
 Draw VT through Aa' 
 and perpendicular to G-L. 
 Then draw HT to make 
 the required angle, say 45 
 degrees, with G-L that 
 the figure in space is to" 
 make with H. . When the 
 figure is revolved about 
 VT as an axis, the projec- 
 tions b, c, d move in the 
 arcs of circles to bi-ci-di 
 on ET. As the distance 
 from H of each point of 
 the figure does not change, 
 the new vertical projec- 
 
 Fig. 82. -:The:Third Angle Projection of a Plane Figure ^j^^^g fg^jj ^^ ^}^g intcrSCC- 
 in aPlane Inclined to V and H. ^ 
 
 tions of the ruled projec- 
 tors and the construction Hnes E-Bb', etc., drawn parallel to 
 G-L. If next the horizontal projection is to be determined 
 when the side Aa'-di makes, say 60 degrees, with G-L, transfer 
 (for clearness) and revolve the vertical projection by means of 
 the base line ^o'-Xi-yi-Zi, and the ordinate 6i'-xi, etc., as 
 shown. In this transfer and revolution, the distance of the 
 points from V do not change; hence the distances from G-L 
 of the new horizontal projections are determined by projecting 
 across from a-bi-Ci-di and up from Aa-I-bi-C'l-d-i . 
 
LINE AND PLANE PROBLEMS 
 
 89 
 
 NOTE. Had the plane figure in the preceding problem been 
 a circle it would have been projected by means of assumed points 
 on the original circle (see Fig. 83). 
 
 Fig. 83. — Third Angle Projection of a Circle lying in a Plane Indlned to V and H. 
 
CHAPTER V 
 
 soLros 
 
 100. A solid has length, breadth, and thickness and is com- 
 pletely bounded by surfaces, which may be plane or curved. 
 To project a solid it is necessary to project the surfaces which 
 bound its exterior form. These surfaces are represented by 
 Unes which would generate them if moved according to some 
 fixed law. Since hues are fixed in length and direction by two 
 points, one at each extremity, the projection of soKds really 
 consists of projecting a series of points. 
 
 B^f 
 
 loi. PROBLEM 37. To draw the third angle plan and eleva- 
 tion of a cube when one face is in H and a vertical face makes 
 a given angle with V. 
 
 Analysis. The face in H (i.e., the plan) will show as a square, 
 the true size of one face of the cube. The side of this square 
 nearest to V will make the same angle with 
 G-L that the corresponding' face of the cube 
 makes with V. As four edges of the cube 
 are perpendicular to H (i.e., -parallel to V) 
 they will appear in their true length in the 
 elevation. 
 
 Construction. (See Fig. 84.) Draw the side 
 Aa-Dd to make the required angle, say 60°, 
 with G-L that the face A^D-M-E of the 
 cube makes with V. On the side Aa-Dd con- 
 struct the square to represent the true size of 
 the face in H. The elevation of this square 
 is in G-L and at a'-b'-c'-d' . From these points draw perpen- 
 diculars a'-e' , d'-m' , i'-J', and c'-k' each equal in length to an 
 edge of the cube. Draw e'^m'-f'-k' parallel to a'-b'-c'-d' to 
 represent the lower face of the cube. As the cube is regarded 
 as opaque the edge V-f must be of a hidden-line construction,. 
 
 90 
 
 m' f 
 
 Fig. 84. — Third Angle 
 Projection of a Cube. 
 
SOUDS 
 
 91 
 
 that is, of the dash-space construction, to show that it is invisi- 
 ble to the draftsman (see § 26, page 18). 
 
 Check. To test the accuracy of construction, measure each 
 edge and check to see that the edges parallel in space are parallel 
 in projection. 
 
 102. PROBLEM 38. To draw the third angle plan and ele- 
 vation of a cube when one edge lies in H and makes 30° or 
 any given angle with V and two parallel faces make 60° or any 
 given angle with H. 
 
 Analysis I. Place the cube with one edge B-F in H and per- 
 pendicular to V. Swing the cube around this edge until the 
 required parallel faces A-B-F-E and C-D-M-K make 60° with 
 H. Maintaining this 
 angle with H, swing 
 the edge, F-B, until 
 it makes 30° with V. 
 The projections in this 
 position are the re- 
 quired plan and eleva- 
 tion. 
 
 Construction. (See 
 Fig. 85.) Draw the line 
 a'-b' to make the re- 
 qmred angle (60°) that 
 the parallel faces of 
 the cube make with H. On this hne construct the square 
 a'-b'-c'-d' to represent the elevation of the cube when the par- 
 allel faces make the required angle with H and are perpendicu- 
 lar to V. If the angle of 60° is maintained with H and the edge 
 Bb-Ff is kept in H, this- edge can take any angle to V with- 
 out 'changing the shape of the plan. The elevation, however, 
 changes shape. 
 
 Hence, at some convenient point, draw the 'line R-S to make 
 the required angle (30°) that the edge in H is to make with V, 
 and on this hne lay off the edge Ffi-Bbi and construct the origi- 
 nal plan. Since the angle of the faces (60°) does not change, 
 neither do the distances that the points a', e', m', d', k', and c' 
 
 Fig. 85. — Projection of a Cube Inclined to V and H. 
 
92 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 are from G-L. That is, they move parallel to G-L and are de- 
 termined in their new positions fli', ei, mi, d\, ki and Ci by 
 the ruled projectors from ai, ei, Wi, di, ki, Ci, etc. 
 
 Analysis II. Instead of revolving the plan as in Fig. 85 to 
 make the given angle (30°) with V, take a new V (see Fig. 86) 
 _Ff__m__k at 30° with the edge B-F. 
 Project the plan upon this 
 new Gi-Li. In drawing the 
 new elevation ai'-Bbi'-Ci'-di, 
 etc., it must be noted that 
 the distances on the ruled 
 projectors from ai, Bbi, c/, 
 di, etc., to Gi-Li are the 
 same as the distances from 
 
 Fie. 86. — Projecting a Cube by use of anew Cl' , b' , c' , d' , etC, tO G-L (seS 
 
 ''_'""'" § 22, page 13). 
 
 NOTE. This method saves one drawing of the plan. 
 Check. As in Problem 37, page 91. 
 
 103. PROBLEM 39. To draw the third angle plan and ele- 
 vation of a cube when one edge is inclined at 45° or any given 
 angle to H and 30° or any given angle to V, and the diagonal 
 of one face is perpendicular to V. 
 
 Analysis I. This problem consists of four distinct operations : 
 
 ist. Represent the cube with one face A-B-C~D in H and the 
 diagonal B-D perpendicular to V. Draw the plan and elevation 
 in this position (see Fig. 87 I). 
 
 2nd. Without changing the distance that the points E,A,C,K 
 on the cube are from V incline the cube by swinging the point E 
 about A until the given edge E-A makes the required angle 
 (45°) with H. This changes the position of the elevation but 
 not its shape or size, as all points move in planes parallel to V 
 (see Fig. 87 II). 
 
 3rd. Revolve the given edge A-E until it makes the given 
 angles with V and H (see Fig. 87 III, also § 54, page 46). 
 
 4th. Without changing the distance of any point on the cube 
 from H, revolve the cube about the point A until the edge As-E^ 
 is parallel to the revolved position of the line A^-E^ shown in 
 
SOLIDS 
 
 93 
 
 Fig. 87 III. This revolution changes the position of the plan 
 but not its shape or size, and all points in elevation move par- 
 allel to G-L (see Fig. 87 IV). 
 
 
 I \ / I ^s !X.I ' 
 
 I ./■ 
 n in IV 
 
 Fig. 87. — Projecting a Cube whicli is Inclined to both V and H. 
 
 Construction I. As the base of the cube is in H its plan will 
 be the square Aa-Bb-Cc-Dd. Since the diagonal E-D makes 
 90° with V, Bb-Dd is drawn perpendicular to G-L, as shown in 
 Fig. 87 1. The elevation of the base is in G-L. At a', b', c' , 
 
 
 Hi * 
 
 
 """^W" 
 
 7-1 
 
 e' e' 
 
 Fig. 88. — Projecting a Cube Inclined to V and H by Means of a New Vertical Plane. 
 
 and d' draw perpendiculars the true length of an edge and 
 draw e'-k' parallel to c'-a'. At a convenient point ai draw a 
 line ffli'-e/ to make 45° with G-L. On this line construct the 
 elevation shown in Fig. 87 II and determine the new plan by 
 
94 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY' 
 
 projecting across from the plan 87 1 to meet the ruled projectors 
 from the elevation in 87 II. 
 
 Take a line £4-^2 equal and parallel to the edge E-A and at 
 some convenient point, and revolve it imtil it makes 45° with H 
 and 30° with V (see Fig. 87 III). At a convenient point draw. the 
 line Es-Ai (shown by projections es'-Os' and es-Aa^) parallel to 
 Ef-Ai and on e^-Aas construct the plan drawn in Fig. 87 II, and 
 determine the new elevation by projecting across from the ele- 
 vation in Fig. 87 III and the ruled projectors from Fig. 87 IV. 
 Visualize the cube to determine which are visible edges. 
 Analysis II. (See Fig. 88.) Take a new Gi-Li making the 
 angle a with the edge ei-Aai and determine the new elevation 
 as in the second method of Problem 38, page 92. 
 
 Check. Parallel 
 edges of the cube 
 must be parallel in 
 projection. 
 
 104. PROBLEM 
 40. To draw the 
 third angle plan 
 and elevation of a 
 L cube when one face 
 is inclined to H at 
 a given angle and 
 an edge of that face 
 also makes a given 
 angle with H. 
 
 Analysis. Deter- 
 mine the traces of 
 the plane which 
 contains the given 
 face of the cube. Determine a line in this plane which makes 
 with H the angle of the given edge. Upon this hne (as the 
 given edge) construct the plan of the face and complete the 
 elevation of the cube upon the elevation of this face. 
 
 Construction. Let the given face A-B-C-D make 30° with 
 H; also let the given edge A-D make 15° with H. Draw HT 
 
 Fig. 89. 
 
 - Projection of a Cube with One Face and One Edge 
 definitely Inclined to H. 
 
SOLIDS 95 
 
 and VT (see Fig. 89) of the plane T which makes 30° with 
 G-L. 
 
 Assume any point Qq' in VT and draw Qq'-X to make 15° 
 with H. Swing the line Q-X into the plane T by revolving 
 X into HT at Xxi. Swing Xxi-Q into H and upon Xxi-Qi 
 construct the face of the cube. Revolve this face into T and 
 upon its elevation erect perpendiculars Oi'-ei', di'-mi, h'-fi, and 
 Ci'-ki to the plane T and equal in length to the edges of the cube. 
 These are the elevations of the edges and ei'-fi'-ki'^nii is the ele- 
 vation of the face parallel to the face A-D-B-C. The plan is 
 completed from the elevation. 
 
 Visualize the cube to determine the invisible edges and check 
 the parallelism of the edges. 
 
 105. Definitions. A p3rramid is a soUd bounded by a polygon, 
 called the base, and a series of triangles having a common vertex. 
 The common vertices of the triangular faces is called the apex 
 of the pyramid. 
 
 The altitude is the perpendicular distance from the apex to 
 the plane of the base. 
 
 The axis of a pyramid is the line joining its apex to the center 
 of its base. 
 
 A right pyramid is one with its axis perpendicular to its base. 
 
 An oblique pyramid is one with its axis obUque to its base. 
 
 106. PROBLEM 41. To draw the third angle plan and ele- 
 vation of a right hexagonal pyramid when its axis is inclined at 
 any given angle jS to H and a. to V. 
 
 Analysis I. This problem can best be analyzed by steps as 
 follows: 
 
 (i) Draw the plan and elevation of the pyramid when it is in 
 its simplest position, that is, with its axis perpendicular to H 
 (see Fig. 90 1). 
 
 (2) Move the pyramid to the right (see Fig. 90 II) and swing 
 it around an axis X- Y drawn through the point D and per- 
 pendicular to V, imtil the axis 0-Q assumes the given angle 
 with H. During this rotation every point of the soHd moves 
 in the arc of a circle (equivalent to 90°-/3) and parallel to V. 
 
96 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Hence the elevation moves through the angle 90°-|S but does not 
 change shape or size. 
 
 Draw ai'-di equal to a'-d' and inclined to G~L at the angle 
 90°-/3. Lay ofif the distances ai'-bi, bi'-Ci, etc., equal to a'-b', 
 b'-c', etc. From 0/ draw the vertical projection of the axis 
 oi'-qi making 90° with the base, and complete the elevation by 
 connecting each of the points ai, hi, Ci', etc., with the vertical 
 projection of the apex qi'. 
 
 Draw the ruled projectors upward from the new vertical pro- 
 jection and draw the transfer lines parallel to G-L and toward 
 
 -'-^. 
 
 
 ! i| 
 
 I 1 
 
 
 I 
 
 _ Si!^l'^L^ ■" 
 
 a'b' f'o'C e'd' "Sfi?,! 
 
 ii'Jil. 
 
 I n i m IV 
 
 Fig. 90. — Third Angle Projection of a Right Pyramid whose Axis is Inclined to V and H. 
 
 the right from the original plan. The intersection of the ruled 
 projectors and the transfer lines determine the new plan. 
 
 (3) Draw the projections of a line S-T at the same angles 
 (/3 with H and a with V) that the axis of the pyramid is required 
 to make with the projection planes (see Fig. 90 III). 
 
 (4) Move the pyramid as before (Fig. 90 II) and change its 
 inclination to V by swinging it round an axis M- N which passes 
 through D and is perpendicular to H. This will not change the 
 distance of the pyramid from H and hence the plan will not 
 change shape or size, but only its position to G-L. Place the 
 horizontal projection 02-^2 of the axis parallel to the horizontal 
 projection of a Hne S-T (Fig. 90 III), and transfer the plan in 
 this position. The plan is most easily transferred by means of 
 
SOLIDS 
 
 97 
 
 the base line 1-2-3-4-^-6 and the perpendiculars from ai, h, Ci, 
 etc. Project down and across for the new elevation (Fig. 90 IV). 
 
 Visualize the pyramid in its different positions in space and 
 note which edges are visible. 
 
 Analysis II. Instead of revolving the plan take Gi-Li (see 
 Fig. 91) to make the angle ai with the axis and determine the 
 new elevation as in the second method of Problem 38, page 92. 
 
 Check. The accuracy of construction may be ascertained by 
 noting the parallehsm of the projections of Hnes which are parallel 
 in space. 
 
 a' b' fo'c' e' d 
 
 Fig. 91. — Projection of a Pyramid whose Axis is Inclined to V and H by use of a Kew 
 
 V Plane. 
 
 107. PROBLEM 42. To draw the third angle plan and 
 elevation of a right pyramid of given base and altitude and with 
 its base in a given oblique plane. 
 
 Analysis. This problem can best be analyzed by steps as 
 follows : 
 
 (i) Revolve the center of the base into V about VT as an 
 axis.. On the revolved position Oi draw the base in its true 
 shape and size (see Fig. 92). Revolve the base A-B-C-D-E-F 
 into the given plane T by any convenient method (see Fig. 78, 
 page 82, Fig. 79, page 83, and Fig. 81, page 86). 
 
 (2) Pass a plane R through 0, perpendicular to T and V. 
 Revolve the line of intersection /-/ of these planes R and T 
 into H. This carries into H. Erect the perpendicular Ooi-Q 
 equal in length to the true altitude of the p3T:amid. Counter- 
 
98 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 rotate Ooi-Q into its true position shown by the projections 
 o'-q' and o-q. The projections of the apex of the pyramid is 
 
 then q,q', when it 
 occupies the required 
 position in space. 
 
 (3) Connect the pro- 
 jections of the apex 
 with the projections of 
 the comers of the base 
 to complete the figure. 
 
 (4) Visualize the 
 pyramid to determine 
 the invisible edges. 
 
 108. PROBLEM 43. 
 To draw the third angle 
 plan and elevation of 
 an oblique hexagonal 
 pjrramid when its axis 
 is inclined to both V 
 and H (see Fig 93). 
 
 The solution of this 
 problem is similar to 
 that of Problem 41, 
 page 95. 
 
 Fig. 92. — Projection of a Right Pyramid whose base is in 
 an Oblique Plane. 
 
 109. PROBLEM 44. 
 
 To draw the third angle 
 
 plan and elevation of a right pentagonal (five-angled) pyramid 
 
 when one slant face C-Q-D is parallel to H and the horizontal 
 
 edge C-D of the base is inclined at any angle to G-L. 
 
 Analysis. This problem can best be analyzed by steps as 
 follows: 
 
 (i) Draw the plan and elevation of the p)n:amid in its simplest 
 position. This will be with a base edge C-D perpendicular to 
 V and the axis perpendicular to H (see Fig. 94 I). 
 
 (2) Swing the pyramid around C-D as an axis until the face 
 C-Q-D is parallel to H (see Fig. 94 II). Every point of the 
 
SOLIDS 
 
 99 
 
 pyramid revolves in the arc of a circle which in each case is 
 parallel to V, hence the new elevation is precisely the same as in 
 
 h c 
 
 /i --^ 
 
 6 
 
 /I \ i\ 
 
 
 \ I 
 
 f\7 1 > I ! 
 
 'z^;?^' 
 z-!''^// 
 
 ^r/' 
 
 ^ 
 
 a' b'/^' o'e' d' 
 
 /' 
 
 
 U 
 
 ii 
 
 Fig. 93. — Projection of an Oblique Pyramid whose Axis is Inclined to V and H. 
 
 Fig. 94 I, but its position is changed relative to G-L. The plan 
 changes shape and position. 
 
 (3) Move the pjnramid to the right (see Fig. 94 III) and with- 
 
 t_A v._ 
 
 Fig. 94. — Projection of a Pentagonal Pyramid with Axis Inclined to V and H. 
 
 out changing the distance of its face C-Q-D from H swing it 
 until the giv^n edge C-D makes the required angle a with G-L. 
 The new plan will be precisely the same as in Fig. 94 II, but its 
 
lOO ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 position relative to G-L is changed. Hence draw the line c-r-d^ 
 at the proper angle to G-L and upon this Une construct the plan 
 exactly as in Fig. 94 II. Project down and across to determine 
 the new elevation. 
 
 Check. The accuracy of the construction may be tested by 
 finding the true lengths of the edges and comparing them with 
 the true length of A-Q of the original figure shown in Fig. 94 II. 
 
 110. Definitions. A conical surface is a curved surface gen- 
 erated by a line moving so that it touches a given circular or 
 elliptical curve and passes through a fixed point not in the plane 
 of the curve. 
 
 The moving line in any of its positions is called an element. 
 A cone is a solid bounded by a conical surface and a plane that 
 cuts' all of its elements. The plane is the base of the cone. The 
 altitude of a cone is the perpendicular distance from the apex 
 to the plane of the base. A circular cone is one whose base is 
 a circle. The axis of a circular cone is a line from the center of 
 the base circle to the apex. 
 
 A right circular cone is a circular cone in which the axis is 
 perpendicular to the base. 
 
 An oblique circular cone is a circular cone in which the axis 
 is not perpendicular to the base. 
 
 111. PROBLEM 45. To draw a cone in any position similar 
 to the pyramids of Problems 41, 42, 43, and 44. 
 
 Project a sufiicient number of points on the base to estab- 
 hsh the ellipse which represents the projection of the base, and 
 complete as in the above problems. 
 
 For example see Figs. 95 and 96. 
 
 112. Definitions. A prism is a solid having equal and parallel 
 faces for bases, and parallelograms for its other sides. 
 
 The altitude of a prism is the perpendicular distance between 
 the planes of its bases. 
 
 A right prism is one whose lateral faces are perpendicular to 
 its bases. A regular prism is also termed a right prism. 
 
 An oblique prism is one whose lateral edges are not perpendicu- 
 lar to its bases. 
 
SOLIDS 
 
 lOI 
 
 G-4 
 
 1 — l-t 
 
 /ll 
 
 
 
 
 m's„;i.2 
 
 
 
 a 
 
 p' I 
 
 / \: 
 
 r' k',0'., i' 
 
 t^m' 
 
 Fig. 95. — Projection of a Right Cone whose Axis is Inclined to V and H by use of a New 
 
 V-Plane. 
 
 / ] oj/zt; m 
 
 Fig. 96. — Projection of an Oblique Cone whose Axis is Inclined to the Planes of Projection. 
 
 113. PROBLEM 46. To draw the third angle plan and ele- 
 vation of a hexagonal prism when the axis is parallel to V, but 
 inclined to H at any given angle, and an edge of one end is 
 parallel to H and perpendicular to V. 
 
 Analysis. This problem can best be analyzed by steps as 
 follows : 
 
 (i) Draw the plan and elevation in its simplest position, 
 that is, with two parallel faces perpendicular to V (see Fig. 97 I). 
 
102 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (2) Move the prism to the right (for convenience) and swing 
 it about the line B-C as an axis until the face C-B-F-E makes 
 
 the given angle a 
 with G-L. In this 
 rotation all points 
 on the prism move 
 in the arc of cir- 
 cles parallel to V, 
 that is, no point 
 on the prism 
 changes its dis- 
 tance from V. 
 Hence, the new 
 elevation (Fig. 97 
 II) is precisely the 
 same in shape and 
 size as at Fig. 97 
 I, only its posi- 
 tion to G-L has 
 Project up and across for the new plan (see Fig. 
 
 Fig. 97. — Projection of a Prism whose Axis is parallel to V'and 
 Inclined to H. 
 
 changed. 
 
 97 n). 
 
 114. PROBLEM 47. To draw the third angle plan and ele- 
 vation of a hexagonal prism, when one edge of an end is in H 
 and inclined at any given angle to V and the axis is inclined at 
 any given angle to H. 
 
 Analysis. This problem can best be analyzed by steps as 
 follows: 
 
 (i) Determine, as in the problem 46, page loi, the plan and 
 elevation when the axis is parallel to V but inclined at the given 
 angle a with H (see Fig. 98 I). 
 
 (2) Draw Gi-Li at any convenient point and at the angle (3 
 with Bbi-Cci that the edge B-C is to make with V. The eleva- 
 tion on the new vertical plane is obtained by drawing the per- 
 pendiculars to Gi-Li from the points on the plan at Fig. 98 II; 
 measuring on these perpendiculars from Gi-Li, the correspond- 
 ing distances from G-L on the elevation Fig. 98 II. 
 
SOLIDS 
 
 103 
 
 Check. The accuracy of the construction may be checked 
 by comparing the true lengths of several edges shown in the 
 final plan and elevation with those shown at Fig. 98 I. 
 
 Ee,^. 
 
 ^/K" 
 
 ^^ 
 
 dr 
 
 .Cc e 
 
 Mb 
 
 A<^ r 
 
 A'a 
 
 I 
 
 Oc, 
 
 -frt 
 
 •% 
 
 
 X 
 
 1^ ^s I XI % ^^ %. 
 
 >>"^ 
 
 Fig. gS. — Projection of a Prism whose Axis is Inclined to V and H. 
 
 I IS- PROBLEM 48. To draw the third angle plan and ele^ 
 vation of a right cylinder when its axis is parallel to V but inclined 
 to H at any given angle. 
 
 Analysis. This problem can best be analyzed by steps as 
 follows : 
 
 (i) Draw the plan and elevation of the cylinder in its simplest 
 position, that is, when one end or face is parallel to H. The 
 plan will be a circle with a diameter equal to the diameter of the 
 cylinder base; the elevation will be a rectangle of length equal 
 to the length of the cylinder and width equal to the diameter 
 of the cylinder (see Fig. 99 I). 
 
 (2) Divide the cylinder into twelve (or any convenient num- 
 ber) equal parts by lines parallel to the axis. (See elements 
 through A, B, C, D, E, etc., in Fig. 99 I.) Move the elevation 
 to the right and incline its base at 90° minus the given angle a. 
 Determine the horizontal projection of the points on the circum- 
 
I04 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 ference exactly as was done in finding the plan of end of prism 
 (see Piroblem 47, page 102) and join these points by a curved 
 line, in place of straight lines as in the previous problems. 
 
 .4zi_i\£: 
 
 "TTT 
 
 1 f—ak -S^r^^ ^ -K s 
 
 K I-* t^ I I |\\ II 
 
 I . I I I I Ji'n> 
 
 I I ' ' I' 
 
 il ' ! I : 
 
 1 1 I I ,11 
 
 Il I I I II 
 I , I , ii 
 
 a_Li_ij 
 
 Fig. 99. — Third Angle Projection of a Right Cylinder whose Axis is Parallel to V 
 and Inclined to H. 
 
 116. PROBLEM 49. To draw the third angle plan and ele- 
 vation of a right cylinder when its axis is inclined to both V and H. 
 
 Analysis. Assume that the axis makes the angle a with H 
 and determine the new elevation on a new vertical plane which 
 is inclined to the original V at an angle of /3. See Fig. 99. 
 
 (i) Proceed as in Problem 48, page 103. 
 
 (2) Draw Gi-Li to make the angle |S with G-L that the new 
 vertical plane Vi makes with V and determine the new vertical 
 projection as in Fig. 95, page loi, and Fig. 98, page 103. 
 
 117. Definition. A helix is a curve generated by a point 
 which moves along the surface of a cylinder in such a way that 
 a constant ratio is maintained between its travel around the 
 cylinder and along its length. 
 
 Thus in Fig. 100 let the moving point start from A and travel 
 around the cylinder and at the same time along its length. 
 Divide the circumference of the base into twelve (any number) 
 
SOLIDS 
 
 105 
 
 equal parts, and divide the length into the same number of equal 
 parts, and through these divisions draw horizontal lines. 
 
 Assume the point starting at a moves to & in plan, it must then 
 move to b' in elevation, or when it has moved through one- 
 twelfth of a revolution, it has also moved through one-twelfth 
 the distance »'-?■'. When it has moved through one-half revo- 
 lution, to i, it has moved 
 through one-half of a'-r' or 
 to i'. When it has moved 
 through a complete revolu- 
 tion in plan, it has moved 
 in elevation to r'. The dis- 
 tance a'-y' through which the 
 moving point travels along 
 
 Fig. 100. — Construction of a Helix. 
 
 Fig. loi. — Projection of Screw Thread. 
 
 the axis while it describes one complete revolution is termed the 
 pitch of the helix. 
 
 A helicoid is a surface generated by a straight line which glides 
 along a helix and maintains an invariable angle with relation 
 to the axis of the curve. Its most common use is in screw 
 threads. 
 
Io6 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 ii8. PROBLEM 50. To project a V screw thread. 
 
 Analysis. Let the section of the thread be the triangle U-S-T 
 shown at Fig. loi I. The side U-T is the pitch of the thread. 
 The bottom and top edges of the thread are each a helix. The 
 helix for the top of the thread is generated on a cylinder having 
 a diameter a-i, while the one to represent the bottom of the thread 
 is on a cylinder having a diaineter 1-7. The difference between 
 these two diameters is equal to twice the perpendicular distance 
 S-W, or twice the depth of the thread. 
 
 Construction. The plan of the thread will be the plan of the 
 two cylinders on which the helices that are to represent the top 
 and bottom of the thread are generated. 
 
 , The elevation is obtained by marking off lengths q'-i' and 
 j'-y' each equal to the pitch, and dividing these distances into 
 the same number of equal parts as the circumference. Next 
 draw horizontal lines through the points of division, and deter- 
 mine the helicoidal lines on the outer and inner cylinders. (See 
 § 117, page 104.) Finish by properly joining the bottoms of the 
 threads to the tops (see Fig. loi II). 
 
 The square thread is similarly drawn (see Fig. loi III). 
 
CHAPTER VI 
 
 TANGENT PLANES AND DOUBLE-CURVED SURFACES OF 
 
 REVOLUTION 
 
 119. Definitions and General Considerations. J^et A-F-E-B 
 be any plane curve in space (see Fig. 102). If the point E of 
 the secant F-E remains fixed 
 at E and the poini F of this 
 secant is moved along the curve 
 imtil it coincides with E, then 
 the line F-E coincides with 
 C-E-D and becomes a tangent 
 to the curve at the point E. 
 By further reference to this fig- 
 ure it is seen that if a straight 
 line C-D is tangent to a plane 
 
 curve A-E-B then the tangent ^* "2. -Projection of a PUne Curve and a 
 
 ° Line Tangent to the Curve. 
 
 will lie in the plane of the 
 
 curve, because the secant F-E moved in the plane of the curve 
 
 as it changed its position to C-E-D where it became the tangent. 
 
 As the point of tangency £ is a point, and the only point, 
 
 common to both the curve and the tangent, the projections of 
 
 the curve and of the tangent on V are tangent, as also are their 
 
 projections on H. In this connection it is self 
 
 evident that two straight lines tangent to each 
 
 other will coincide. 
 
 A straight line is tangent to a surface at a 
 
 given point when it is tangent to a line of the 
 
 surface at that point. Thus, the line A-B 
 
 (Fig. 103) is tangent to the sphere at the 
 
 point E, because it is tangent to the great 
 
 circle E-F-I-J at the point E. 
 
 A plane is tangent to a surface at a given point when it contains 
 
 all the straight lines tangent to the surface at that point. The 
 
 107 
 
 Fig. 103. — A Line, also 
 a Plane, sliown Tan- 
 gent to a Surface. 
 
io8 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 tangent plane has one point in common with the surface to which 
 it is tangent, which is thie point of tangency. Thus, the plane 
 T is tangent to the sphere 0, at the pouit E because it contains 
 
 all the straight lines 
 tangent to the sphere at 
 this point. The point 
 E is common to the 
 sphere and the plane. 
 
 If a surface has rec- 
 tilinear elements, the 
 tangent plane must con- 
 tain the element pass- 
 ing through the point 
 of tangency. This ele- 
 ment is called the ele- 
 ment of tangency. In 
 Fig. 104 the plane T- 
 is tangent to the cone at the point A , hence tangent to the cone 
 along the element of tangency B-D. Figure 105 shows the 
 plane T tangent to the cylinder at the point A , hence along the 
 element of tangency B-D through the point 
 A. A straight line is normal to a surface /^ ^ 
 
 at a given point when it is perpendicular to 
 the plane which is tangent to the surface at 
 that point. Thus, the line E-A (Fig. 104) 
 is perpendicular to the plane T at the point 
 A, hence it is normal to the cone at the Fig.ios.-APiane which 
 
 point A . is Tangent to a Cylinder. 
 
 Fig. 104. — A Plane which is Tangent to a Cone. 
 
 
 N^D.^1 
 
 T 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 ^ ! 
 
 
 1 
 
 1 
 
 120. PROBLEM 51. To pass a plane tangent to a cone 
 {•whose base is on H) and through a given point on the surface. 
 Analysis. An element of the cone through the given point 
 will be a line in the tangent plane and will pierce H in a point 
 on the horizontal trace of the required plane (see Fig. 106) . Also 
 since this trace and the cone base are both in H they will be 
 tangent at the point where the element of tangency pierces H. 
 The vertical piercing point of the element of tangency (when pro- 
 duced) gives a point in the vertical trace of the reqmred plane. 
 
TANGENT PLANTS AND DOUBLE-CURVED SURFACES 109 
 
 Construction. Let the cone be given in the third angle by 
 its projections as in Fig. 106, and let a, a' be the projections of 
 the given point 
 on the surface 
 through which 
 the tangent 
 plane T is to 
 pass. Through 
 a, a' and the apex 
 of the cone b, b' 
 draw the element 
 of tangency,. 
 which is a line 
 of the reqmred 
 plane. This el- 
 ement A -B 
 pierces H at D, 
 and the tangent 
 
 TJ'T If, \\\p base ^^' ""*•"* Plane drawn Tangent to a Cone through a Given Point 
 
 on an Element of the Cone. 
 
 (drawn perpen- 
 dicular to the radius C-D) at this point is the horizontal trace 
 of the required plane. Produce the element of tangency until 
 it pierces V at E, which point is therefore on the vertical trace 
 of the required plane; hence VT drawn through E and is the 
 required vertical trace. 
 
 Check. Any line drawn through any point on the element of 
 tangency (other than .4) and parallel to the vertical trace VT will 
 be a line in the required plane and hence must pierce H on HT. 
 
 121. PROBLEM 52. To pass a plane tangent to a cone 
 through a point without the surface. 
 
 Analysis. A hne drawn through the given point and the apex 
 of the cone must be a line of the required plane; hence it will 
 pierce V and H in the vertical and horizontal traces of the re- 
 quired plane. The horizontal trace of the required plane will 
 be tangent to the horizontal projection of the base of the cone 
 and will pass through the horizontal piercing point of the 
 line. 
 
no 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Construction. Let the cone be given' by its projections as in 
 Fig. 107, and let a, a' be the given point through which the tan- 
 gent plane must pass. Draw the line A-E through this given 
 
 point and the apex E 
 of the cone; determine 
 the piercing points C and 
 B of the line A-E ex- 
 tended. The horizontal 
 trace HT is drawn 
 through Cc and tangent 
 to the projection of the 
 base of the cone. The 
 vertical trace is drawn 
 through Bh and inter- 
 sects HT on G-L. 
 
 NOTE. The plane T 
 is tangent to the cone 
 on the near side, and a 
 second plane could be 
 determined tangent on 
 the far side of the cone. 
 Check. Pass a Hne 
 through A parallel to ET. This line must pierce V on VT and 
 intersect the element of tangency. 
 
 Fig. Z07. — To pass a Plane Tangent to a Given Cone 
 and thiougli a Given Point without its Surface. 
 
 122. PROBLEM 53. To determine the traces of a plane 
 which shall contain a given line and make a given angle with H. 
 
 Analysis. Such a plane must contain the given line and be 
 tangent to a right cone, whose base is in H and whose elements 
 make the given angle with the base. 
 
 Construction. Let A-B (Fig, 108) represent the given line, 
 and Aa and Bh' its piercing points. From any convenient point 
 C on this line draw the line C-D making the given angle a with 
 G-L. This line C-D represents an element of a right cone 
 which is parallel to V. Draw the horizontal projection of the 
 cone and HT will be tangent to the horizontal projection of 
 the base outline and will pass through A . The trace VT will 
 meet HT on G-L and will pass through B. 
 
TANGENT PLANES AND DOUBLE-CURVED SURFACES ill 
 
 Aa 
 
 Fig. io8. — To determine a Plane which nill 
 make a Given Angle with H and contain 
 a Given Line. 
 
 Check. The plane T must contain the given line since its 
 traces are passed through the piercing points of the line and so 
 as to meet on G-L. Deter- 
 mine the angle the plane 
 makes with H (see Problem 
 27, page 72). 
 
 123. PROBLEM 54. To 
 pass a plane tangent to a 
 cone and parallel to a given 
 straight line. 
 
 Analysis. Since the re- 
 quired tangent plane must 
 contain the vertex of the cone, 
 a straight line through the 
 vertex and parallel to the 
 given line will be a line of 
 the required plane. The hor- 
 izontal trace of the required plane must pass through the hori- 
 zontal piercing point 
 of this line and be tan- 
 gent to the cone base. 
 Construction. Let 
 the cone and the given 
 line A-B be repre- 
 sented by their pro- 
 jections in Fig. 109. 
 Draw F-E through 
 the vertex of the cone 
 and parallel to A-B. 
 Through Dd the hor- 
 izontal piercing point 
 of F-E draw the hor- 
 izontal trace HT oi 
 
 Fig. 109.— APlane passed jangent to a Cone and Parallel to the required plane tan- 
 a Given Straight Line. ^^^^ ^^ ^j^^ ^^^^ ^^^_ 
 
 The vertical trace FT is drawn through the vertical piercing 
 point Ee' of the line and to meet ET on G-L. 
 
 Check. The element of tangency must pierce V in VT. 
 
112 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 124. PROBLEM 55. To pass a plane tangent to a cylinder 
 through a given point on the cylinder. 
 
 Analysis. Since the plane is tangent to the cylinder at a point, 
 it is tangent all along the element through this point. Hence 
 
 the element through 
 the given point is a 
 line of the tangent 
 plane and will pierce 
 V and H in the traces 
 of this plane. If the 
 base of the cylinder 
 is on H, the horizontal 
 trace of the tangent 
 plane must be tangent 
 to the base and pass 
 through the horizontal 
 piercing point of the 
 element of tangency. 
 Construction. Let 
 the cylinder be given by its projections in Fig. no, and let E 
 be the given point on its surface. Through the projections e, 
 e', of the point E, draw the projections of the element A-B and 
 determine its piercing points Aa and Dd'. The horizontal trace 
 of the tangent plane will be perpendicular to the radius A-C, 
 that is, tangent to the base at the point Aa. The vertical trace 
 of the plane will pass through the vertical piercing point Dd' of 
 the element of tangency A-B and meet the horizontal trace on 
 G-L. 
 
 Check. Any line intersecting the element of tangency and 
 parallel to ET must pierce V in VT. 
 
 Fig. no. — A Plane passed thiough a Given Point on a 
 Cylinder and Tangent to it. 
 
 125. PROBLEM 56. To pass a plane tangent to a cylinder 
 and through a given point without the surface. 
 
 Analysis I. Since a plane which is tangent to a cylinder is 
 tangent all along an element, the problem is solved by passing 
 a plane through the element of tangency and the given point. 
 
 Construction I. Let the cyhnder be given by its projections 
 in Fig. Ill and let A be the given point. Draw the line A-D 
 
TANGENT PLANES AND DOUBLE-CURVED SURFACES 
 
 "3 
 
 through the given point A and parallel to the elements of the 
 cylinder; extend A-D to pierce H at DcL. A Une through Dd 
 tangent to the base is the 
 horizontal trace of the re- 
 quired plane. Through A 
 draw the line A-E parallel to 
 ET and it will pierce V at E, 
 which is a point on VT. The 
 element of tangency is B-F. 
 
 Check. The element of 
 tangency B-F must pierce V 
 inFr. 
 
 SPECIAL CASE. When 
 the cylinder is one of revolu- 
 tion and its axis is parallel to 
 G~L. ^ 
 
 Construction 11. Let A-B 
 in Fig. 112 be the axis of 
 the cylinder and / the given point. A straight Une through J 
 
 and parallel to A-B must be 
 
 /^ ¥ 
 
 Fig. III. — A Line drawn through a Point and 
 Tangent to a Cylinder. 
 
 Del 
 
 K 
 
 WP 
 
 §! 
 
 \ 
 
 \ 
 
 —fy^-f^-^' 
 
 a line of the required plane, 
 both traces of which must be 
 parallel to G-L. 
 
 Pass a profile plane R through 
 the point / and it wiU cut a 
 circle, with center at C, from 
 the cylinder. Revolve the plane 
 R around its horizontal trace 
 into H. The center of the cir- 
 cle wiU fall at Cc\, and the 
 point / at Jj\. Next draw the 
 circle in its true size about the 
 center Cci and the line Dd-Ei 
 drawn tangent to this- circle is 
 
 Point and Tangent to a Cylinder^hose Axis & line of the required plane rC- 
 
 isParaUeitoG-L. volvcdiutoH. Revolve Dd-Ei 
 
 to' its true position and as it pierces H at Dd, and V at Ee', the 
 line HT through Dd and parallel to G-L is the horizontal trace 
 
 Ee'\ VT 
 
 Fig. 112. — A Plane passed through a Given 
 
114 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 and VT thi;ougli Ee' and parallel to G-L is the vertical trace of 
 the required plane. 
 
 To determine the element of tangency revolve the point of 
 tangency Fji, /i' to its true position and through it draw the 
 projections of K-M parallel to G-L. 
 
 126. PROBLEM 57. To pass a plane tangent to a sphere at 
 a given point on its surface. 
 
 Analysis. A sphere shows in projection on V and H as a circle 
 with a dianleter equal to the diameter of the sphere. The 
 radius of the sphere drawn to the point of contact of the tangent 
 
 plane is perpendicular to 
 this plane. Hence, if a 
 plane is passed through 
 the given point and per- 
 pendicular to the radius 
 of the sphere drawn to 
 the tangent point, it wiU 
 be the required plane. 
 
 Construction. Let the 
 sphere be given by its 
 projections shown in Fig. 
 113, and let i, b' be 
 the given point on the 
 sphere. Through the 
 point B draw B-D per- 
 pendicular to the radius 
 B-C and parallel to H. 
 The vertical projection 
 of B-D will then be parallel to G-L and its horizontal projection 
 will be parallel to the required horizontal trace HT. Through 
 Dd' draw VT perpendicular to c'-b'. Through the intersection 
 of VT with G-L draw HT parallel to d-b. 
 
 Check. Swing the sphere about a vertical axis C-A until the 
 radius B-C is parallel to V. Draw a line tangent to the great 
 circle of the sphere at 61, &i', and determine its horizontal piercing 
 point Ffi. When the line B-F is revolved to its true position 
 the point F must fall on HT. 
 
 Fig. 113.- 
 
 ■To pass a Plane Tangent to any Given Point 
 on the Surface of a Sphere. 
 
TANGENT PLANES AND DOUBLE-CURVED SURFACES 115 
 
 127. PROBLEM 58. To pass a plane tangent to a sphere 
 and through a given straight line without the sphere. 
 
 Analysis. A plane passed through the center of the sphere and 
 perpendicular to the given line will be pierced by this line (pro- 
 
 Fig. 114. — To pass a Plane through a dlveii Straight Line and Tangent to a Sphere. 
 (.0-1 Is equal to o'-m' In length; IV-V is equal to Ul-k.) 
 
 duced if necessary) and it will cut a great circle from the sphere. 
 An auxiliary line drawn from the point in which the given line 
 pierces this plane, and tangent to the circle cut from the sphere 
 by the auxiliary plane, will be a line of the required plane. 
 This hne with the given line will determine the required plane. 
 
Il6 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Construction. Let the sphere be given by its projections in 
 Fig. 114, and let a-c, a'-c' be the projections of the given line. 
 Pass the plane R through the center of the sphere 0, and per- 
 pendicular to the given line A-C (see Problem 26, page 69). 
 By use of the plane S, determine the piercing point K of the line 
 A-C on the plane R (see Problem 23, page 64). Revolve the 
 plane R, and with it the center of the sphere, around HR as 
 an axis into H. Draw a great circle of the sphere about Ooy (the 
 revolved position of O) and through the revolved position lLk{ of 
 K draw tC^i-Dci tangent at F/i to the great circle. The pro- 
 jection Dd must be a point on HT. Determine the horizontal 
 piercing point B of the given line A-C and draw the required 
 horizontal trace ET through Dd and Bb. Determine Ee' the 
 vertical piercing point of A-C and draw VT through Ee' and 
 the point of intersection of HT with G-L. 
 
 The plane T touches the sphere at F and is perpendicular 
 to A-C. 
 
 128. Definitions. An hyperboloid of revolution is a warped 
 surface of revolution generated by revolving a straight line 
 around an axis at a constant distance from and kept at a con- 
 stant angle with the axis. 
 
 An hyperboloid of revolution of two nappes is a double curved 
 surface of revolution generated by revolving an hyperbola about 
 its transverse axis. 
 
 A torus is a double curved surface of revolution generated by 
 revolving a circumference about a line which is in the plane of 
 and outside the circumference. 
 
 129. PROBLEM 59. To draw the third angle plan and ele- 
 vation of an hyperboloid of revolution with its' axis perpendicular 
 to H and to assume a point on its surface. 
 
 Analysis. In order to draw the hyperboloid, the distance of 
 the generating line from the axis must be known as well as its 
 inclination to a plane perpendicular to the axis. As the gen- 
 erating line revolves, the path of the horizontal projection of 
 each point is a circle and the vertical projection is a straight 
 line parallel to G-L. Hence, by finding the projections of points 
 
TANGENT PLANES AND DOUBLE-CURVED SURFACES 117 
 
 on the generating line in the proper position, the projections of 
 the hyperboloid can be determined. To assume a point on the 
 surface draw the two projections of any generating line; assiune 
 a point on one projection of 
 the line and draw the ruled 
 projector to the other projec- 
 tion of the line. 
 
 Construction. In Fig. 115 
 let a-b, a'-b' represent the given 
 axis. Let m-k, m'-k' repre- 
 sent the generating line, at the 
 proper distance from A-B, and 
 making the correct angle with 
 H. Mark off any number of 
 convenient points such as C, 
 D, E, F, and M. As these 
 points revolve about A-B their 
 horizontal projections move in 
 circles and their vertical pro- 
 jections in straight line paral- 
 lel to G-L. Revolve all points 
 xmtil they are in a plane paral- 
 lel to V as at CiDiEiFi and if i 
 and then draw in the curve. 
 
 Fig. IIS.— Projection of an Hyperboloid of 
 Revolutions. 
 
 130. PROBLEM 60. To 
 draw the third angle plan and 
 elevation of a torus which is inclined to both V and H. 
 
 Analysis I. The projection of the torus consists of projec- 
 tions of lines tangent to the generating circle. 
 
 Construction I. Draw the torus in its simplest position, that 
 is, with its axis perpendicular to H [see Fig. 116(a)]. Several 
 positions of the generating circle are shown, the center being 
 at C, Ci, C2, etc. Fig. 116(b) shows the construction when the 
 axis is inclined to H and parallel to V. Fig. 116 (c) shows the 
 construction when the axis is inclined to both H and V. 
 
 Construction 11. The method described above makes it 
 necessary to use a generating circle, which in most positions 
 
ii8 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 projects as an ellipse." A simpler construction results by assum- 
 ing the torus to be generated by a sphere instead of a circle. 
 This generating sphere (center at S) projects in all positions as 
 
 
 c c4 c'3 
 (a) 
 
 Fig. 116. — Projection of a Torus Inclined to V and H. 
 
 a circle, hence it is only necessary to determine the projections 
 s, s', Si, Si, of several positions of the center S of the sphere, 
 and then describe circles about these projections and draw the 
 projections of the torus tangent to these circles. See Fig. 116. 
 
CHAPTER VII 
 SECTIONS 
 
 131. Definitions and General Considerations. A section is a 
 plane figure cut from an object by a plane passing through it 
 (see Figs. 117, 118, and 119). Such a plane as T or R is termed 
 a cutting plane. The section illustrates the appearance of the 
 cut surface of the object with one of the "cut parts" removed. 
 
 Fig. 1 17. — Sections that can be cut from a Cylinder by a Plane. 
 
 The section is usually shown in its true form, but it may be 
 represented by its plan and elevation. The line in which the 
 cutting plane intersects the object is termed the line of inter- 
 section, and is common to both the solid and the cutting plane; 
 hence, whatever the form of the object, the outline of inter- 
 section must be a plane line, and the section a flat surface (see 
 Figs. 117 II and III). The lines outlining the form of a sec- 
 tion may be straight, curved, or a combination of these two, 
 depending upon the form of the solid cut and the position ai 
 the cutting plane. Thus in Fig. 117 I the right cylinder can be 
 cut in any one of three general positions, each position giving 
 
 a definite form of section. The section cut by a plane T, 
 
 119 
 
120 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. Ii8. 
 
 - Sections that can be cut fiom a Cone by a 
 Plane. 
 
 through or parallel to the axis, is termed a longitudinal section 
 and is a rectangle in form. The section cut by a plane R, per- 
 pendicular to the axis, is termed a transverse section and in 
 
 form is a circle. The 
 section cut by a plane 
 S, inclined to the axis, 
 is termed an oblique 
 section and in form is 
 an ellipse. 
 
 Five sections may be 
 cut from a cone (see 
 Fig. ii8); a circle, if 
 the cutting plane is 
 perpendicular to the 
 axis; a triangle, if the 
 cutting plane contains 
 the vertex; an ellipse, if the cutting plane makes an angle with 
 the axis of the cone greater than that of an element; an hyper- 
 bola, if the angle the cutting plane makes with the axis is less 
 than that of the element; and 
 a parabola, if the angle the 
 cutting plane makes with the 
 axis is equal to that of the 
 element. 
 
 But one form of section can 
 be cut from a sphere, namely, 
 a circle (see Fig. -119). The 
 diameter of this circle depends 
 upon the distance the cutting 
 plane is from the center of 
 the sphere. When the cutting 
 plane T passes through the center of the sphere, the section is a 
 great circle of the sphere. That is, the diameter of the circle is 
 the same as that of the sphere. When. the cutting plane H is 
 tangent to the sphere the section has diminished to a point. 
 
 Fig. 119. 
 
 -The Section cut from a Sphere by 
 a Plane. 
 
 132. PROBLEM 61. To determine any section of a right 
 square prism. 
 
SECTIONS 
 
 121 
 
 Nni 
 
 Analysis. Place the prism with a base against H and a diago- 
 nal of the base perpendicular to V. Take a cutting plane per- 
 pendicular to V and inclined to H. The plane will cut the 
 edges of the prism in points the projections of which are known. 
 By connecting these points with straight lines, 'the line of inter- 
 section of the cutting plane with the prism is determined. By 
 revolving this outline 
 of intersection into V 
 or H the true form 
 of the section is de- 
 termined. 
 
 Construction. Let 
 the prisfti be shovfrn 
 by its projections in 
 Fig. 1 20, and let VT 
 and HT represent the 
 cutting plane. The 
 plane T cuts the edge 
 B-F of the prism in 
 the point iV, the ver- 
 tical projection n' of 
 which is determined 
 by the intersection of 
 the vertical trace VT 
 with the vertical pro- 
 jection b'-f of the 
 edge. Since B-F is 
 perpendicular to H the horizontal projection of all its points 
 must .be in a single point, hence the horizontal projection n of 
 N. is the same point as the horizontal projections b of the point 
 B and / of F. Sirnilarly, the points Q and are found. The 
 point / must have its horizontal projection on the ruled projec- 
 tor fromy and on the edge E-K, hence at y. Similarly, M 
 must be on the edge E-F. 
 
 Therefore j-m-n-o-q must represent the plan of the section 
 and j'-m'-n'-o'-q' its elevation. To determine the true form 
 of the section revolve the polygon j^m-n-o-q about HT into H 
 as shown in Ooi-Nfir-Mmi-Jji-Qqi. 
 
 Fig. 120. — Section cut from a Right Prism by a Plane 
 Perpendicular to V and Inclined to H. 
 
122 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Check. Revolve the section into V- about VT as an axis, and 
 the resulting polygon Oo^-Nn^-Mm^-Jj^-Qq^ will check with 
 Ooi-Nni-Mnh-Jji-Qqi. if the construction is accurate. 
 
 Fig. HI. —To determine the Section cut from any Right Prism by a Plane. 
 
 133. PROBLEM 62. To determine any section of any right 
 prism. Proceed as in Problem 61, page 120. See Fig. 121. 
 
 134. PROBLEM 63. To determine the section cut from a 
 right cylinder by a plane making a given angle with its axis. 
 
SECTIONS 
 
 123 
 
 Analysis. Divide the circumference of the cylinder into any 
 convenient number of parts; draw elements through these points; 
 
 ^ — 5 
 
 Fig. 122. — To determine the Section cut from a Cylinder by a Plane. 
 
 determine the intersection of the elements and the cutting plane 
 and revolve the section so found into V or H. 
 
 Construction. Let the cylinder be given as in Fig. 122, and 
 let VT and HT be the traces of the cutting plane. Divide the 
 
124 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 circumference of the base of the cylinder into any convenient 
 number of parts (1-12 as shown in Fig. 122) and draw the pro- 
 jections of elements of the cylinder through these points. Thus 
 A-B is one such element. The others are drawn, but, to avoid 
 confusion, are not lettered. The plan of the section will fall in 
 the plan of the cylinder and the elevation will fall in VT. Re- 
 volve the points i to 13 into H about HT and join these points 
 (in revolved position) with a curved line. 
 Check. Revolve into V about VT. 
 
 135. PROBLEM 64. To determine the section of a pyramid 
 when cut by a plane making a given angle with its axis. 
 
 Analysis. Determine the intersection or piercing point of 
 the edges of the pyramid with the cutting plane. Connect these 
 points to form a polygon and revolve the polygon into V or H. 
 
 Construction. Let the pyramid be given as in Fig. 123, and 
 let VT and HT represent the cutting plane. 
 
 Frcuts the vertical projection of the edges in the points i', k', 
 j', m', n', r' , s', and the corresponding horizontal projections of 
 all points, except the point / on the edge B-Q, can be deter- 
 mined by the ruled projectors. The edge B-Q coincides with 
 the ruled projector, hence B-Q can be revolved parallel to V; 
 the point 7i determined, and then counter-revolved to its correct 
 position. The polygon i-k-j-m-^n—r-s is the plan of the sec- 
 tion and Hi- Kki-Jji-Mmi- Nfii-RriSsiis its true form and size 
 fotmd by revolving the section into H around HT. 
 
 Check. The polygon lii'-Kki'-Jji'-Mmi'-Nni'-Rri'-Ss^' is 
 the section revolved into V and should check with the section in H. 
 
 136. PROBLEM 65. To determine the section of a right cone 
 when cut by a plane making a given angle with its axis. 
 
 Analysis I. Divide the base of the cone into any conven- 
 ient number of parts and draw elements of the cone through 
 these points. Determine the points of intersection of these ele- 
 ments with the cutting plane. Draw a curved line through 
 these points of intersection and revolve this figure into V or H. 
 
 Construction I. Let the cone be given as in Fig. 1 24 and let 
 HT and VT repres'ent the cutting plane. Let a, a' and c, c' be 
 
SECTIONS 
 
 125 
 
 Fig. 123. — To determine tlie Section cut from a Pyramid by a Plane. 
 
 the projection of any two points of division on the base of the 
 cone, and let a-q, a'-q' and c-q, c'-q' represent the elements 
 through these points. The vertical trace VT cuts a'-q' at 
 b'; then b is found by drawing the ruled projector through b' 
 
126 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. 124. — Section cut from a Right Cone by a Plane Inclined to its Axis. 
 
 to cut a-q, hence is at b. Similarly, determine other points and 
 draw the curve as shown. Revolve this figure, which is an ellipse, 
 into H at Bbi-Ddi or revolve into V at Bhi-Ddt'. 
 
 Analysis II. Pass a series of horizontal planes through the 
 cone. .Such planes will cut circles from the cone, which are 
 shown in plan as circles and in elevation as straight lines paral- 
 lel to G-L. The points of -intersection of these circles with the 
 cutting plane marks points on the required section. 
 
 Construction II. Let the cone be given as in Fig. 125, and let 
 the plane cut the cone parallel to a single element, so that the 
 
 i\ 
 
SECTIONS 
 
 127 
 
 section cut is a parabola. ET and VT represent the cutting 
 plane. Take any horizontal plane S and it will cut from the cone 
 a circle of radius 0~C horizontally projected in the circle c-h-a 
 and vertically projected in the trace VS. The horizontal pro- 
 jections a and h must be on the horizontal projection a-h-c of the 
 circle, and on the ruled projector from a', b', etc., hence at 
 
 Fig. 125. — Section of a Right Cone cut by a Plane Parallel to an Element. 
 
 a and b. Other points are determined in the same manner 
 and all these points are revolved into either V or H to determine 
 the true shape of the section. 
 
 137. PROBLEM 66. To determine the section of an annular 
 torus when cut by a plane making a given angle with H. 
 
 Construction. See Fig. 126. Cut the torus by planes par- 
 allel to H, such as S. This plane S cuts two circles from the torus, 
 one of radius 0-C, shown in vertical projection in the trace VS 
 and in horizontal projection as the circle a-c-b, the other of 
 radius 0-D, shown also in the vertical projection VS and in the 
 horizontal projection e-d-f. The cutting plane T intersects each 
 
128 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 of these circles in two points; the circle A-C-B in the points 
 A and B and the circle E-D-F in the points E and F, and the 
 four points are located in the required section. In a similar 
 
 Fig. 126. — Section cut from a Torus by a Plane Inclined to its Axis. 
 
 manner a sufficient number of points can be foimd so as to de- 
 termine the projections of the section, and then by revolving the 
 section about HT into T or about VT into V it can be fully 
 determined. 
 
CHAPTER VIII 
 INTERSECTIONS AND DEVELOPMENTS 
 
 I 
 
 138. Definitions and General Considerations. When two sur- 
 faces meet or penetrate, the line along which they meet is termed 
 their line of intersection. The line of intersection is therefore 
 a line common to both intersecting surfaces, and its form is de- 
 termined by the character of the surfaces and their relative 
 size and position. It is necessary to determine the projections 
 of the line of intersection in order to represent the surfaces, and 
 the true form of the line of intersection is necessary in order to 
 develop the surfaces. By the development of a stirface is meant 
 that it is laid ont flat without any part of it being " stretched." 
 
 When the intersecting surfaces are solids, the general method 
 of determining the line of intersection is to pass planes through 
 the solids which will cut intersecting sections. The boundaries 
 of these sections intersect in points which are common to both 
 solids, hence are points on their Hne of intersection. 
 
 The development of a solid is effected when every line of its 
 surface has been brought into a single plane and it is shown in its 
 true size and shape. The development can then be cut out of 
 cardboard or sheet metal and when properly folded or shaped 
 it will make a surface exactly Hke the surface of the solid illus- 
 trated. Only those bodies which are bounded by planes, such 
 as cubes, prisms, and pyramids, and single curved siurfaces 
 such as cylinders and cones, can be developed. Double curved 
 surfaces such as spheres and ellipsoids can not be developed. 
 The cube, prism, and pyramid are developed by placing one 
 face in contact with a plane, and then turning the solid about its 
 edges to bring aU of the remaining faces of the solid in contact 
 with the plane so that their true size and shape can be outlined. 
 It is such an outline of all the faces of the solid shown in true 
 
 size and shape which constitutes the development of the solid. 
 
 129 
 
I30 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 To develop a cone or cylinder, one element is placed in con- 
 tact with a plane and then the solid is rolled on this plane until 
 every other element has touched the plane. 
 . That portion of the plane covered by the solid in its revoli^- 
 tion represents the development of the surface of the solid. 
 
 139. PROBLEM 67. To determine the true form of the 
 line of intersection of an hexagonal i»:ism when cut by a plane 
 meeting its axis at any given angle, and to develop the surface 
 of the prism. 
 
 K J 
 
 Fig. 127. — To determine the True Form of the Line of Intersection of a Plane and a Prism 
 and to develop the Surface of the Prism. 
 
 Analysis. Since the prism is a solid bounded by planes, the 
 problem is to find the line of intersection of two planes; the cut- 
 ting plane, and the planes of the prism sides. This line of inter- 
 section is 'therefore a straight Hne, and since the prism is repre- 
 sented by its edges, the problem is really the determination of 
 the points where the edges of the prism pierce the cutting plan?. 
 Such points are common. to both the prism and the cutting 
 plane, hence, when connected in the proper order by straight 
 lines, these points determine the line of intersection. 
 
 Construction. Draw the plan and elevation of the prism and 
 the traces of the plane as shown in Fig. 127. It is evident that 
 
INTERSECTIONS AND DEVELOPMENTS 131 
 
 the edges of the prism pierce the cutting plane T in the points, 
 ^j 2, J, '4, 5, and 6. Lay off on G-L produced (or any convenient 
 straight line) the distance N-0 = n-o; O-I = o-i; etc. Draw 
 the perpendiculars N-i, O-2, etc. (equal in length to n'-z' and 
 o'-2' respectively), to represent the edges of the prisms in de- 
 velopment. 
 
 Project the points i' to i, 2' to 2, etc., to represent the posi- 
 tion of I, 2, J, etc., in development. Connect the points 1-2,- 
 2-5, etc., to get the true Jorm of the line of intersection. 
 
 To draw the development of the section, revolve the plane T 
 into H about the trace HT. Transfer this figure 1-2-3-4-5-6 
 to the development by building it up on the side 6-1, making it 
 exactly the same size and shape as the section would be if re- 
 volved into H. Since the plan of the prism shows the base in its 
 true size and shape, the developed end or base N-0-I-J-K-M 
 can be laid off on the side N-O, and the development is com- 
 plete. 
 
 Check. The top portion of the prism can be developed and 
 the true size and shape of the section can be determined by re- 
 volving T into V about VT. The simi of the developed seg- 
 ments of any given edge should equal the height of the prism. 
 
 140. PROBLEM 68. To determine the true form of the 
 line of intersection of a right cylinder cut by a plane meeting 
 its axis at any given angle, and to develop the surface of the 
 cylinder. 
 
 Analysis. Let the cylinder and plane be given as in Fig. 128. 
 Divide the base of the cylinder into a sufficient number of equal 
 parts (say twelve), and through these points of division draw 
 elements of the cylinder. Determine the piercing points of these 
 elements on the cutting plane, exactly as if they were the edges 
 of a prism. Revolve the piercing points into H about ET and 
 connect them with a smooth curve. Develop the surface as in 
 Problem 67, page 130. 
 
 Construction. Draw a straight line B-D-F-B equal in length 
 to the circumference of the cylinder. Divide this line into the 
 same ntunber of equal parts as the base of the cylinder. Erect 
 perpendiculars at the points of division, determine the points 
 
132 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 1-2-3, etc., as in Problem 67, and through these points draw a 
 smooth curve. Complete and check as in Problem 67, page 130. 
 
 Fig 128. — To determine the True Form of the Line of Intersection of a Plane and a Cylinder 
 and to Develop the Surface of the Cylinder. 
 
 141. PROBLEM 69. To determine the true form of the 
 line of intersection of a p3nramid cut by a plane which meets the 
 axis of the pyramid at any given angle, and to develop the sur- 
 face of the pyramid. 
 
 Analysis. The .vertical projections of the points where the 
 edges- pierce the cutting plane are found by inspection; the hori- 
 zontal projections are determined by drawing ruled projectors 
 from the vertical projection of the point, to the horizontal pro- 
 jection of the edge, upon which the point is located. Connect- 
 ing the horizontal projections of the points in proper order 
 determines the horizontal projection of the section cut by the 
 plane. The true shape of this section is determined by revolv^ 
 ing it into H as in the preceding problems. 
 
 To develop the surface of a right pyramid, draw an arc, having 
 as its radius the true length of an edge. On this arc lay off 
 chords equal in length and equal in number to the base edges 
 of the pjnramid. At the extremities of each of these chords 
 draw straight lines to the center of the circular arc. These 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 133 
 
 lines represent the developed edges of the pyramid. On each 
 developed edge mark off the true distance from the base to the 
 piercing point on that edge. Connect these points in proper 
 order for the true development of the pyramid surface. The 
 development of the base and of the section are brought into 
 proper relation as in problem 67, page 130. 
 
 Construction. See Fig. 129. Determine the projections, i', /; 
 2' , 2; etc., of the piercing points and revolve the plane T into H 
 
 a' b'f 01 ei 
 
 Fig. 129. — To determine the True Form of the Line of Intersection of a Plane and a Pjnamid 
 and to develop the Surface of the Pyramid. 
 
 about HT to get the true section 1-2-3, etc. Swing the in- 
 clined edges of the pyramid parallel to V to determine the true 
 distances V to 2', d to j', etc., a'-g' being a true length projec- 
 tion; draw an arc with 5' as a center and cf-a' as a radius. Step 
 off a'-B = a-b, B-C = h-c, etc., until the complete base line is 
 stepped off, and connect these points in successive order. From 
 these points draw B-q' , C-q', etc., and measure off the distances 
 B-2 = a'-2i; C-j = a'Si, etc., to get the true distances from 
 the base along the edges to the piercing points. Connect the 
 points I to 2, 2 to J, etc., and on the edge 1-6 construct the 
 
134 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 section 1-2-3-4-5-6 making it the same in all respects as the 
 section which was revolved into H about HT. Also construct 
 the base A-B-C-D-E-F the same in all respects as a-b-c-d-e-f. 
 Check. The portion of the pyramid from the apex to the 
 plane could be developed and should check up with the devel- 
 oped lower portion. 
 
 142. PROBLEM 70. To determine the true form of the 
 line of intersection of a right cone when cut by a plane meeting 
 the cone axis at any given angle, and to develop the surface of 
 the cone. 
 
 Analysis. Divide the base line into a sufi&cient number of 
 equal parts (say twelve). Through these points of division draw 
 elements of the cone. Proceed as in Problem 69, page 132, 
 treating the elements as if they were edges of a pyramid, with 
 the single exception that the points are connected by a smooth 
 curved line. 
 
 Construction. Divide the base circle into twelve equal divi- 
 sions, A-B, B-C, etc., as shown in Fig. 130. Draw the elements 
 A-Q, B-Q, etc., projected at a-q, a'-q', b-q, b'-q', etc. Proceed 
 as explained in Problem 69, page 132. 
 
 Check. Same general method as in Problem 69. 
 
 NOTE. The horizontal projections of the points 4 and 10 
 (see Fig. 130) cannot be determined by ruled projectors, since 
 they He in a plane perpendicular to G-L. To determine these 
 projections, pass a plane R through the cone at these points 
 (2 and 10), and at right angles to the cone axis. This plane will 
 cut a circle from the cone, and this circle will pierce the plane 
 T in the required points, because those points are common to 
 both the cone and the plane T, hence on their line of intersec- 
 tion; thus the points are completely determined. 
 
 Check. The horizontal projections of the points i; 2, 12; 
 J, II, etc., can also be determined in the above way. 
 
 143. PROBLEM 71. To determine the line of intersec- 
 tion of a prism when cut by a plane inclined to V and H. Also 
 to develop the surface and determine the true form of the line 
 of intersection. 
 
INTERSECTIONS AND DEVELOPMENTS 135 
 
 [ 
 
 a' b' c' di e' f i' 
 o' n' ml fc' y 
 
 Fig. 130. — To determine tlie True Form of the Line of Intersection of a Plane and a Cone 
 and to develop the Suiface of the Cone. 
 
 Analysis. Pass a series of planes through the edges of the 
 prism so as to intersect the given plane. Each of these planes 
 will cut a line from \he given plane, which line intersects the 
 edge or edges of the prism through which the airxiliary plane was 
 passed. The projections of the points of intersection are the 
 projections of the piercing point of the edges of the prism on the 
 given plane, since they represent points which are common to 
 the cutting plane and the edge of the prism. The development 
 is made as in Problems 67 to 70 inclusive, page 130. 
 
 Construction. Assume one base of the prism in H and let the 
 plane T be given by its traces HT and VT (see Fig. 131). 
 Pass a series of planes (such as R) through the vertical edges 
 (such as M-N), cutting out the lines of intersection (such as 
 K-I) on the plane T. The vertical projection w'-w' of the 
 edge intersects the vertical projection k'-Ii' at the point a', 
 which is the vertical projection of the piercing point of the 
 
136 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 edge M- N on the plane T. In like manner, the projection of 
 all the piercing points are determined and joined in proper order 
 for the line of intersection of the given plane and prism. The 
 
 Fig. 131, — To determine the Line of Intersection cut by a Plane from a Prism and to develop 
 the Surface of the Prism. 
 
 development is foimd and checked as in Problems 67 to 70 in- 
 clusive, page 130. 
 
 144. PROBLEM 72. To determine the line of intersection 
 of a cylinder cut by a plane inclined to V and H. Also to de- 
 velop the surface and determine the true form of the line of 
 intersection. 
 
 ' I Analysis. Divide the base of the cylinder into any number 
 of equal parts, and through the points of division draw elements 
 of the cylinder. Treat these elements of the cylinder exactly 
 as if they were edges of a prism, and determine points of the 
 line of intersection as in Problem 71, page 134; these points are 
 connected with a smooth curve. 
 
 Construction. See Fig. 132. Pass several auxiliary planes 
 R, S, etc., through the cylinder and perpendicular to H. De- 
 termine the projections of the line of intersection as in Prob- 
 lem 71, page 134. 
 
 Revolve X-Y until parallel to H, and draw lines through the 
 projections, i, 2, 3, 4, etc., each parallel to Xx-Xi, to get the true 
 distances Y-i, Y-2, etc. Knowing the true length of the major 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 137 
 
 axis {Xot-y being perpendicular to HT) and the true length and 
 location of ordinates to this axis, the true section Xx^A- Y-B 
 can be constructed. 
 
 Check. The section can be checked by revolving it about 
 HT into H. The surface is developed similarly to that in 
 Problem 71. 
 
 Fjg. 132. — To determine the Line of Intersection of a Cylinder and a Plane and to develop 
 the Surface of the Cylinder. 
 
 145. PROBLEM 73. To determine the line of intersection 
 of a pyramid cut by a plane which is inclined to V and H. Also 
 to develop the surface and determine the true form of the line 
 of intersection. 
 
 Analysis. Pass auxiliary planes through the edge of the 
 pyramid (these planes will be perpendicular to H), and deter- 
 mine the line of intersection as in Problem 72, page 136. De- 
 velop the surface as in Problems 69 and 70, page 132. 
 
 Construction. See Fig. 133. Pass the auxiliary plane S 
 through the edges as shown and determine the projection of the 
 two points D and E where the edges X-Y and X-Z pierce the 
 ■plane T. There are two points on the section. 
 
 Similarly determine all other points on the section. 
 
 Check. The Hnes of intersection of all auxiliary planes and 
 T must pass through the point C. 
 
138 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Develop the surface of the pyramid as in Problem 69, 
 page 132. 
 
 V^V 
 
 Fig. 133. — Intersection of a Plane and a Psrramid and Development of the Pyramid. 
 
 146. PROBELM 74. To determine the line of intersection 
 of a cone cut by a plane inclined to V and H. Also to develop 
 the surface of the cone and determine the true form of the line 
 of intersection. 
 
 Analysis. Divide the base into a sufficient number of parts 
 (say twelve), and through these points draw elements of the 
 cone. Proceed as in Problems 70 and 73, page 134, with the 
 exception that projections and developed points oi the section 
 are to be connected by a. smooth curve. 
 
 Construction. See Fig. 134. The general construction is 
 similar to that of Problem 73, page 137. 
 
 147. PROBLEM 75. To determine the true form of the 
 line of intersection of two square prisms, meeting at any angle, 
 and to develop the surfaces of the prisms. 
 
 Analysis. Since the faces of the prisms are planes, all lines 
 
INTERSECTIONS AND DEVELOPMENIS 
 
 139 
 
 Fig. 134.— Intersection of a Plane and a Cone and Development of the Cone. 
 
 of intersection are straight lines, and these are determined by 
 finding points common to both prisms and connecting these 
 points by straight lines. 
 
 Construction. Let the principal prism be vertical, that is, 
 let it have its base parallel to H, and let it be intersected first 
 by a prism (on the left) which has its axis parallel to H, and 
 next by a prism (on the right) which has its axis at any angle 
 to H (see Fig. 135). 
 
 The vertical prism. will show ia plan as a square a-b-c-d, 
 and in elevation as two tangent rectangles, d'-j'-i'-c' and 
 i'~c'-b'-j' . To draw the vertical projection of the prism inter- 
 secting on the left, draw projections of the square K-L-M- N 
 to represent the base of the prism in its proper vertical position 
 when viewed in the direction of the arrow. Project the edges 
 m'-r' ; k'-o' ; I'-q' and n'-f, against the elevation of the vertical 
 prism. The plan of the prism can be drawn in a similar manner. 
 It is then evident that the projections p,q, r, and are the hori- 
 zontal projections of points common to both prisms. The vertical 
 projections of these points are determined by the ruled projectors. 
 Thus, p' must be at the intersection of the ruled projector from 
 p and the vertical projection of the edge n'-p', that is at p'. 
 
140 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Connect the points thus found to determine the h'ne of 
 intersection of the prisms. The lower prism X-W-Y-Z is pro- 
 jected by determining the position of its axis and making the 
 construction as in the previo^s case. To develop the prism 
 A-B-C-D, use Fr-Ji produced as a base line; lay off the dis- 
 
 Fig. 135.— To determine the True Fonn of Intersection of Two Prisms and to develop tbe 
 Surfaces of the Prisms. 
 
 tances Cy-Bi, Bi-Ai, Ay-Di, Di-Ci, each equal to a side of the 
 prism. Draw the perpendiculars C1-/1, Bi-Fi, etc., making each 
 equal to the length of the prism. These represent the edges of 
 the prism in the development. As all vertical heights are shown 
 true length in elevation^ the points q' and r' can be projected 
 directly to the edge Dr-Ji upon which they are located. Thus 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 141 
 
 the points Qi and i?i are determined. The points Oi and Pi are 
 determined by drawing the elements 1-2 and 3-4, making E1-2 
 
 ii JVi Ml Ki Li 
 
 3 Oi, 
 
 p-c=fi > 
 
 Ii 
 
 n 
 
 Fig. I3S- — Continued. — Development of the Surfaces. 
 
 h 
 
 equal to o-a, and 4-I1 equal to p-c. Also, Si-Ui-Ti-Vi is 
 found in the same manner. ' 
 
 The development of the prism W-X- Y-Z is determined by- 
 laying off Xi-Zi, Zr- Yi, etc., each equal to a side of the prism 
 base, and erecting the perpendicular Xi-Fi equal to x'^'; Zi-Si 
 equal to z'-s', etc.; xf^', z'-s', etc., being true length projections 
 of these edges. 
 
142 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 148. PROBLEM 76. To draw the plan and elevation of two 
 right cylinders, the axes of which meet at any given angle, and 
 to develop the surfaces and determine the true form of the line 
 of intersection. 
 
 Analysis. Assume that the axes of the cylinders lie in a plane 
 parallel to V. Also, that the axis of one cylinder is perpendicular 
 
 Fig. 136. — To determine the True Form o{ Intersection of Two Cylinders and to develop the 
 Surfaces of the Cylinders. 
 
 to H, another is parallel to H, and a third makes 60 degrees with 
 the first cylinder. To determine the Unes of intersection, pass a 
 series of planes through the cylinders parallel to V. Such plalies 
 cut elements from the cylinders, and the intersection of these 
 elements determines points on the line of intersection of the 
 cylinders. Develop the surfaces by laying ofif rectangles 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 143 
 
 having a height equal to the height of the cylinder and a 
 length equal to the circumference of the base of the cylin- 
 der; determine the position of elements of the cylinder in this 
 
 41 Ci 
 
 Fig. 136 Continued. — Development of tlie Surfaces. 
 
 development and locate the points of intersection on these 
 elements. 
 
 Construction. See Fig. 136. Draw the projections of the 
 axes 1-2, 3-4, and 5-6. AU are parallel to V in plan, and in 
 elevation i'-2' is perpendicular to G-L, ^'-4' makes 90 degrees 
 with i'-2' , and 5'-(5' makes 60 degrees with i'-2' . The cylinder 
 1-2 will show in plan as a circle having a diameter equal to the 
 
144 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 diameter of the cylinder, and in elevation as a rectangle having 
 a width equal the diameter of the cylinder and a length equal 
 the length of the cyhnder. To find the projections of the 
 cylinder" j-^, draw its base on j-4 produced as shown. This 
 represents the base when looking in the direction of the arrow. 
 Similarly draw the base in elevation. Divide this base iato 
 any number of equal parts (say twelve) and through these 
 points of division pass planes R, S, etc., parallel to V. These 
 planes cut from the cylinders elements which intersect at points 
 common to both cylinders, heijce on the line of intersection. 
 Thus, the plane S cuts the elements E-F and M-N from the 
 cylinder 1-2 and V-W and K-R from the cylinder 5-15. The 
 elements M-N and K-R intersect at K, therefore the point K 
 is a point on the line of intersection. Similarly determine all 
 other points and draw a smooth curve through them. The line 
 of intersection of the cylinders j-4 and 1-2 is similarly obtained. 
 
 To develop the cylinder 1-2, use Fi-Ni produced as a base line. 
 Draw a vertical line at Qi-Pi to represent the' element Q-P. 
 Lay off the length Qi-Ni-Fi-Qi equal to the circumference of 
 the cylinder and draw the element Q2--P2, which coincides with 
 Qi-Pi when the development is rolled together to form the 
 cylinder. 
 
 Next locate on the development the element Mi-Ni, etc., 
 of the cylinder which passes through the points K, V, etc., of the 
 line of intersection by laying off the distance Qi-Ni equal to 
 the arc g^-«, etc. The altitudes of the points K, V, etc., can be 
 transferred with dividers or in this special case projected directly 
 across by the horizontal hnes k'-Ki, v'-Vi, etc. 
 
 To obtain the development of the cylinders 3-4 and s~<^, 
 draw at any convenient position a base line equal in length to 
 the circumference of the cylinder and proceed as in the case of 
 the prisms of Problem 75, page 138. 
 
 149. PROBLEM 77. To draw the plan and elevation of a 
 pyramid and prism which intersect; to develop the surfaces 
 and determine the true form of the line of intersection. 
 
 Analysis. Assume the axis of the pyramid perpendicular to 
 H and the axis of the prism parallel to H. Pass auxiliary planes 
 
INTERSECTIONS AND DEVELOPMENTS 145 
 
 through the pyramid so as to be perpendicular to V and to 
 contain the edges of the prism and pyramid, and determine the 
 points in which the boundaries of th^ sections intersect. Connect 
 these points in proper order for the hne of intersection. Develop 
 the surfaces as in Problem 69, page 132, and Problem 75, page 138. 
 
 Construction. Let the pyramid and prism be given as in 
 Fig. 137 I. Pass the horizontal plane S through the edge E-F 
 of the prism. The plane S must cut a square section from the 
 pyramid. This square must have one corner at the point P 
 where VS intersects the edge 0-M, since P is common to both 
 the plane S and the edge 0-M. 
 
 This square section is shown in plan at p-i-2-3, and the 
 projection k where e-f intersects p-i determines the horizontal 
 projection of a point which is conmion to both pyramid and 
 prism, hence a point of their line of intersection. 
 
 The vertical projection k' of this point is determined by the 
 ruled projector k-k'. Since the point K is located in the face 
 M-O-N , it shows that the edge E-F pierces the pyramid in 
 this face. In a similar manner, pass the plane T through the 
 edge C-D and determine the point /. Also pass the plane U 
 through the edge A-B and determine the point X. The points 
 K and X are both in the face M-O-N, and hence can be con- 
 nected. The point / cannot be connected to either- isT or X, 
 as it lies on the opposite side of the edge 0-M and in a different 
 face of the prism. To determine the points where the edge 0-M 
 pierces the prism, pass the plane R through the edge 0-M and 
 perpendicular to V. The plane R cuts the edges of the prism 
 at points vertically projected at s'-d'-j' , hence S'~^~7 is the 
 horizontal projection of the section cut from the prism by this 
 plane. Also y and z are both horizontal projections of points 
 commati to the prism and p}a:amid; their vertical projections 
 are determined by ruled projectors. The point F, being on the 
 edge 0-M, can be connected to both X and /, and the point 
 Z can be connected to both K and /. This determines the line 
 of intersection on the right, and that on the left is determined 
 in a similar manner. 
 
 Check. Produce the edge 8-9 of the square section cut from 
 the pyramid by the plane T until it intersects the edge C-D of 
 
146 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 the prism. The lines S-g and C-D must intersect, as both lie 
 in the plane T and are not parallel. Also, as the line 8-p lies 
 
 Fig. 137. — To determine the Projection of the Line of Intersection of a 
 Pyramid and Prism. 
 
 in the plane of the pyramid face, N-O-M, it must intersect 
 C-D in the plane of this face or in the plane produced; hence 
 at the point 10. As the points 10 and K both lie in the plane 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 147 
 
 of N-O-M they can be connected, and if the line K-io inter- 
 sects the edge 0-M at Z, this part of the construction is correct. 
 
 K 
 
 F 
 
 
 _xl 
 
 B 
 
 
 ll 
 
 
 \ 
 
 \. 
 
 D 
 
 J! 
 
 /z 
 
 
 
 K 
 
 F 
 
 
 in 
 
 Fig. 137. — Continued. — To develop the Surfaces of the Pyramid and Prism. 
 
 Also, as the point X lies in the plane of N-O-M, the point Y 
 can be similarly checked. 
 
148 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Development. Lay out the surface of the pyramid as in 
 Problem 69, page 132, and determine the location of the square 
 sections i-P-j; 8-g, etc. 
 
 To determine the true form of the line of intersection, locate 
 the point Z on the edge 0-M by swinging 0~M parallel to V, 
 when z' will fall at Zi, and mi'-Zi is the true distance that the 
 point Z is from M (see Fig. 137 I). Similarly locate the point 
 Y on the edge 0-M in development (see Fig. 137 II). To 
 determine the point /, draw the element 0-W which will con- 
 tain the point I. As the edge J-M of the pyramid is parallel 
 to H, the distance j-^ is a true length, and hence J-W in the de- 
 velopment is known. Join W-0 in the development (Fig. 137 II), 
 and where this line intersects the edge ii-g the point / is located. 
 All other points are similarly determined and connected in proper 
 order. 
 
 To develop the prism, draw Gi-Li (Fig. 137 I) parallel to the 
 prism base. Determine the new vertical projection of the prism 
 and di'-bi'-fi shows each edge in its true length. Hence 
 draw F-B, Fig. 137 III, equal to fi'-bi of Fig. 137 I, etc. 
 The edge F-K is shown in true length in f-k, Fig. 137 I, hence 
 can be transferred at once to the development. Similarly de- 
 termine B-X. To determine the point Y of the development, 
 draw the projections of the prism element Y-V (see Fig. 137 I). 
 The length bi'-Vi is a true length and determines B-V of the 
 development. The element V- Y is parallel to B-X, and hence 
 can be drawn. The projection v-y is a true length, hence the 
 length V- Y is known. 
 
 Similarly, all other points are determined and connected in 
 proper order for the true form of the line of intersection. 
 
 Check. Similar lines in Figs. 137 II and 137 III must be 
 equal. That is, K-X of Fig. 137 II must equal K-X of Fig. 
 137 III, etc. as these lines coincide on the objects which intersect. 
 
 150. EROBLEM 78. To draw the plan and elevation of a 
 cone and cylinder which intersect and to develop the surfaces 
 and determine the true form of the curve of intersection. 
 
 Analysis I. Assimie the axis of the cone to be perpendicular 
 to H and the, axis of the cylinder parallel to V and H. Pass a 
 
INTERSECTIONS AND DEVELOPMENTS 
 
 149 
 
 series of auxiliary horizontal planes through the cone and cylin- 
 der. These planes cut circular sections from the cone and 
 rectangular sections from the cylinder. The points of inter- 
 section of these circles and rectangles which are cut by the 
 same plane are points common to both solids, and hence are 
 
 Fig. 138. — To determine the Intersection of a Cone and Cylinder. 
 
 points on their curve of intersection. Develop the surfaces as 
 in Problem 70, page 134, and Problem 68, page 131. 
 
 Construction. Draw 1-2, I'-s' and 3-4, 3' -4' (see Fig. 138 I), 
 to represent the projections of the axes of the cone and cylin- 
 der respectively. Upon these axes draw the plan and elevation 
 of the solids. Assxmie a new vertical plane (shown by Gi-Li, 
 Fig. 138 1) parallel to the base of the cyHnder and draw in the 
 new elevation on 11-21 and 31-41'. Pass the horizontal plane 
 
ISO 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 S through the solids. This plane cuts the cylinder along the 
 elements A-C and B-D, and from the cone it cuts a circle of 
 radius ei'-ji'. This circle, shown in plan at/-m-w, cuts the hori- 
 zontal projection of the element B-D at m and n. Hence these 
 
 G 
 
 D 
 
 y 
 
 \ 1 
 1 
 
 i 
 1 A' 
 
 1 1 
 
 1 
 
 1 A 
 
 -f\^\ 
 
 ' ! ' \ 
 
 i1 
 
 ! 1 
 
 i 1 
 
 i ! 
 
 1 \ 
 
 1 ' 
 
 1 1 
 1 1 
 
 ' ' 1 
 i 
 
 in 
 
 w 
 
 Fig. 138. — Contmaed.— To develop the Surfaces of the Cone and Of Under which Intersect. 
 
 are horizontal projections of two points on the curve of inter- 
 section. Their vertical projections are determined by ruled 
 projectors. The projection /-?w-« of the circle does not reach 
 the horizontal projection of the element A-C, therefore the cone 
 does not intersect the cylinder on this element. This is also 
 seen by inspection, as k\ marks the last point on top-and- 
 near-side of the curve of intersection. The points x and y are 
 
INTERSECTIONS AND DEVELOPMENTS 151 
 
 not on the element A-C, but are on V-W, an element cut by tbe 
 plane U. This plane also cuts from the cone the circle dra-wn 
 in plan through x and y. Similarly determine all points neces- 
 sary and draw the curve of intersection. 
 
 To develop the surface of the cone, proceed as in Problem 70, 
 page 134. To develop the cylinder, proceed as with the prism 
 in Problem 68, page 131, using a suflSicient number of elements 
 of the cylinder in the same manner as the edges of the prism 
 were used in problem 67. For developments see Figs. 138 I and 
 138 II. The bases of cone and cylinder have been omitted in 
 this development. 
 
 Check. The curve of intersection 0-K-P-Y-X on the de- 
 velopment of the cone, Fig. 138 II, must be equal in length 
 to 0-K-P-Y-X on the development of the cylinder, Fig. 
 138 III. 
 
 151. PROBLEM 79. To determine the curve of intersection 
 of two double-curved surfaces of revolution whose axes intersect. 
 
 Analysis. Let the two surfaces be those of an ellipsoid and a 
 sphere given by their projections in Fig. 139. Intersect the sur- 
 faces of the elHpsoid and sphere by a series of auxiliary spheres 
 whose centers are at some point on the major axis of the ellip- 
 soid. The section line between such an auxiliary sphere and the 
 original sphere, as well as between the auxihary sphere and ellip- 
 soid, are both circles. The intersection of two such circular sec- 
 tions detei;mines points on the Hne of intersection of the surfaces. 
 
 Construction. Let the axes of both the elhpsoid and sphere 
 intersect at O and be parallel to V. Let c'-e'-d'-f represent 
 the vertical projection of an auxiliary sphere having its center 
 at O; d-d! is the vertical projection of the circle in which the 
 auxiliary sphere meets the given sphere, and e'-j' is the vertical 
 projection in which the auxihary sphere meets the ellipsoid. 
 These two projections intersect at Q and 'N . Draw the hori- 
 zontal projection e-j of the circle also the ruled projector from 
 5' and n' to intersect the horizontal projection of this circle at 
 two points q and w, hence these are the horizontal projections of 
 two points on the required ciurve of intersection. The horizontal 
 projections of the auxiliary sphere and the circular section cut 
 
152 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. 139. — To deteimine the Intersection of an Ellipsoid and a Sphere. 
 
 from the sphere 0-A are not necessary in the construction and 
 hence are omitted. By the above method any number of points 
 such as k, k' and m, m' can be determined and the projections 
 of the curve of intersection can then be drawn. 
 
CHAPTER rx 
 
 ISOMETRIC PROJECTION 
 
 152. Introductory. Isometric drawings can frequently be 
 used to give those not skilled in the interpretation of mechanical 
 drawings a clear idea of the construction and appearance of a 
 structure or naachine. Also, it frequently happens that pecu- 
 liarities of construction can be more clearly presented by this 
 method than by the use of the ordinary plan, elevation, and 
 sections. Isometric drawings resemble perspective drawings in 
 that they show the three principal dimensions — length, width, 
 and height — of an object in a single view. A perspective 
 drawing, however, cannot be completely dimensioned, whereas 
 an isometric drawing is 
 almost as easily dimen- ■ 
 sioned as is a mechani- 
 cal drawing. 
 
 The method by which 
 isometric ^drawings are 
 made is termed isometric 
 projection, and this con- 
 sists in projecting an ob- 
 ject upon a vertical plane, 
 to which its three princi 
 pal dimension lines 
 equally inclined. For ex- 
 ample, let 0-X, 0-Y, and 0-Z, in Fig. 140, represent the three 
 edges of a right trihedral angle in space. Let 0-Y be in a ver- 
 tical plane that is in a plane perpendicular to V, and let each of 
 the other two edges make the same angle with V. 
 
 When occupying this position, if the three edges are projected 
 upon V, their projections o'-x'. o'-y, and o'-z' will radiate from 
 0' and form three equal angles of 1 20 degrees each. 
 
 The projection o'-y' will be vertical, while o'-x' and o'-z' will 
 
 IS3 
 
 „j.„ Fig. 140. — Fundamental Principle of Isometric Pro- 
 jection, Illustrated. 
 
154 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 each make 30 degrees, in opposite directions, with a horizontal 
 line through 0' . 
 
 The lines in space, 0-X, 0-Y, and 0-Z, are termed coordi- 
 nate axes, and the planes determined by the coordinate axes 
 are termed coordinate planes. 
 
 The isometric origin is 0', and o'-x', o'-y', and o'-z' are the 
 isometric axes. 
 
 If 0-A, 0-B, and 0-C coincide with 0-Y, 0-X, and 0-Z re- 
 spectively, and represent the height, width, and length of a cube 
 itself, the three faces boimded by these lines can be shown in 
 projection on V, and the projections will be bounded by o'-a', 
 o'-V , and o'-c' . 
 
 Every point located on one of the faces of the cube would 
 have its projection in the angle formed by the projections of 
 the edges which terminate or boimd that face. Thus, the poiat 
 D is on the face A-O-B-E, and its isometric projection is in 
 the angle a'-o'-b'. If a second point E on this same plane had 
 been in such a position that the line D-E joining the points 
 were parallel to 0-X, then the isometric projection d'-e' of the 
 line. will be parallel to the isometric axes o'-x'. Furthermore, 
 if two lines in space, D-E and F-G, are parallel, their isometric 
 projections will be parallel. It should here be observed that 
 any solid' whatever could be placed in the right trihedral angle 
 so that its three principal lines of dimensions will either coin- 
 cide or be parallel to as many of the three axes 0-Y, 0-X, 
 and 0-Z as possible, and when in this position it can be iso- 
 metrically projected on V. Theoretically, it is necessary to use 
 a special scale, called an isometric scale, to measure off distances 
 on the plane V. This is because the projection of a line when 
 " inclined" is always shorter than the line itself. In the case of 
 isometric projections, the three axes 0-X, 0-Y, and 0-Z are each 
 inclined to V at an angle of 35° 16' and hence an inch length on 
 one of these axes will be represented by the product resulting 
 from multiplying i inch by the natural cosine of this angle, which 
 is 0.816. That is, i inch actual measurement on the object 
 projects only 0.816 of an inch, and a scale constructed upon this 
 basis could be used to measure all distances on V parallel to one 
 of the isometric axes. The isometric scale is mentioned h'ere as 
 
ISOMETRIC PROJECTION 
 
 155 
 
 Fig. 141. —Isometric Projection Illustrated. 
 
 a matter of theoretical interest, since such scales are now seldom 
 used in practice. 
 
 Isometric drawings are usually made with the same scales 
 as are used in the making of mechanical drawings, in order to 
 facilitate scaling and dimensioning. 
 
 153. Fundamental Principles. Based on the discussion in 
 §152, page 153, the cube shown in Fig. 141 has been drawn. 
 By reference to this figure the 
 
 following five fundamental prin- 
 ciples will be understood. 
 
 (i) There are three isometric 
 axes which radiate from a com- 
 mon point termed the origin. 
 One axis is drawn vertical, one 
 is drawn at 30 degrees toward 
 the right, and one at 30 degrees 
 toward the left. 
 
 (2) The isometric axes repre- 
 sent lines mutually perpendicular 
 to each other in space. ■ They correspond to the three dimen- 
 sions, — height, length, and width, — and on the drawing meas- 
 urements can only be laid off on or parallel to one of these axes. 
 
 (3) Lines vertical on the object are vertical on the drawing. 
 
 (4) Lines parallel on the object are parallel on the drawing. 
 
 (5) Right angles on the object are usually either 60 degrees 
 or 1 2Q degrees on the drawing. 
 
 154. PROBLEM 80. To determine the isometric projection of 
 a point which lies in one coordinate plane and at a given distance 
 from the other two coordinate planes. 
 
 Analysis. Since the point lies in one of the coordinate planes, 
 the isometric projection of the point must be at the intersection 
 of isometric ruled projectors which lie in this same plane. 
 
 From the origin measiu-e off distances along the proper co- 
 ordinate axes to locate the foot of the isometric ruled projectors. 
 
 Construction. See Fig. 142. Let the point lie in the plane 
 X-O-Z, it being | inch from the plane Y-O-Z and | inch from 
 the plane Y-O-X, Let o'-a;', o'-y', and o'-g' be the projections 
 
iS6 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 of the isometric axes. Measure off o'-a^' equal to | inch (the 
 distance the point is from the coordinate plane X-O-Y), and 
 through aj draw the ruled projector 
 a J -a' parallel to o'-x'. 
 
 Make o'-aj equal to f inch (the dis- 
 tance the point is from the coordinate 
 plane Z-O-Y), and through a J draw the 
 ruled projector aj-a' parallel to o'-z'. 
 
 The lines aj-a' and aj-a' are the 
 isometric ruled projectors for the point 
 A in space, and a' is the required iso- 
 metric Jirojection of the point. 
 
 Fig. 142. — ToDetennine the Iso- 
 metric Projection of a Point on 
 a Coordinate Plape. 
 
 155. PROBLEM 81. To determine the isometric projection 
 of any point in space when its position relative to the coordinate 
 planes is known. 
 
 Construction. See Fig. 143. Let the 
 point A be located as in Problem 80, 
 page 155, with reference to the planes 
 X-O-Y and Y-O-Z but let it be I inch 
 below the plane X-O-Z. Its position rel- 
 ative to the plane X-O-Z is determined 
 as in Problem 80. From 0' measure o'-ay 
 off equal to § inch along o'-y'. At ay 
 draw Uy'-j parallel to o'-z', and lay off 
 
 Fig. 143. — Isometric Projection 
 of Any Point in Space. 
 
 Fig. 144. — To Determine the Isometric 
 Projection of Any Straight Line. 
 
 the distance ay'-j equal to finch. 
 Similarly, from a/ lay off o/-r 
 equal to f inch and parallel to o'-x'. 
 
 This method of constructing an 
 " offset " parallelogram i-ay-j-a' 
 illustrates one way of determining 
 the projection a'. 
 
 From 0' proper distances could 
 be measured off along o'-z' and 
 o'-x' to locate 2 from which a' 
 could be located. 
 
 156. PROBLEM 82. To determine the isometric projection 
 of any straight line when the position of any two of its points 
 is known with reference to the coordinate planes. 
 
ISOMETRIC PROJECTION 
 
 157 
 
 Analysis and Construction. See Fig. 144. Determine the pro- 
 jections a' and V of the two given points A and B, as in Prob- 
 lem 81, page 156, and the line connecting the projections of the 
 points is the required projection of the hne. 
 
 157. PROBLEM 83. To draw the isometric projection of 
 a square figure of given dimensions which lies in one coordi- 
 nate plane and at known distances from the other two. 
 
 Analysis and Construction. Let the figure be i inch square. 
 Assume that it lies in X-O-Z, with one side j inch from and par- 
 allel to 0-Z, and another 
 side I inch from and 
 parallel to 0-X, that is 
 the sides are respectively 
 parallel to these two axes. 
 
 Determine the projec- 
 tion d' (see Fig. 145) of 
 one corner, which is i inch 
 from o'-x' and J inch from 
 o'-z' (see Problem 80). 
 Draw d'-c' parallel to o'-z' 
 
 and measure off along this Rg. 14s.— To Draw the isometric Projection of a 
 . T . , , • Square Which Lies in a Definite Position in One of 
 
 line I mch to determme the coordinate Planes. 
 
 c'. Draw d'-a' parallel 
 
 to o'-x' and measure off along this hne i inch to determine a'. 
 Draw c'-b' parallel to o'-x', and a'-b' parallel to o'-z'. This 
 locates b', and a'-b'-c'-d' is the projection required. 
 
 158. PROBLEM 84. To draw the isometric projection of a 
 cube. 
 
 Analysis. Consider a vertex of the cube as coinciding with the 
 origin of coordinates, 0, and three adjacent edges coinciding with 
 the coordinate axes. In this position these three edges of the 
 cube wiU be isometrically projected on the isometric axes, and the 
 remaining edges wiU be projected in lines parallel to these axes. 
 
 Construction. Lay o&.o'-a' and o'-c' (see Fig. 146), each 
 equal to an edge of the cube. Draw a'-b' parallel to o'-z', and 
 b'-c' parallel to o'-x'. The intersection b' of these two lines 
 
iS8 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. 146,— Isometric Projection of a Cube. 
 
 determines another vertex of the cube. Lay off o'-g' equal to 
 an edge of the cube and draw ^-i' and g'-/' parallel to o'-x' 
 and o'-z' respectively. From a' and c' draw lines parallel to 
 
 o'-y'. Then a'-b'-c'-o' repre- 
 sents the top, d'-a'-o'-g' the 
 left-hand front face, and g'-o'- 
 d-j' the right-hand front face 
 of the cube. 
 
 159. PROBLEM 85. To 
 draw the isometric projection 
 of an hexagonal pyramid. 
 
 Analysis and Construction. 
 Let the plan of the p3n:a- 
 mid be shown in Fig. 147 II. 
 Assume the pyramid to be 
 circimiscribed by a rectangular prism, the base of which circum- 
 scribes the base of the pyramid, and the altitude of which 
 equals the altitude of the pyramid. Draw the isometric pro- 
 jection of the right 
 prism as in Problem 
 84. Draw the di- 
 agonals I'-j' and 
 2'-4' to determine 
 the center g' of the 
 top face of the 
 prism. The point 
 Q is the apex of 
 the pyramid. De- 
 termine the vertices 
 a'-V-c'-d'-e'-i' of 
 the base as in Prob- 
 lem 84. 
 
 To draw the projection of the base of the pyramid, make 
 8'-a' in Fig. 147 I equal to 8-A in Fig. 147 II and a'-f equal 
 to A-F. Make 8'-e' equal to 8-E and -connect/ and e'. The 
 projections of other vertices of the base can be similarly de- 
 termined. 
 
 Fig. 147. — Isometric Projection of a Pyramid. 
 
ISOMETRIC PROJECTION 
 
 159 
 
 Connecting each vertex? of the base with the apex (( determines 
 the projections of the edges of the pyramid. The edges that 
 are invisible can be de- 
 termined by inspection. 
 
 160. PROBLEM 86. 
 
 To make an isometric 
 
 drawing of a circle. 
 
 Analysis and Con- 
 struction. Circumscribe 
 
 a square about the circle 
 
 and locate any number 
 
 of points as shown in 
 
 Fig. 148 II. The distances x'-f , f-6', 6'-$', etc., in Fig. 148 1, 
 
 are respectively equal to 
 1-7, 7-6, 6-5, etc., in Fig. 
 148 II, etc. 
 
 161. PROBLEM 87, To 
 make an isometric draw- 
 ing of any plane curve. 
 
 Fig. 149 illustrates clearly 
 
 Isometric Drawing of a Ciicle. 
 
 m 
 
 n 
 
 Fig. 149. —Isometric Drawing of a Plane Curve. 
 
 how a plane curved figure can be 
 shown in isometric. 
 
 162. PROBLEM 88. To make 
 an isometric drawing of a mor- 
 tise and tenon. 
 
 Fig. 150 clearly shows how this 
 can be done. 
 
 163. PROBLEM 89, 
 To make an isometric 
 drawing of a piping 
 system. 
 
 Fig. 151 clearly shows 
 
 how this can be done. pig. iso.— Isometnc Drawing of a Mortise and Tenon. 
 
i6o 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 164. Cavalier or Cabinet Projection. Sketches made in cav- 
 alier or cabinet projection are sometimes preferred to those made 
 
 in isometric projection (see 
 §152, page 153). This sys- 
 tem is similar to the iso- 
 metric in that it makes use 
 of three axes along or parallel 
 to which measurements- cor- 
 responding to the three space 
 dimensions — length, width, 
 and height — can be made. 
 The axes, however, do not 
 occupy the same position as 
 in isometric projection, one 
 axis being vertical, one hori- 
 zontal, and one 45 degrees 
 to the horizontal (see Fig. 
 152). Measurements are 
 made trtie size on and parallel to the vertical and horizontal 
 axes, and half-size on and parallel to the 45-degree axis. This 
 
 Fig. JSi. — Isometric Drawing of a Piping Layout. 
 
 ioL- 
 
 Fig. IS2. — Cavalier or Cabinet Projection of a Cube. 
 
 foreshortens the depth of the drawing and gives the sketch a 
 perspective effect. The front face and all others parallel to it 
 are drawn in their true shape and size as in a mechanical 
 drawing. 
 
 165. Pseudoperspective. This method is in principle a modi- 
 fication of cabinet projection (see § 164). Both vertical and 
 
ISOMETRIC PROJECTION 
 
 l6l 
 
 horizontal axes are used the same as in cabinet projection, 
 but the third axis is drawn at a greater angle than 45 degrees. 
 In this system it is necessary that measurements made in the 
 plane of the paper or parallel to the paper be made in their 
 
 Fig. 153. — Pseudoperspective. 
 
 true size, while those representing lines perpendicular to the plane 
 of the paper are reduced to some assumed ratio of their actual 
 length the exact ratio in each case being that which gives the 
 best results. See Fig. 153. This method gives the drawing a 
 sense of depth similar to a perspective and can be used to 
 good advantage in making illustrations and sketches. 
 
CHAPTER X 
 
 SET OF DRAWING EXERCISES IN DESCRIPTIVE 
 GEOMETRY 
 
 166, Introductory. The following exercises have been 
 arranged to follow the text as indicated. When the student finds 
 it necessary to refer to the text matter in the solution of an 
 exercise time will be conserved by first referring to the index. 
 The exercises are designed to be solved on paper measuring 
 i2"xi8" and within border lines measuring ii"xi6". See 
 Fig. 154- 
 
 -18" 
 
 
 
 
 5 + 
 
 . , irr" ' 
 
 BORDER LLN£~*- 
 
 4^ 
 
 
 *• 
 
 
 
 
 STAMP TITLE-FORM 
 
 HERE 
 
 -^ 
 
 
 \ 
 
 f 
 
 
 
 
 
 C^ 
 
 EDGE OF PAPER ' 
 
 Fig. 154. —Layout of "Plate" for the Drawing-board Exercises. 
 
 The title form is shown in Fig. 155 and is usually provided in 
 the form of a rubber stamp. The title form is located in the 
 lower right Jiand corner of the plate as indicated in Fig. 154. 
 
 The student should proceed with the work systematically; 
 (a) tack down the paper; (b) draw in the border line; (c) stamp 
 in the title form; (d) write in with ink name, date of beginning, 
 
 162 
 
SET OF DRAWING EXERCISES 
 
 163 
 
 section of the class to which the student belongs, and the number 
 of the plate; (e) draw in the ground lines; (f) work the exer- 
 cises; (g) ink in date of finishing plate and total time consumed. 
 
 ^ 
 
 I 
 
 DESCRIPTIVE GEOMETRY 
 M.D. PERT., SIBLEY COLLEGE 
 
 NAME 
 
 BEGUN- FINISHED. : 
 
 Total Hours 
 
 Section No. 
 
 Fig. 155.— Title Foim to be used in this Work. 
 
 Letters and Numerals. Unless otherwise instructed the free 
 hand Slant Gothic Upper Case Alphabet \" high will be used for 
 general lettering including traces of planes, abbreviations for 
 true lengths, true angles, etc. Gothic numerals the same size 
 will be used in numbering the exercises. The numeral is located 
 at a convenient place above each exercise. The projections of 
 .points will be lettered with the lower case slant Gothic letters 
 about -§2" high. The plates will be numbered with \" Roman 
 numerals. 
 
 A 4 H drawing pencil sharpened to a cone point will be found 
 best for lettering. 
 
 The Mechanical Penciled Line. All mechanical lines are to 
 be "clear cut" and of the correct construction and weight. A 6 H 
 pencil is to be used and must be kept well sharpened to a cone 
 point and not to a chisel edge. 
 
 Locations of Ground Lines on a plate are stated thus: G-L, 
 2i and 8 inches, which indicates that there are two ground Unes 
 on the plate in question. The first ground line is 2 i inches from 
 and parallel to the top border lifte, and the second is 8 inches 
 from, and parallel to the top'border line. 
 
l64 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Character of the ground line i^ an unbroken line of light weight 
 with the capital G priated at one extremity and the capital L 
 at the other, thus: 
 
 Ruled Projectors. The location of the ruled projector con- 
 taining the projections of a point is always given along G-L and 
 from the left hand border line; thus: [yi] A indicates that the 
 ruled projector on the drawing, for the point A in space, is 7 J" 
 from the left hand borderline. Wherever a dimension is given 
 in the bracket [ ], it is to be understood that this is a measure- 
 ment along the ground line and from the left-hand border line. 
 All dimensions unless otherwise noted are in inches. 
 
 A ruled projector is a broken line of light weight and is made tip 
 of a series of dashes Ye inch long separated by spaces -^ inch 
 
 long, thus : The ruled projector should be 
 
 drawn with a sharp cone pointed 6 H pencil and made lighter 
 in weight than the G-L. 
 
 The Location of Points on the plate may be given in any one 
 of several ways. First, the location of the ruled projector may 
 be given and following this the distance the point is from H and 
 V. Or the point may be located thus: A (i + i J + 2) which 
 indicates that the ruled projector for the point .4 is i" from the 
 left hand border line; the vertical projection is ij" above G-L 
 and the horizontal projection is 2" below G-L. Note that the 
 letter outside the bracket indicates the point in space; the first 
 dimension within the bracket locates the ruled projector; the 
 second dimension locates the vertical projection and is above 
 G-L if preceded by a + sign and below G-L if preceded by the 
 — sign; the third dimension locates the horizontal projection, 
 below G-L if preceded by the plus sign and above G-L if preceded 
 by the minus sign. 
 
 An extension of this nethod locates a line by two points; 
 thus, A~B (6 i + 1 1 + 2) (8 + i + |) ; the dimensions within the 
 first set of parenthesis refer to the point A while those in the 
 second refer to the point B. 
 
 The projections of points should be shown by a fine lead pencil 
 
SET OF DRAWING EXERCISES 165 
 
 dot on the ruled projector, and indicated by its proper letter 
 made of the lower case Gothic alphabet about •^" high. 
 
 167. PLATE I. Exercises i to 25. Text §1 to §39 inclusive. 
 G-L 2h" and 7 J" 
 
 NOTE. All exercises are to be numbered with Gothic numer- 
 als i" in height, and placed at a convenient distance above 
 the exercise. 
 
 1. — Indicate the point [|] A ij" above H and ij" in 
 front of V. 
 
 2. — Move the point ^4 to the right J" and also i" 
 nearer H. 
 
 NOTE. When a point in space changes its position, the pro- 
 jections of the path of motion of the point must be indicated. 
 Seep. 18, §26 (f)and §26 (h). 
 
 3. — Move ^1 to the right \" and i" nearer V. 
 
 4. — Move Ai to the right \" and into the ground line. 
 
 5. — Draw the projections of the line joining A and^s. 
 
 NOTE. The student must be able to teU the instructor how 
 the Une A-Az lies with reference to H and V. This process is 
 termed visualizing the line A-A^. As the instructor will be con- 
 stantly testing th^ student's knowledge of the work in this manner, 
 no exercise should be passed over until the student has visualized 
 it. 
 
 6. — Through [2 i] draw Gi-Li below G-L at 45 left, and 
 
 2 1" long. Determine the vertical projections of A ; 
 Ai] A2 and A3 on Vi. 
 
 NOTE. The above notation indicates that d-ii is drawn from 
 a point on G-L which is 2i" from the left border line; Gi-Li is 
 drawn toward this left border line; it is below G-L and makes 
 45 degrees with G-L. For the sake of simplicity, a new vertical 
 plane is always folded away from the main problem. To illus- 
 trate; the new vertical projections in this problem are drawn to 
 the right of Gi-Li. 
 
l66 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Notation. To avoid confusion, Roman numerals are used to 
 designate new vertical planes, and their corresponding ground 
 ' lines, when the data of the problem has already used subscripts. 
 Thus the vertical projection of 4 2 on Vi becomes a\\. 
 
 7.' — Join the vertical projections of A and A^. Notice 
 the difference in length of the vertical projection of 
 A-Az on the original V and on Vi. Note also, the 
 angles that these two projections make with their 
 respective grounds Unes. If the drawing has been 
 accurately constructed the vertical projection on 
 Vi is equal in length to the line A-A% in space. 
 See Theoreni III p. 28. Also the angle this pro- 
 jection makes with Gi-Li is the true angle that the 
 line in space A-A3 makes with its horizontal pro- 
 jection. See §37, p. 29. 
 
 Notation. When the projection of a line is equal in length to 
 the line in space, this fact is noted by placing TL on the pro- 
 jection. The letters TL are i" in height. Likewise the true 
 angle is marked TA . 
 
 Question. Can the projection of a line ever be longer than the 
 line in space? " Can the angle a projection makes with its ground 
 line ever be greater than the angle the line in space makes with 
 the opposite projection? 
 
 8. — [3] A is f" above H and i" in front of V; B is i" 
 to the right of A. Indicate the line A-B parallel 
 to V and to H. 
 
 9. — [4 1] C is i" in front. of V and %" above H. Indicate 
 the line C-D perpendicular to V,. in the first angle, 
 and i" long. 
 
 10. — [5] E is f" above H and f" in front of V. Indi- 
 
 cate the line E-F perpendicular to H in the first 
 angle and i" long. Mark a point A on the line 
 E-F and f " from E. 
 
 11. — Indicate the point A (5 J -f- 1 -I- |) and the point 
 
 B {sh+ f + 1). Indicate the line joining A and B. 
 
SET OF DRAWING EXERCISES 167 
 
 Take a profile plane (that is a new vertical plane 
 that is perpendicular to V) through [6] with Gi-Li 
 below G-L and determine the new vertical pro- 
 jection of A-B on the profile plane. Locate the 
 point C on A-B and i" from B and give the pro- 
 jections of C on all planes. 
 
 12. — Indicate the line C-D (7 J -M -1- 1) (8i + J -|- j^). 
 
 Draw Gi-Li below [9] 45 left, 2^" long. Deter- 
 mine the projections of C-D on Vi and locate the 
 point A, i" from where the line C-D continued 
 pierces Vi. How far is the point A and also the 
 piercing point from V? 
 
 13. — Indicate the line E-F (9I -I- i + |) (io§ + j6 + i) 
 
 and find its projections on a profile plane through 
 [11] and extending ij" below G-L. Find the pro- 
 jection on this plane of a point C which lies on E-F 
 and is f " from F. ' 
 
 14. — The projections of the line A-B, C-D and E-F 
 
 are parallel. A-B {12^+ ii+ i) (14 -f- i-i- ij); 
 C-Dli2i + 2 + i) (14 + ? + ?);E-F (i2i + I J +1^) 
 (14 -h ? + ?) Indicate the lines by their pro- 
 jections. Take a new Gi-Li 3!" long, below, 
 [15 a] 45 left. Determine the vertical projections 
 
 on Vi. 
 
 / 
 
 NOTE. When + ? or — ? appears in the data of a problem it 
 means that the dimension thus indicated is determined by the 
 conditions of the problem. 
 
 Question. How are the lines A-B, C-D and E-F in space, 
 located relative to each other? 
 
 15. — Theline/-iir (14 + 1^ + 0) (i2i+o 4- ?) intersects 
 C-D at the point P to be determined. Indicate 
 the line J-K by its projections. Visualize these 
 Unes. 
 
 Question. If fines in space are parallel must their projections 
 always be parallel lines? 
 
i68 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Indicate the following points. 
 
 16. — [\]A I i back of V and i i" below H. 
 
 17. — £(i + ii-i|). 
 
 18. — C [i|] in the fourth angle is li" from H and if" 
 
 from V. 
 
 19. — Z)(2 + Ii+ i). 
 
 20. — Join the point A of Ex. 16 to the point D of Ex. 19 
 
 and indicate the angles that the "different portions 
 of the line A-D hes in. 
 
 21. — [2i] j4 is in the third angle ij" from H and i|" 
 
 from V. [4] Z> is in the first angle li" from H and 
 h" from V. 
 Indicate the line A-D and find its vertical projection on 
 a new profile plane through [4!], d-Zi extending 
 1 1" above and 1 1" below G-L. 
 
 NOTE. Swing d\ to the left; that is it falls between a and d'. 
 
 22. — Indicate the lines. 
 
 A-B (6j+ii+2) (61+ i+ i). 
 C-D (6^+1+ i) (6i+i + 2). 
 E-F (6§ + IT6 + 1) (64 + T6 + 2). 
 /-Z(6i+ i+ii) (64 + if+ii). 
 
 NOTE. Observe how difficult it is to visualize the relative 
 position of these lines. 
 
 23. — Determine the projections of the lines of Ex. 22 
 
 on a profile plane at [7]. Gi-Li below G-L and 2|" 
 long. 
 
 NOTE. Observe the ease with which these lines can now be 
 visualized. This illustrates the practical value of new planes of 
 projection. 
 
 24. — Transfer the points of Ex. 23 to [9J] and the 
 
 profile plane to [9 j]. Determine the projections of 
 
 ' the points in the profile plane as before. Using 
 
SET OF DRAWING EXERCISES 169 
 
 the point of intersection of the projections, as a 
 center of rotation, swing the projections counter 
 clockwise through 45 degrees and determine the 
 new horizontal and vertical projections. 
 
 Question. How has this operation affected the lines in space? 
 
 25. — The points 4 (i2|— li— li), B (12J— f— j), 
 C (12 J — 2 — ?) are the vertices of a triangular 
 figure. A-C is I" long. By the use of a profile 
 plane determine the shape of the figure and its 
 projection in both planes. [13] Gi-ii above G-L 
 and 2l" long. 
 
 168. PLATE II. Exercises 26 to 32. Text to §44, page 32. 
 
 G-Z3"and8" 
 
 26. — Given the line A-B (i + 2 + 2 j) (2 J + i + j). 
 
 The point [i ^] E is on the line A-B, and is also the 
 point of intersection of the lines A-B and C-D 
 
 (i+4+f) (2j+?+?). 
 
 (I). — Indicate the line F-J, parallel to, and |" from 
 
 H, and intersecting A-B and C-D. 
 (II). — Is the Une F-J parallel to V? 
 (III). — Visuahze the figure E-F-J. 
 
 27. — Given the Une A-B (3 + i^ + i|) (4* + I + o). 
 
 Extend A-B to [6] C. 
 (I). — Indicate on the drawing what angles the three 
 
 parts of the line A-C lie in. 
 (II). — Indicate the hne [3] Z)— [4J] F parallel to G-L, 
 
 intersecting A-B and h" from V. 
 (III). — How far is D-F from H? 
 
 (IV). — Indicate the Hne [4I] L— [sf] M parallel to V 
 and H, iatersecting the Une A-C and \" from H. 
 
 (V). — Indicate the Une N-P from N (6 + + 0) to 
 bisect that part of A-C which Ues in the. third 
 angle. 
 
I70 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 28. — The projections of the lines A-B aiid C-D are 
 
 perpendicular to G-L at [6|]. A is 2^" above H 
 
 and i" in front of V; 5 is J" above H and if" in 
 
 front of V; C is i" above H and i" in front of V. 
 (I). — By the use of a profile plane at [7] determine the 
 
 point of intersection E of A-B and C-D; E is i" 
 
 and D is 2" from V. 
 (n). — Bisect the line C-D with the point F and find 
 
 /and/'. 
 (m). — Is a profile plane necessary for the solution 
 
 of both (I) and (II)? State reasons. 
 
 29. — Given the line .4-5 (ioH-,2i + ij) (i2| + \ +f) 
 
 and the point E (13J + 2\+ \). 
 (I). — Determine the vertical and horizontal piercing 
 
 points of the line A-B. 
 (II). — Check the location of the piercing points by 
 
 means of a new vertical plane through [15]. 
 (III). — How long is the line A-Bl 
 (IV). — From the point E draw the line E-F to make 
 
 60 degrees with A-B and to intersect it in the first 
 
 angle. 
 (V). — What is the true distance E-K from the point 
 
 E to the line A-Bl 
 (VI). — Where would the line E-F intersect • ^-5 if 
 
 drawn on the opposite side of E-K7 
 
 30. — Given the line A-B (i + i J + 2i) (4 — i— 2i) and 
 - the point £ (2+1+ 2\), 
 
 (I). — Determine the piercing points of the line. 
 
 (II). — With a new V through [li] determine the true 
 length of the part of the line that lies in the second 
 angle, and also the angle it makes with H. 
 
 (ni). — With a new V through [2f] determine the true 
 length of the part of the line that lies in the third 
 angle, and the true angle that this line makes with 
 H. 
 
SET OF DRAWING EXERCISES 171 
 
 Question. How do the two trueapgles (II) and (III) compare 
 in size and why? 
 
 (IV) . — Through the point iSdraw a line to i? (4 ± ? ± ?) 
 so that E-F is parallel to and passes through the 
 same angles as A-B. Explain and visualize Ex. 30. 
 
 31. — Given the line A-B in the third angle. [5] A is 
 
 1 1" from H and 1 5" from V ; a-b and a'-b' each make 
 30 degrees with G-L. [7] B is approximately 
 i" from V and js" approximately from H. 
 
 (I). — Indicate a line C-D in the first angle parallel 
 to A-B. [5] C is I" from H and i" from V; [7] 
 D to be determined. 
 
 (II). — Indicate the line E-F in the second angle 
 parallel to A-B; [5] E is 2" fromV and f'fromH; 
 [7] F to be determined. 
 
 (III). — Indicate the Hne X-Y in the fourth angle 
 parallel to A-B. [5] X is i^" from H and f" from 
 V; [7] F to be determined. 
 
 (IV). — Join the points E and B; also, E and Y and 
 determine what planes are pierced, and what angles 
 passed through by the lines E-B and E-Y. Visual- 
 ize this problem. 
 
 (V). — Check this problem by the use of a profile plane 
 P through [91]. 
 
 NOTE. Fold F to the right of Gp-Lp for the first and second 
 angles; this throws the third and fourth angles to the left of 
 
 Crp l^p • 
 
 Question. If lines in space are parallel must their projections 
 be parallel on all planes? Give reasons. 
 
 32. — Indicate theline^-5 (11^—2^+ |) (14J— ? + i) 
 
 such that the line makes 30 degrees with H. 
 (I). — Determine the piercing point C of A-B. 
 (II). — Locate a point D on A-B that is ij" from H 
 
 and a second point E that is f" from H;' what is 
 
 the distance from D to E? 
 
172 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (III). — Indicate the point J 14] P in the second angle 
 I J" from V and ij" from H; indicate a line F-K 
 to intersect A-B at K, and 2" from H. 
 
 (IV.) — Through what angles does F-K pass? 
 
 169. PLATE m. Exercises 33 to 42. Text to §5 j Page 41. 
 
 G-L 2 i" axidyi". 
 
 Notation. In rotation problems reserve the letter C to indi- 
 cate the center of rotation. Also, always show both projections 
 of the radius of rotation; across the horizontal projection of the 
 radius place the capital R and across the vertical projection 
 place R'. 
 
 33. — Indicate the point A (j + i + |); the line D-E 
 
 (ii + 5^+ J) (li + l^ + ii). 
 
 (I). — Revolve the point A clockwise through 90 
 degrees and show its new position. 
 
 (II). — Show the projections of the radius of rotation 
 for each 30 degrees passed through in (I). 
 
 (III). — Revolve A from its original position until 
 a^ is at [i\; mark the angle through which the 
 radius of rotation has swept 6 and indicate the 
 angle that A lies in. 
 
 (IV). — Ditto when A is revolved from its second 
 position in (I) to 05' at [21-6"]. 
 
 34. — Indicate the line A-B (3 1 + 2 + i i) (3 1 + o +1 i) 
 
 and£(3f -|-;i|+ ij). 
 
 (I). — Indicate the line E-D towards H, to the left of, 
 and making 30 degrees with A-B. D is j" from H. 
 
 (n). — Revolve E-D clockwise about A-B as an axis, 
 through 150 degrees. 
 
 (III). — From the last position continue the revolution 
 through 210 degrees and show the projections of the 
 the line E-D for each 30 degrees passed through. 
 
SET OF DRAWING EXERCISES 173 
 
 Question. What solid is generated by E-D as it passes 
 through 360 degrees? Indicate on the drawing the positions 
 when E-D is shown in true length. 
 
 Notation. A line may be given by one point and the direction 
 of the line to the other point. Thus A-B (2 + i + i) (45 up 
 right [5] ) indicates that from the point A, the Hne is drawn at 
 45 degrees upward, toward the right to meet the ruled projector 
 through [5]. The projection of a line may also be indicated in 
 this manner. 
 
 35. — Indicate the point A (sl + if + o) and the line 
 
 D-E (41 + o + i) (30 up right to G-L). 
 
 (I). — Revolve the point A about the line D-E as an 
 axis and into H in the first angle. Construction 
 at [6f] right. This indicates that the right triangle 
 constructed to find the true radius of rotation has 
 its altitude from the G-L up at [6f] and its hypo- 
 thenuse to the right of this altitude. 
 
 (II). — Revolve A from its position in (I) through 180 
 degrees. 
 
 (III). — Show where A pierces V and H when revolved 
 through 360 degrees. 
 
 (IV). — Revolve the radius of rotation A-C into V to 
 find the true radius of rotation. 
 
 36. — Indicate the point A (8| + i| + |) and the line 
 
 D-E (71 + o + If) (30 up right to G-L). 
 (I). — Revolve the point A about D-E and \ from H 
 
 in the first angle. Construction [loi] right. 
 (II). — On this same construction determine where A 
 
 pierces V when revolved into the second angle. 
 (in). — Ditto, determine the projections of A when the 
 
 radius of rotation makes 30 and 45 degrees with 
 
 H. 
 (rV). — Revolve A into H in the second angle. 
 
 37. — Indicate the point A (15+2 + ii) and the line 
 
 Z>-£(i3f+i+ii) (i5i+ i+i). 
 
174 ELEMENTS OF DESCRIPTIVE GOEMETRY 
 
 (I). — Revolve the point A about D-E until the radius 
 of rotation A-C is parallel to H in the first angle. 
 Construction [12 f] left. 
 
 (II). — On same construction revolve the point A 
 from its original position until it touches V in the 
 first angle, and indicate the true angle through 
 which the radius of rotation has revolved. 
 
 (III). — When the point A is revolved into V in the 
 first angle indicate the angle that the radius of 
 rotation A-C makes with V and with H. 
 
 38. — Indicate the point A (if + .1 J + f) and the line 
 
 D-E (i+i + o) (i|+ j + if). 
 
 (I). — Revolve A about D-E through an angle of 30 
 
 degrees and away from V. Take Gi-Li through 
 
 [2]. 
 (II). — Now revolve Ai back until it touches V and 
 
 determine the angle through which the radius of 
 
 rotation A-C has revolved. 
 (III). — Revolve A until it is i" from H in the first 
 
 angle. 
 (IV). — From the last position revolve A through 180 
 
 degrees showing its position for each 30 degrees 
 
 passed through. 
 (V). — Show the path of the vertical projection of A 
 
 for the above 180 degrees revolution; draw the 
 
 path in the first angle as a solid hne the part in the 
 
 second angle I" dash and ys" space construction. 
 (VI). — Continue the revolution of A to complete 360 
 
 degrees and draw the path as in (V). 
 
 39. — Indicate the'line A-B (3 J + if + ij) a'-b' makes 
 
 45 down to ruled projector [5]; a-b makes 30 up. 
 Indicate also D-E (4J + t + i§) (4i + 2 + i j). 
 (I). — Revolve A-B about D-E clockwise to find its 
 true length. 
 
SET OF DRAWING EXERCISES 
 
 175 
 
 (II). — Continue the revolution until A-B has com- 
 pleted 90 degrees. 
 
 (III). — Continue the revolution until A-B lies in a 
 profile plane. 
 
 (IV). — How far is this profile plane from the Hne D-E? 
 
 40. — Indicate the two lines intersecting on ruled pro- 
 
 jector [7 J] A-B (6 + I J + I) (8§ + i| + 25^) and 
 
 the line D-E (6 1+2^+1 f) (8i-f-f approx. 
 
 + If). 
 (I). — Revolve the line D-E about A-B until it is 
 
 parallel to H; move D toward V. 
 (11). — Determine the angle d through which D has 
 
 revolved. Take auxihary plane through point of 
 
 intersection of the lines. 
 
 41. — Indicate the hne A-B (9I + i + if) a'-b[ makes 
 
 30 up to ruled projector [11] and a-b makes 45 up. 
 (I). — Without moving B determine the true length 
 
 of A-B in V. 
 (II). — Ditto in H as a check on (I). 
 (ni). — What is the true angle that A-B makes with 
 
 V and with H? 
 (IV). — Check (II) and the true angle with H by an 
 
 auxiliary plane through A-B. 
 
 42. — Indicate the line ^-5 (12 + ii+ i) (144— ij— i). 
 (I). — With A fixed, determine the true length of A-B 
 
 inH. 
 (II). — Ditto in V as a check. 
 (III). — Indicate the piercing points of A-B before and 
 
 after each revolution. 
 (IV). — Indicate the angles that the parts of A-B 
 
 pass through both before and after revolving. 
 
 170. PLATE IV. Exercises 43-49- Text through §54 
 
 Page 48. 
 
 G-L2i" and^i". 
 
176 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 43. — ^-5-C (i-f-l) (if-if-if) (2i-i-i) rep- 
 
 resent the corners of a triangular cardboard figure. 
 (I) — Draw the plan and elevation of the figure; 
 visualize it. 
 
 NOTE. In mechanical drawing the vertical projection is 
 called the elevation and the horizontal projection the plan. 
 
 (II). — Take Gi-Li through c-b and determine the 
 
 true length of C-B on Vi; 
 (III). — On this true length construct the balance of 
 
 the figure so as to show its true shape in Vi. 
 
 44. — A-B-C are the three corners of a triangular 
 
 piece of cardboard lying in the third angle. 
 The edge A-B rests in V. A (43^— if— o) and 
 B (6^6 —1—0); B-C is 2 -^ and A-C is 2f" long. , 
 (I). — Draw the plan and elevation of the triangular 
 cardboard when the vertex C rests against H. 
 Use a profile plane at [6.f]. 
 
 45. — A cable in a factory bmlding runs from A (10 + 
 
 2i+ through the floor to 5 (13 — | + i f ) in the 
 room below. 
 
 NOTE. Scale of drawing i" = i'— o". 
 
 (I). — Determine the length of the cable between A 
 and B. 
 
 (II). — Determine the point in the floor (H) where 
 the hole must be bored to let the cable pass through. 
 
 (III). — At [10] and [10 §] two profile planes limit the 
 two sides of a wall 2 feet thick. Determine the 
 angle at which the hole must be bored to let the 
 cable pass through this wall and also the length 
 of the hole. 
 
 46. — A-C (131 + 0+1) (isi + o + 2|) is the diagonal 
 
 of the base of a square pyramid. The edge A-B 
 of the base is adjacent to V. The pyramid is 
 2i" high. 
 
SET OF DRAWING EXERCISES 
 
 177 
 
 (I). — Determine the length of the slanting edges of 
 
 the pyramid. 
 (II). — The point F is on the edge A-E and f above H; 
 
 the point K is on the edge B-E and |" above H. 
 
 Indicate the line K-F and determine its true 
 
 length. 
 (III). — In what angle is the pyramid located and if 
 
 it were projected as objects are always represented 
 
 in mechanical drawing, where should it be located, 
 
 and what would be the relation of its views? 
 
 4T.—A-B ii+i+i) (ii+2i+2) and B-D (iH- 
 2 a + 2) (3 J + I + i) are two lines intersecting at B. 
 
 (I). — Show on V the true angle between these two 
 lines. 
 
 48. — The line A-B is 3" long; [4J] A is i" above H 
 
 and 2 i" in front of V. 
 (I). — Indicate the line when it makes simultaneously 
 30 degrees with H and 45 degrees with V. 
 
 49. — Fig. 156 shows the third angle plan and elevation 
 
 of a plain hip roof. Make this drawing to the 
 scale of i" = I'-o", and omit all point notation 
 excepting that called for. A (7i— 1|— |). Draw 
 the end view with Gi-Li at [13]. 
 
 Fig. 156. 
 
178 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (I). — Determine the length of the hip rafter A-B. 
 (II). — Locate the point M on the surface A-B-D-E, 
 
 3 feet below the ridge and 10 feet from the hip 
 
 A-B. 
 (III). — Locate the point N on the rafter E-F and 3 
 
 feet from F. 
 (IV). — ■ Show on V the true shape of the roof plane 
 
 A-B-E-D and on P the true shape of D-E-F. 
 
 171. PLATE V. Exercises 50-59. Text through §75 Page 
 64. 
 
 Notation. The location of a plane on the drawing is given by 
 the point on G-L where its traces intersect; by the direction of 
 the traces from this point and the angle that the traces make 
 with G-L. Thus. VS [| + R^o] ES [| + i?4S] indicates that the 
 vertical trace VS starts \" from the left border of the drawing; 
 + indicates that the vertical trace is above G-L. i?30 indicates 
 that the general direction of the trace is toward the right at 30 
 degrees vrith G-L. Likewise ES starts from the same point 
 on G-L and + for the horizontal is below G-L; i?45 indicates 
 to the right at 45 degrees with G-L. 
 
 Note that — would place the VS below G-L and the ES 
 above as in the line notation. For general notation of the plane 
 see page 49. 
 
 G-L 2h"and Ji". 
 
 50. — Indicate the plane 5 with VS [j + R30] ES 
 
 [i +R45I 
 (I). — Determine the projections of the point [2] A 
 
 which is I" in front of V and lies in the plane 5. 
 (II). — The point [3.I] £ is i" above H and on the plane 
 
 S. Indicate the line A-E and prove that it lies 
 
 in the plane S. 
 
 51. — Through the point B (s^ + J + f) pass a plane 
 
 Q parallel to the plane S of Ex. 50. 
 (I). — Determine the perpendicular .distance from the 
 point B to HQ. 
 
SET OF DRAWING EXERCISES 
 
 179 
 
 (II). — Ditto to VQ. 
 
 52. — Indicate the plane T with VT [6^ + ^245] HT 
 
 [64 + i?3o]. Through the point A (8j— J— f) 
 pass a plane Q parallel to the plane T. 
 
 (I). — In what angles are the planes T and Q? Visual- 
 ize them. 
 
 (II). — What is the true distance between HT and HQ? 
 
 (III). — Determine the point [8] B on the plane T 
 
 and i from V. 
 (IV). — Determine the angles through which the line 
 
 A-B passes. 
 
 53. — Indicate the plane 5 with VS [11 + 2245] HS 
 
 [11 + Rso]. 
 (I). — Determine the Hne A-B i|" long in the plane 
 
 S, and also parallel to, and J" from V. 
 (II). — What relation in direction does a'-b' bear to 
 
 VS and why? 
 (Ill) . — Determine the perpendicular distance from 
 
 a point on the line A-B to VS. 
 
 54. — Indicate EQ [15! + ^45] and the point D 
 
 (14 + f + i) which lies in the plane Q. 
 (I).— Determine VQ. 
 (II). — Determine the line of intersection of the planes 
 
 5 and Q. 
 
 55. — Indicate VS [i + R30] HS [i + R45] and VQ 
 
 [I + R45] HQ [i + R30]. 
 (I). — Determine the line of intersection of planes Q 
 
 and S. Use profile plane at [21]. Visualize these 
 
 planes. 
 
 56. — Check the line of intersection of the planes Q 
 
 and S in Ex. 55 by the use of another method for 
 determining the line of intersection of two planes. 
 
 57. — Indicate VQ [Ai + R6o] HR [44+^4S] and VS 
 
 [6| + i275] HS [6i + R6o\. Make all traces 2^" long. 
 
l8o ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (I). — Determine the line of intersection of Q and 5 
 without lengthening the traces. 
 
 58. — Determine the traces of the plane R containing 
 
 the lines A-B and C-D. A-B (iif+f+if) 
 (i2f + ij+ f) and C (iif + ji+ I); C-D inter- 
 sects A-B at J" from V. 
 
 59. — Indicate VS [14 f +i6o] ES is a continuation of VS. 
 (I). — Determine the line of intersection of plane S 
 
 with plane R of Ex. 58. 
 
 Notation. Traces of planes extending back across G-L into 
 other angles are shown as construction lines. See §26 p. 18. 
 
 172. PLATE VI. Exercises 60-69. Text through §90 Page 
 78. 
 
 Notation. When a point such as 4, is projected on a plane, 
 such as S, the projection on 5 becomes A^, its V projection 
 a's and its H projection a's. If the point had been A^, its pro- 
 jection on 5 would have become A-^^, etc. 
 
 G-L 3" and 8". 
 
 60. — Indicate the plane S with VS [3 + ^45] HS 
 
 [3 -t- Z4s] and the point ^ (i f -h 2i + j|). 
 
 (I). — Indicate the perpendicular A-B from the point 
 A to the plane S. 
 
 (II.) — Indicate the vertical (perpendicular to H) dis- 
 tance A-D from the point A to the plane S. 
 
 (III.) — Measure the angle B that the vertical hne 
 A-D makes with the plane 5. 
 
 (rV.) — Measure the angle <i> that A-B makes with 
 A-D. 
 
 (V). — What h6 ■\-4> equal and why? 
 
 61. —Indicate the plane 5 with VS [4i+R4s] ES 
 [4J-|-J?3o] and the points A (41+ii+ii); B 
 (Si+if+il); C(4f+2-l-i-B.) 
 
SET OF DRAAVING EXERCISES i8i 
 
 (I.) — Indicate the triangular figure A-B-C and pro- 
 ject it upon the plane 5. 
 
 (II). — Prove that the Hne A^—Cs, is in the plane 5. 
 
 (III). ^Determine the V piercing point D and the H 
 piercing point E of the hne A-C and measure the 
 distance of these points from VS and ES. 
 
 (IV). — Visualize the data of Ex. 6i and the line D-E. 
 
 62. — Indicate the hne A-B (91+2^+2) (9I + | +2) 
 
 and the plane S with F5 [7 f + Ras\ 55 [7 f + ^230]. 
 (I). — Project the Hne A-B on the plane S. 
 (II). — Prove the accuracy of the construction by 
 
 revolving the vertical plane Q containing the line 
 
 A-B into V. Revolve Q to the left. 
 (III). — What is the true distance from 5 to 5s? 
 
 63. — Indicate the Irae A-B (11I+2I+2) (12J + 
 
 ij + i) and the point C (13 + i + i|). 
 
 (I). — Pass the plane S through the point C and per- 
 pendicular to the line A-B. 
 
 (II). — Indicate a second perpendicular line D-E of 
 equal length to A-B and piercing the plane S, 
 \" from the point F where A-B pierces 5. 
 
 64. — Indicate the point M (i5i+ir6 + ile)- Pass 
 
 the plane Q through the point M and parallel to 
 the plane 5 of Ex. 63. 
 
 (I). — Determine the distance between the planes S 
 and Q. 
 
 (II). — Determine whether or not the line A-B con- 
 tinued, would cut the plane Q. If so where; 
 if not, give reason. 
 
 65. — Indicate ES \\-\- i?3o] VS passes through D 
 
 (2f +2-ho). 
 (I). — Determine the angle that the plane 5 makes 
 
 with H. 
 (II.),— Ditto with V. 
 
l82 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 66. — Indicate F5 [5^ + L45]. 
 
 (I). — Determine BS such that the plane 5 makes 
 
 45 with V. 
 (n). — Through the point A (3J + + 0) draw the 
 
 line A-B perpendicular to and touching the plane 
 
 S atB. 
 (III). — Prove that the angles that the line A-B makes 
 
 with V and H are the complements of the angles 
 
 that the plane 5 makes with V and H. 
 
 67. — Indicate the point ^ (7 f + 4 + f). 
 
 (I). — Pass a plane 5 through A such that S makes 
 45 with H and 60 with V. Begin construction 
 at [6]. 
 
 NOTE. These angles may be changed at the instructor's 
 discretion. 
 
 68. — Indicate the point A {xo+ i+ ire)- 
 
 (T). — Pass a plane Q through A and parallel to the 
 plane 5 (Ex. 67) and with avixiliary planes T and 
 U prove accuracy of the construction of Ex. 67. 
 
 69. — Indicate the planes Q with VQ [13 f + R45] 
 
 EQ [131— i7S] and S with VS [15^ + ^60] HS 
 
 parallel to HQ. 
 (t). — Determine the true angle between the plane Q 
 
 and the plane 5. Mark the true angle TA. 
 (n). — Indicate the point A which is i" back of V, 
 
 i" above H and on the plane Q. 
 (m). — Check (II). 
 (rV). — Visualize Ex. 69. 
 
 173. PLATE Vn. Exercises 70-76. Text through §99, page 
 88. 
 
 G-L2h"andTl". 
 
 70. — Indicate the planes 5 and T with F5 [| + R45] 
 
 ES [i + R^o] and VT \zi + L6o] ET [ii + Rysl- 
 
SET OF DRAWING EXERCISES 183 
 
 (10 • — Determine the true angle between the planes S 
 
 and r. Take auxiliary plane R through [4!]. 
 
 Mark the vertex of this angle E. 
 (II). — Take a vertical plane P through E and at 60 
 
 left; determine the angle that the plane P cuts 
 
 on S and T. 
 Question. Is this angle larger or smaller than the true angle 
 between 5 and T and why? 
 
 (III). — Ditto (II) with a vertical plane Q "taken with 
 
 HQ coinciding with HR. 
 
 NOTE. In the following problems use a ruled projector for 
 the altitudes of the triangle in determining the T. R. That is, 
 make the construction on the Exerjcise. 
 
 71. — Indicate A-B (6 + 2 + if) (7i + I + li) and 
 
 F5 [5 + R30] HS [5 + R45]- 
 
 (I) . — Determine the true angle that the line A-B makes 
 with the plane R. 
 
 (n). — Check the accuracy of the above angle by the 
 following method. From the point [6i] E on 
 A-B draw the perpendicular E-N to pierce the 
 plane 5 at iV; determine the point K, where A-B 
 pierces the plane S; join N and K; revolve the 
 triangle N-E-K into H. 
 
 72. — Indicate the plane S with VS [H + R^o] HS 
 
 [8i + 2^45] and the line A-B (8i + i + ij) parallel 
 
 to G-L and 2 J" long. 
 (I). — Determine the true angle that A-B makes with 
 
 5. 
 (n). — What angle does A-B make with the line E-Ki 
 
 The line E-K being in R and parallel to H. 
 
 73. — Indicate the line A-B (i2f + | + f) (i4i + i + 2) 
 
 and the point C (13 + I + if). 
 (I). — Through the point C draw the line C-D to make 
 an angle of 45 with the line A-B. 
 
l84 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 74. — Indicate the plane 5 with F5 [f + 2?3o] HS 
 
 [f + i2'4s] and the point C (2 1+ ? + |) in the plane 
 
 5. 
 (I). — Determine the plan and elevation of a circle 
 
 in the plane 5 with the center C and i i" diameter. 
 
 Determine eight points on the circumference of 
 
 the circle. 
 (II). — Draw the plan and elevation of a cone with the 
 
 circle as a base and an altitude of i". 
 (III). — Visualize the cone and tell why it does not 
 
 represent good practice in mechanical drawing. 
 
 75. — Draw a third angle plan and elevation of the 
 
 hexagonal plane figure A-B-C-D-E-F-A of i" 
 sides. The center is located at C (5 1 — li — i). 
 The figure is parallel to V and the side A-B is par- 
 allel to H. 
 
 (I). — Move the center C of Ex. 75, 2f" to the right a'nd 
 in this position draw the projections of the plane 
 hexagonal figure, when it has revolved counter 
 clockwise through an angle of 45°, about an axis 
 through Ci and perpendicular to H. 
 
 (II). — Move the center Ci 2f" to the right. On the 
 projection c'2 construct the same elevation as at 
 ■ Ci but revolved counter clockwise through 60°, 
 and determine the new horizontal projection. 
 On this figure as a base draw the plan and eleva- 
 tion of the hexagonal prism of Ex. 75 (I). 
 
 76. — A triangular figure A-B-C is projected at A 
 
 (^3l-i-i)B (141-i-if) C (141-i-f). 
 (I). — Determine the true size and shape of the figure 
 inH. 
 
 (II). — On A-B-C as a base construct a pyramid of 
 I" altitude and determine its plan and elevation 
 in the original position.. 
 
SET OF DRAWING EXERCISES 185 
 
 174. PLATE Vin. Exercises 77-81. Text through Chap- 
 ter V, page 106. See Fig. 157, Page 187, for general layout. 
 
 G-L, 2\". 
 
 77. — Construct right-hand helix of radius i J" and pitch 
 
 T 1" 
 
 78. — Make a detail drawing of tread for spiral stairway 
 
 (see dimensions in data for Exercise 79). 
 
 79. — Draw the plan and elevation of a spiral stair of 
 
 dimensions as foUows: 
 Total rise, 8 Ft. 8". Single rise, 8". 
 Radius of tread, 4 Ft. Thickness of metal, i". 
 Center-post, 4I" diam. Solid balustrade* 3 Ft. high. 
 Draw assembly of stairway to a scale of \" = i foot, and 
 details to a scale of i" = i foot. 
 
 80. — ^Make a drawing of a right-hand V-threaded 
 
 screw, having a pitch of |" and sides at an angle of 
 60-degrees. 
 
 NOTE. Use twelve points on the circumferences for Exer- 
 cises 80 and 81. The Yi" and ri" spaces will be found on the 
 scale of i" = I foot. 
 
 81. — Make a drawing of a right-hand square-threaded 
 
 screw having pitch of i" and depth of thread the 
 same as the V thread of Exercise 80. 
 
 174 (a). May be substituted for Plate VHI. 
 G-L 3". 
 
 77. — Indicate C (if— o— ij).' 
 
 (I). — With C as the center of the top make a third 
 angle drawing of ij" cube with one face in H 
 and a face parallel to V. Designate the right hand 
 edge that is parallel to, and fartherest from V 
 as A-B; A being in H. 
 
 (II). — Move C to Ci (4f— o-f i|); keeping the face 
 C in H revolve the cube clockwise through 30 and 
 draw its plan and elevation. 
 
l86 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (III). — Keeping AiinH. move it parallel to V and to 
 Ai (8 — o — ?); keeping A-B and the edges parallel 
 to A-B in their same vertical planes, revolve the 
 cube from its position in (II), counter-clockwise 
 about A, until the face lying in H in (II) makes 
 60 with H. 
 
 (IV). — Bisect the three edges having their vertex at 
 A with the points D, E and F; connect D, E and F 
 and change (III) to show the cube with corner 
 D-E-F removed. 
 
 78. — With A at 10 1 and with B to the right, draw the 
 
 projection of the edge A-B when it makes 45 with 
 V and 30 with H. 
 
 79. — With comer A at As (134—0+ if) draw the 
 
 plan and elevation of the cube when the edge A-B 
 makes 30 with H and 45 with V. 
 
 Question. With reference to the position of the cube in 77 
 (III) about what axis has it been revolved to obtain its position 
 as shown in Ex. 79? 
 
 (I). — Take out the same comer D-E-F as in Ex. 78. 
 
 80. — A section of "spiral" rivetted steel pipe 24" in 
 diameter and 3! feet long has its center line 
 located i i from the left border line. Constmct the 
 heb'cal .center line of the rivetted joint. Scale 
 i" = I ft. Pitch 3 1 feet. Base of cylinder i" 
 from lower border. 
 
 81. — Show a layout for the flat steel sheet before it is 
 rolled into the cylindrical shape to form the pipe 
 of Ex. 80. Scale i" = i ft. 
 
 NOTE. Start with the lower comer of the sheet 3" from the 
 left border line and lay out toward the right. Make no allow- 
 ance of metal for lap at riveted joint. 
 
 175. PLATE IX. Exercises 82-88. Text through Chapter 
 VI, page 118. 
 
SET OF DRAWING EXERCISES 
 
 187 
 
 
 Hi 
 I 
 
1 88 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 NOTE. Stamp this plate above the lower G-L at [13]. 
 
 G-L 3" and 8". 
 
 82. — Indicate A (i|+ii+i) and C (i| + o+i); 
 show a right cone in the first angle with its base of 
 of 1 1 diameter in H and its center at C. Its ver- 
 tex is at A._ The point [if] 5 is a point on the 
 surface of the cone; B is also h from H .and ap- 
 proximately f from V. 
 
 (I). — Determine the traces of the plane S tangent to 
 the cone and containing the point B. 
 
 (II). — Prove that A and\B are both on the element of 
 tangency. 
 
 (III). — What angle does the plane R make with H? 
 
 83. — Indicate the points C (31—0— i); A (4I— if 
 
 — 1 1) ; 5 (5 f — I — i) . Ai& the vertex of an oblique 
 cone in the third angle. The base is i i" diameter 
 and rests on H with its center at C. 
 
 (I). — Pass the place 5 through the point B and tangent 
 to the cone on the side away from V. 
 
 (II)'. — Pass the plane T through the point B and tan- 
 gent to the cone on the opposite side. 
 
 (III). — Check the accuracy of the work by deter- 
 mining the piercing points of the elements of tan- 
 gency. 
 
 84. — Indicate A (8i-c— 1 1) and C (8 J-o— |). • 
 
 An obUque cylinder Ues in the third angle with its 
 base in H; the base is li" diameter and its center 
 is at C. The horizontal projection of the axis of 
 the cylinder makes 30 degrees left and the vertical ' 
 projection 45 degrees left with GL. The altitude 
 of the cylinder is 1 1". The point A is on the sur- 
 face of the cylinder. 
 (I). — Determine the traces of the plane 5 tangent to 
 the cylinder at the element of the cylinder passing 
 through the point A. 
 
SET OF DRAWING EXERCISES 189 
 
 (II). — Check construction by determining if the point 
 A lies in the plane S. 
 
 85. — Indicate A (15 + i + li) and B (13! + o + li). 
 
 B is the center of an oblique cylinder with its base 
 
 in H. The base is ij" diameter and the altitude 
 
 of the cyHnder is i §". 
 (I). — Pass the plane 5 through A and tangent to the 
 
 cylinder along an element on the V side of the 
 
 cylinder. 
 (II). — M (13I + 0+?) is the horizontal piercing 
 
 point of an element of the cylinder on its side away 
 
 from V. Find the traces of the plane T tangent 
 
 to the cylinder along this elements 
 
 ■86. — Indicate C (li— f— i). 
 
 A sphere of ij" diameter with its center at C lies in 
 
 the third angle. The point B is if" from V and 
 
 is on the sphere je" from, and to the right of an 
 
 axis through the sphere and perpendicular to V. 
 (I). — The plane 5 is tangent to the sphere at the point 
 
 B; determine VS and ES. 
 (II). — If the point B is revolved clockwise until B\ 
 
 is f from V, determine the traces of the plane T 
 
 tangent to the sphere at 5i. 
 (III). — Locate a point D on the sphere if " from H 
 
 and on the same great circle as B. 
 
 87. — Indicate the line AC (6f— |— if) (6f— 2f— if). 
 A-C is the axis and altitude of an hyperboloid 
 of revolution; the generating line is f" from the 
 axis and makes an incKnation of 45 degrees with H. 
 
 (I). — Determine the projections of the hyperboloid 
 by finding at least 10 points on each side of the 
 axis for the elevation. 
 
 (II). — Determine a point B on the surface of the 
 hyperboloid; 5 is 2" from the top and 6 is in the 
 upper right hand quadrant of the plan. 
 
I go 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (III). — Check the location of B by passing a second 
 position of the generating line through it. 
 
 NOTE. Where alternate exercises are given, the Instructor 
 will assign the one the student is to work. 
 
 87. — Alternate. [4I] and [7I] limit the end planes of 
 a cylinder in the third angle. The cylinder is 
 I \" in diameter and is tangent to both V and H. 
 
 (I). — Determine the traces of the plane S tangent to 
 the cylinder along an element which is i" from V. 
 
 (II). — Draw in the projections of the elements of the 
 cylinder that are nearest VS and HS. 
 
 88. — Indicate C (9I— i— ij). The point C is the 
 center of a torus generated by a sphere |" in 
 diameter. The locus of the sphere centers is in a 
 circle of if" diameter. The torus is in the third 
 angle. 
 
 (I). — -Draw its plan and elevation when it is parallel 
 toH. 
 
 (II). — Move C to [ii|] swing the elevation counter- 
 clockwise about an axis through Ci and perpen- 
 dicular to V until it makes 60 degrees with G-L 
 and draw its new plan and elevation. 
 
 (III). — Take a new Gi-L\ at [14! — L45] and deter- 
 mine the new vertical projection. 
 
 88. — Alternate. Indicate the line A-B (ioH-2t + 
 iH) (iif+l+ii); C (lof+f-Hi). The 
 point C is the center of a sphere of i" diameter. 
 Determine the traces of the plane T containing the 
 line A-B and tangent to the sphere. 
 
 1 7S (a) . May be substituted for Plate IX. 
 G-L, 3" and 8". 
 
SET OF DRAWING EXERCISES 
 
 191 
 
 82. — The base of an oblique cone rests on H in the 
 
 third angle; it is li" in diameter and has its center 
 [2f] A, i" back of V. The altitude of the cone is 
 i^" and its vertex is at [i|] B, f" back of V. De- 
 termine the traces of a plane R passing through the 
 point [1] C, i" back of V and f" below H and tan- 
 gent to the cone. 
 
 83. — The line A-C passes through [41] 4 , i j|" back of 
 
 V and re " below H. [5 i] C is i i" back of V and i" 
 below H. Determine the traces of a plane R pass- 
 ing through A-C and making 45 degrees with H. 
 
 Suggestion. Take axis of cone through C. 
 
 84. — The base of an oblique cylinder is i i" diameter 
 
 and rests against H in the third angle. The center 
 of the base is at [10] C, |" back of V. The horizon- 
 tal projection of the cylinder axis makes 30 degrees 
 (right) and the vertical projection 45 degrees 
 (right) with G-L. The altitude of the cylinder is 
 I i". Determine the traces of the plane R tangent 
 to the cylinder at the point [10 1] A, if" a. 
 
 85. — [12 f] and [15 j] limit the end planes of a cylinder 
 
 in the third angle. The cylinder is i i" in diameter 
 and is tangent to both V and H. Determine the 
 traces of the plane R tangent to the cyhnder along 
 an element which is i" from V. 
 
 86. — [4I] i" back of V and f" below H represents the 
 
 center of a sphere of i|" diameter. [3I] B, if" 
 back of V is a point on the sphere. Determine the 
 traces of the plane R tangent to the sphere at B. 
 
 87. — [6f] C is on H and i§" in front of V and is the 
 
 center of the base of an obhque cylinder resting on 
 H. The diameter of the base of the cylinder is i" 
 and the altitude is |". The vertical projection of 
 the cyUnder axis makes 60 degrees (left) with G-L 
 and the horizontal projection is parallel to the 
 
192 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 vertical. [7] A is |" above H. Determine the traces 
 of a plane R tangent to the cylinder at the point A. 
 
 88. — [10 f] A is 2 1" above H and ijf " in front of V. 
 
 [i 1 1] 5 is f " above H and i |" in front of V. [10 f] 
 C is f" above H|and i" in front of V, and is the 
 center of a sphere of i" diameter. Determine the 
 traces of a plane R containing the line A-B and 
 tangent to the sphere. 
 
 176. PLATE X. Exercises 89-90. Text through Chapter 
 VIII, page 152. 
 
 For location of G-L and layout of plate see Fig. 158, Page 
 193, also Fig. 159, Page 194. 
 
 NOTE. In the exercises to foUow, neatness and accuracy are 
 of prime importance. Keep pencil sharp and use Ught con- 
 tinuous lines for aU construction work; leave this work on 
 drawing to show method, and draw in the outlines heavy enough 
 to easily distinguish them from construction lines. When it is 
 necessary to divide a circle into a number of equal parts, twelve 
 are suggested as this can be easily done with the 30-60 triangle. 
 A system of numbering points should be used in order to easily 
 recognize the points in the various views. 
 
 89. — A hexagonal psrramid having an altitude of 
 
 2 1" and the sides i|" at the base, has a conical 
 hole cut through its central portion. The diameter 
 of the hole at the base of the pyramid is 2", and at 
 the vertex is o". Cut the pyramid by a plane 
 R as shown in layout and draw its plan and eleva- 
 tion. 
 
 t 
 
 (I). — Draw the side elevation of the pyramid of 
 Exercise 89. 
 
 (II). — Show" the true shape of the section cut from the 
 figure by the plane in Exercise 89. 
 
SET OF DRAWING EXERCISES 
 
 193 
 
194 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 (III). — Develop the surface of the figure shown in 
 Exercise 89. 
 
 BOBCgg VnTE 
 
 Fig- IS9. 
 
 89. Alternate Fig. 159 represents two square prisms 
 intersecting as shown. 
 
 (I). — Locate these prisms on the sheet as indicated 
 and make a complete mechanical drawing includ- 
 ing the line of intersection of the two prisms. 
 
 (II). — Develop the surface of the larger prism. Start 
 the development with one corner of the surface 
 i" from the left and je" from the bottom border 
 Kne. 
 
 go. — A pentagonal pyramid of 2 J" altitude and having 
 a 2|" base diameter is cut by a plane as shown in 
 layout. A hole of i" diameter is drilled as shown 
 (that is perpendicular to the plane and through 
 the axis of the pyramid). 
 
 (I).^ — Draw the side elevation of the pyramid of 
 Exercise 90. 
 
 (II). — ^Show the true shape of the section cut from 
 the figure by the plane in Exercise 90. 
 
SET OF DRAWING EXERCISES 
 
 195 
 
 (III). — Develop the surface of the figure shown in 
 Exercise 90. 
 
 177. PLATE XI. Exercises 91-93. Text through Chapter 
 VIII, page 152. 
 
 G-L See layout of sheet Fig. 160, Page 196. 
 
 Note. Length of vertical prism 2\". Length of inclined 
 prism 2 J". 
 
 91. — Draw complete plan and elevation of the inter- 
 
 secting prisms showing Unes of intersection. 
 (I). — Develop the surfaces of Exercise 91. 
 
 92. — Draw complete plan and elevation of the inter- 
 
 secting prisms showing Unes of intersection. 
 (I). — Develop the surfaces of Exercises 92. 
 
 93. — Draw complete plan and elevation of the inter- 
 
 secting cylinders showing their line of intersection. 
 Use 12 elements in the solution of this exercise. 
 (I) . — Develop the surfaces of Exercise 93. 
 
 Note. This plate may be varied by changing the angle 
 of intersection of the axes. Also, the axes may be taken so as 
 not to intersect. 
 
 177. (a) May be substituted for Plate XI. 
 
 Fig. 161 represents a three piece sheet metal, 90 degrees 
 cylindrical elbow with a conical connecting piece. Diameter of 
 cyhndrical piece = 8". Inner radius of elbow + 6". The hori- 
 zontal- and vertical elbow pieces are the same length. Other 
 dimensions are those in brackets on drawing, Fig. 161. Scale 
 of drawing 3" = i ft. 
 
 91. — Dimensions not in brackets are for locating the 
 
 drawing on the sheet. 
 (I). — Dra:w plan and elevation including the Hne of 
 
 intersection of B and C. 
 (II). — Develop conical base A. 
 
 92. — Develop Section C. 
 
 93. — Develop Section B. 
 
196 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
SET OF DRAWING EXERCISES 
 
 197 
 
 B 
 
igS 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 178. PLATE XII. Exercise 94. 
 
 94. — The section of a sheet metal gutter, joined to a vertical 
 pipe, by a conical connection is shown in Fig. 162. 
 
 (I). — Determine the curve of intersection of the 
 gutter and conical connecting piece. 
 
 (II). — Develop the pattern for the conical connecting 
 
 piece. Scale | size. 
 
 BOaPBR UNB 
 
 Fig. 162. 
 
 179. PLATE XIII. Exercise 9s. 
 
 95. — The section of a sheet 
 metal pipe, intersected by a 
 smaller pipe is shown in Fig. 
 163. 
 
 (I) . — Determine the line of 
 intersection of the two pipes. 
 
 (II). — Develop the pattern 
 for the larger pipe. Scale 3" 
 = I ft. 
 
 Fig. 163. 
 
SET OF DRAWING EXERCISES 
 
 199 
 
 180. PLATE XIV. Exercises 96-97. 
 
 96. — A 36" standard cast iron flanged elbow with a 
 24" back outlet is shown in Fig. 164. 
 
 (I). -^Determine the Kne of intersection A-B. Scale 
 1" = I ft. 
 
 Fig. 164. 
 
 NOTE. This problem involves the intersection of a cyUnder 
 and a torus. Pass planes to cut rectilinear elements from the 
 cylinder and circular elements from the torus. The intersection 
 of these elements determine the required line of intersection. 
 
 97. — The end of a connecting rod is shown in Fig. 165. 
 (I). — Determine the line of intersection formed by the 
 flat side and the curved surface. Scale 4" = i ft. 
 NOTE. This is a problem of 
 finding the line of intersection of 
 a plane the flat side) with a sur- 
 face of revolution; this surface 
 being generated by the arc of 
 2" radius revolving about the 
 center line of the rod as an axis. 
 Pass a series of planes perpen- 
 dicular to the center line of the 
 rod. 
 
 '"y 
 
 Fig. i6s. 
 
 These planes will cut circles from the surface of revolu- 
 tion, which will intersect the flat side in points on the required 
 line of intersection. 
 
200 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
SET OF DRAWING EXERCISES 
 
 20I 
 
 i8i. PLATE XV. Exercises 98-100. Text through Chapter 
 rx, page 161. 
 
 See Fig. 166 for layout of sheet. Also Figs. 167 and 168. 
 
 98. — Make an isometric projection of a 2" cube having 
 a 2'' diameter circle on each face. 
 
 Fig. 167. 
 
 Fig. 168. 
 
 99. — Make an isometric projection of Fig. 167 
 
 100. — Make an isometric projection of Fig. 168. 
 
202 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 i8i (a). May be substituted for Plate XV. 
 
 98. 
 
 — (I). — Make an isometric drawing of a i|" cube. 
 Locate the lower corner of the cube 2 f " from the 
 left and 4f from the top border line. 
 
 99. — (II) . — Show a circle inscribed in each of the 
 three visible faces of the cube. Use approximate 
 method for circles. 
 (III). — Make a cabinet drawing of the cube, with the 
 lower left hand comer of the cube located 6" from 
 the left and 4I" from the top border line. 
 Question. Why the difference in the size of 98 (I) and 98 
 (III)? 
 
 99. — (I). — The tool rest support slide of the Sibley 
 lathe is shown in Fig. 169. Make and fully dimen- 
 sion an isometric drawing of the sHde. Scale half 
 size. Locate the lower comer of the sUde iif" 
 from the left and 3 1" from the top border line. 
 
 ± 
 
 4, 
 
 T 
 
 He, 
 
 Ml 
 ■ I 
 
 lii 
 
 -3"- 
 
 1 
 
 Mo 
 
 I I I 
 
 ih'M 
 
 
 NOTE. The general rule to 
 follow in putting dimensions on 
 an isometric drawing is to draw 
 extension and dimension Hnes 
 parallel to the isometric axes 
 and in the plane of the surface 
 which is being dimensioned. 
 
 (II). — Make a cabinet draw- 
 ing with all necessary dimen- 
 Fig. 169. sionsof Fig. 169. Scale half size. 
 
 Locate lower left hand comer iiV' from the left 
 and 7 i" from the top border line. 
 
 100. - — The tail stock clamp of the Sibley lathe is shown 
 in Fig. 170. Make and fully dimension an isomet- 
 ric drawing of the clamp. Use approximate 
 method for circles and do not show invisible lines. 
 Scale full size. Locate comer A, at 54" from the 
 left and 6 J" from the top border line. 
 
SET OF DRAWING EXERCISES 
 
 203 
 
 NOTE. If this drawing were 
 made in the usual way, with the 
 top side showing, the drawing 
 would not be clear. In cases 
 like this the isometric axes are 
 inverted, so that the irregular 
 outline of the under side will be 
 visible. 
 
 Fig. 170. 
 
 ADDITIONAL PLATES 
 
 182. PLATE XVI. Roof and Stack Problem. Text through 
 Chapter IV, page 89. 
 
 See Fig. 171, Page 204, for layout of this plate. 
 
 The rafters of a boiler-house roof make 30 degrees 
 with the joist, and the eave of the roof makes 45 
 degrees with a front vertical wall. A smokestack 
 of 30-iiich diameter passes through the roof (center 
 at the point "C") and extends 14 feet above the 
 hne of the joist. Four feet from the top of the 
 stack 3 guy wires 120 degrees apart are attached 
 and extend down to the roof to act as braces. 
 The wire No. i is to have a direct pull against the 
 roof (that is, it will be perpendicular to the roof.) 
 The other two wires are to make, the same angle 
 ■with the stack that No. i makes. A mechanical 
 drawing is to be made of the above in the first 
 angle, using a scale of |" = i foot. Also deter- 
 mine the length of the guy wires and the angle each 
 makes with the roof. 
 
 183. PLATE XVII. Belt Problem. Text through Chapter 
 VII, page 128. 
 
 See Fig. 172, Page 205, for layout of this plate. 
 
204 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 STAMP 
 TITLE-FORM HERE 
 
 Fig. 171. —Layout for Plate XVI. 
 
SET OF DRAWING EXERCISES 
 
 205 
 
2o6 ' ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
SET OF DRAWING EXERCISES 
 
 207 
 
 (I) . — The center of a circle of i" diameter is f" 
 behind B and f" below H. Determine the pro- 
 jections of the circle such that it shall lie in a plane 
 whose vertical and horizontal trace each makes 
 45 degrees with G-L. 
 
 (II). — Make a third-angle mechanical drawing from 
 the sketch and the following data. 'The pulleys 
 are 7 ft. — 3" from center to center; they are on 
 shafts at right angles to each other, and located 
 as shown in Fig. 172. A guide pulley is to be 
 introduced between pulley No. i and No. 2 in 
 such a manner as to change the direction of the 
 belt at a point 3 ft. — |" from the center of pulley 
 No. I. Diameter of guide pulley = 24"; diameter 
 of pulley No. i = 30"; diameter of pulley No. 
 2 = 42"; face of all puUeys = 8"; thickness of 
 rim = I J"; length of hubs = 11"; diameter of 
 hubs = 8",; width of belt = 7". Show thickness 
 of belt \"; thickness of web 2"; diameter of shaft 
 4". Use a scale i" = i ft.— o". 
 
 184. PLATE XVIII. Development of Transition Piece. 
 
 (Between a rectangular opening and a round pipe.) Text 
 through Chapter VIII, page 152. 
 
 See Fig. 173. Take elements £-1, £-2, £-3, etc.; determine, 
 true lengths of elements and also of arcs a-i, 1-2, 2-3, etc., and 
 proceed with development. 
 
 185. PLATE XIX. Locomotive Slope Sheet, Text through 
 Chapter VIII, page 152. 
 
 See Fig. 174. This is a warped surface and divisions on the 
 circumferences such as 8-7, 7-6 and h-g, g-f must be small 
 enough to make such lines as 8-G, 7-F, 6-£, etc. approximately 
 straight Unes. Determine the true size of the triangles, such as 
 h-8-g and 8-^-7, etc., and lay them out on the development as 
 at E-8-G and S-G-j, etc. 
 
208 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 
 Hi 
 I 
 
SET OF DRAWING EXERCISES 
 
 209 
 
 186. PLATE XX. Development of the Sphere. 
 
 The surface of the sphere can be developed only approxunately. 
 There are two methods of accompHshing this. The first is the 
 "zone" method, in which the sphere is resolved into a number of 
 frustums of cones, which frustums are then developed as shown 
 in Fig. 175. The second method is the 
 "gore" method in which the sphere is divided 
 into sections such as A-C-B-F, Fig 176, 
 and these sections are then developed. 
 Fig. 177 shows layout for plate XX. 
 
 Fig. 175- Development of the Sphere (Zone Method). 
 
 -irD- 
 
 ============ -^ 
 
 Fig. 176. Development of the Sphere (Gore Method). 
 
2IO 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
SET OF DRAWING EXERCISES 
 
 211 
 
 187. Material for Additional Exercises. 
 
 Figs. 178-187 are added for the In- 
 structors' convenience in making new 
 assignments. 
 
 Fig. 178. Conical Helix. 
 
 Fig. 180. Fimnel. 
 
 —32"d.— 
 
 T 
 
 4- 
 
 j_ 
 
 Fig. 179. Screw Conveyor. 
 
 Fig. 181. Locomotive Stack. 
 
212 
 
 ELEMENTS OF DESCRIPTIVE GEOMETRY 
 
 Fig. 182. Hopper, 
 
 Fig. 184. Scale Scoop. 
 
 Fig. 183. Hood. 
 
 Fig. 185. Boat Ventilator. 
 
 f" 18' 
 
 \Wi- 
 
 J- 
 
 Fig. 186. Exhaust Head. 
 
 Fig. 187. Boiler Breediisg. 
 
INDEX 
 
 A. 
 Altitude of cone, p. loo, § no. 
 prism, p. loo, § 112. 
 pyramid, p. 95, § 105. 
 Angle, base, of right circular cone, p. 46, § 53. 
 
 between line and plane, to determine, (Prob. 31) p. 78, § 91. 
 traces of a plane, to determine, (Prob. 28) p. 73, § 86. 
 two intersecting lines, to determine, p. 83, § 94. 
 
 planes, how measured, p. 72, § 84. 
 
 to determine, (Prob. 29) p. 75, § 87. 
 dihedral, definition of, p. 65, § 77. 
 
 projection of a point in each, p. 17, § 24. 
 first, p. 16, § 24. 
 fourth, p. 16, § 24. 
 given straight line makes with a given plane, to determine, (Prob. 31) 
 
 p. 78, § 91. 
 plane makes with V or H, how determined, (Prob. 27) p. 72, § 85. 
 projection of point in each, p. 17, § 24. 
 second, p. 16, § 24. 
 
 straight line makes with a plane, to determine, (Prob. 31) p. 78, § 91. 
 third, p. 16, § 24. 
 
 to bisect or otherwise divide, (Axiom) p. 83, § 94. 
 true, between two intersecting lines, (Cor.) p. 29, § 38. 
 which a line perpendicular to one of two intersecting planes makes with the 
 other plane is a complement of the angle between the two planes, 
 (Prin.) p. 77, § 88. 
 which projection of a line on a plane makes with the line is the same as 
 angle the line makes with the plane, (Prin.) p. 78, § 90. 
 Angles, four dihedral, p. 15, § 24. 
 Apex of a pyramid, p. 95, § 105. 
 Arc of revolution, p. 35, § 46. 
 Axes, coordinate, p. 154, § 152. 
 
 isometric, p. 154, § 152. 
 Axiom. Angle between two intersecting lines, to determine, p. 83, § 94. 
 Angle, to bisect or otherwise divide, p. 83, § 94. 
 Planes intersect in a common straight line, p. 59, § 71. 
 When projections of intersecting lines are at right angles, etc., p. 34, 
 
 . § 45- 
 Aids of cone, p. 45, § S3". P- 100, § no. 
 pyramid, p. 95, § 105. 
 revolution, p. 35, § 46- 
 rotation, (Exp. Ill) p. 33, § 44- 
 revolution of a point about an, (Prob. 5) p. 35, §47; (Prob. 6) p. 38, §48; 
 (Prob. 7) p. 39, § 49- 
 
 B. 
 Base angle of right circular cone, p. 46, § 53. 
 
 of cone, p. 100, § no. 
 Bisect an angle, to, p. 83, § 94. 
 
 213 
 
214 INDEX 
 
 C. 
 Cabinet or cavalier projection, p. i6o, § 164. 
 Cavalier or cabinet projection, p. 160, § 164. 
 Center of revolution, p. 35, § 46. ' 
 
 Chapters of the text. (See Contents.) 
 Circle, great, of a sphere, p. 120, § 131. 
 
 isometric projection of, (Prob. 86) p. 159, § 160. 
 
 of revolution, p. 35, § 46. 
 
 projection of, to determine, p. 85, § 97. 
 
 third angle projection of, (Prob. 35) p. 85, § 98. 
 Circular cone, obhque, p. 100, § no. 
 
 right, p. 45, § 53; P- io°> § "o. 
 Cone, altitude of, p. 100, § no. 
 
 and cylinder, line of intersection between, to determine, (Prob. 78) p. 148, 
 
 § ISO- 
 plane, line of intersection between, to determine, (Prob. 70) p. 134, 
 § 142; (Prob. 74) p. 138, § 146. 
 axis of, p. 45, § 53; p. 100, § no. 
 base of, p. 100, § no. 
 circular, p. 100, § no. 
 definition of, p. 100, § no. 
 development of, p. 130, § 138; (Prob. 70) p. 134, § 142; (Prob. 74) p. 138, 
 
 § 146; (Prob. 78) p. 148, § 150. 
 element of, p. 45, § 53; p. 100, § no. 
 intersection of, wilJa plane, p. 134, § 142; p. 138, § 146. 
 oblique circular, p. 100, § no. 
 of revolution, p. 45, § 53. 
 plane passed tangent to, and through a given point on the surface, (Prob. 
 
 51) p. 108, § 120. 
 plane passed tangent to, and through a given point without the surface, 
 
 (Prob. 52) p. 109, § 121. 
 plane passed tangent to, and parallel to a given straight line, (Prob. 54), 
 
 p. Ill, §123. 
 projection of, in any position, (Prob. 45) p. 100, §111. 
 right circular, p. 45, §53; p. 100, § no. 
 base angle of, p. 46, § 53. 
 section cut from, by a plane, p. 120, § 131; (Prob. 65) p. 124, § 136. 
 Conical surface, p. 100, §110. 
 Construction line, convention for, p. 18, § 26. 
 
 of helix, p. 104, § 117. 
 Convention for construction line,.p. 18, § 26. 
 direction of sight, p. 19, § 26. 
 ground line, p. 18, § 26. 
 invisible line, p. 19, § 26. 
 path traced by a point, p. 19, § 26. 
 ruled projector, p. 18, § 26. 
 visible line, p. 18, § 26. 
 Conventions, line, p. 12, § 21; p. 18, § 26. 
 
 and notations, p. 49, § 56. 
 Coordinate axes, p. 154, § 152. 
 
 planes, p. 154, § 152. 
 Corollary. Angle a line makes with a plane, p. 29, § 37. 
 
 Line parallel to H and in a plane is parallel to-the horizontal trace of 
 
 the plane, p. 59, § 70. 
 Point on a plane, projections of, p. n, § 19. 
 Projections of a point fall on one and the same straight line per- 
 
 pendicular to G-L, p. n, § 18. 
 Measuring off a true distance on a line, p. 29, § 39. 
 True angle between two intersecting lines, p. 29, § 38. 
 True distance between two points, p. 29, § 36. 
 Cube, development of, p, 129, § 138. 
 
INDEX 215 
 
 Cube, isometric projection of, (Prob. 84) p. 157, § 158. 
 
 projection of, (Piob. 37) p. 90, §101; (Prob. 38) p. 91, §102; (Prob. 39) 
 
 p. 92, § 103; (Prob. 40) p. 94, § 104. 
 third angle plan and elevation of, to determine, (Prob. 37) p. 90, §ior; 
 (Prob. 38) p. 91, §102; (Prob. 39) p. 92, §103; (Prob. 40) p. 94, 
 § 104. 
 Curve, isometric projection of any plane, (Prob. 87) p. 159, § 161. 
 Cutting plane, definition of, p. 119, § 131. 
 
 Cylinder and cone, line of intersection between, to determine, (Prob. 78) p. 148, 
 §150. 
 plane, line of intersection between, (Prob. 68) p. 131, § 140; (Prob. 
 72) p. 136, § 144. 
 development of, p. 130, § 138; (Prob. 68) p. 131, § 140; (Prob. 72) p. 
 
 136, § 144; (Prob. 76) p. 142, § 148; (Prob. 78) p. 148, § 150. 
 intersection of, with plane, p. 136, § 144. 
 plane passed tangent to, and through a given point on its surface, (Prob. 
 
 5S) p. 112, § 124. 
 plane passed tangent to, and through a given point without its surface, 
 
 (Prob. s6) p. 112, § 125. 
 projection of, p. 103, § 115; p. 104, § 116. 
 right, cut by a plane, p. 131, § 140. 
 
 section cut from, by a plane, p. irg, § 131; (Prob. 63) p. 122, § 134. 
 third angle plan and elevation of, (Prob. 48) p. 103, § 115; (Prob. 49) 
 p. 104, § 116. 
 Cylinders, line of intersection between two, (Prob. 76) p. 142, § 148. 
 
 D. 
 Data dealt with in Descriptive Geometry, p. i, § 3. 
 
 for problems in Descriptive Geometry, p. 15, § 23. 
 Definition of cone, p. 100, § no. 
 
 cutting plane, p. 119, § 131. 
 Descriptive Geometry, p. i, § i. 
 dihedral angle, p. 65, § 77. 
 helicoid, p. 105, § 117. 
 helix, p. 104, § 117. 
 "position," p. 2, § 4. 
 prism, p. 100, § 112. 
 pyramid, p. 95, § 105. 
 section, p. 119, § 131. 
 soUd, p. 90, § 100. 
 Definitions, p. 45, §53; p. 65, § 77; P- 95, § los; P- 100, § "o and § 112; p. 104, 
 §117; p. 116, §128. 
 and general considerations, p. 107, § 119; p. 119, § 131; p. 129, § 138. 
 principles, p. 49, § SS- 
 Descriptive Geometry, data dealt with in, p. i, § 3. 
 definition of, p. I, § i. 
 drawing, p. 4, § 5. 
 measurements in, p. 4, § 5. 
 practical value of, p. i, § 2. 
 problems, data for, p. 15, § 23. 
 
 set of drawing exercises in, p. 162, § 166 to § 187 inclusive. 
 Development of cone, p. 130, § 138; (Prob. 70) p. 134, § 142; (Prob. 74) p. 138, 
 § .146; (Prob. 78) p. 148, § 150. 
 cube, p. 129, § 138. 
 cylinder, p. 130, § 138; p. 131, § 140; p. 136, § 144; (Prob. 76) 
 
 p. 142, § 148; (Prob. 78) p. 148, § 150. 
 prism, p. 120, § 138; (Prob. 67) p. 130, § 139; (Prob. 71) p. 134, 
 
 § 143; (Prob. 75) p. 138, § 147; (Prob. 77) p. 144 § 149. 
 pyramid, p. 129, § 138; (Prob. 69) p. 132, § 141; (Prob. 73) 
 
 P- 137, § 145; (Prob. 77) p. I44> § 149- 
 solid, p. 129, § 138. 
 
2l6 INDEX 
 
 Dihedral angle, definition of, p. 65, § 77. 
 
 projection of a point in each, p. 17, § 24. 
 angles, the four, p. 15, § 24. 
 Direction of, and point on, a line will definitely locate the line, (Prin.) p. 25, § 30. 
 
 sight, convention for, p. 19, § 26. 
 Distance, a given, measured off on a given line, (Prob. 10) p. 44, § 52. 
 between two points, to determine, (Prob. 8) p. 40, § 50. 
 from a given point to a given plane to determine, (Prob. 24) p. 68, § 80. 
 true, between two points, (Cor.) p. 29, § 36. 
 
 to measure off, on a given line, (Cor.) p. 29, § 39. 
 Double curved surfaces, p. 129, § 138. 
 
 of revolution, intersection between, (Prob. 79) p. 151, § 151. 
 Drawing board exercises in this work, § 167 to § 187 inclusive. {See also Chapter X, 
 Table of Contents, p. xv.) 
 Descriptive Geometry, p. 4, § 5. 
 
 exercises in Descriptive Geometry, set of, p. 162, § 166 to § 187 in- 
 clusive, 
 isometric, p. 153, § 152. 
 
 of any plane curve, (Prob. 87) p. 159, § 161. 
 circle, (Prob. 86) p. 159, § 160. 
 
 cube, (Prob. 84) p. 157, § 158- 
 
 hexagonal pyramid, (Prob. 85) p. 158, § 159. 
 
 mortise and tenon, (Prob. 88) p. 159, § 162. 
 
 piping system, p. 159, § 163. 
 
 plane curve, p. 159, § 161. 
 
 pyramid, p. 158, § 159. 
 
 square figure, (Prob. 83) p. 157, § 157. 
 
 E. 
 
 Element of cone, p. 45, § 53; p. 100, § no. 
 
 tangency, p. 108, § 119. 
 Elevation, definition of, p. 83, footnote. 
 
 Exercises, drawing, set of, in this work, p. 162, § 166 to § 187 inclusive. {See also 
 Chapter X, Table of Contents, p. xv.) 
 how worked up, p. 162, § i66. 
 in this course, p. 162, § 167 to § 187 inclusive. {See also Chapter X, Table 
 
 of Contents, p. xv.) 
 numbering of, p. 163, § 166. 
 Experiment I, p. 2, § 5. 
 
 II. Projections of line and of intersecting lines, p. 20, § 27. 
 
 III. Rotation of point about a line, p. 32, § 44. 
 Eye, position of, when projecting a point, p. 16, § 24. 
 
 F. 
 Figure, plane, isometric projection of, p. 157, § 157. 
 
 third angle plan and elevation of, to determine, (Prob. 36) p. 88,§ 99. 
 third angle projection of, to determine, (Prob. 34) p. 83, § 95. 
 to determine its true form, (Prob. 33) p. 81, § 93. 
 third angle projection of, p. 88, § 99. 
 square, isometric projection of, (Prob. 83) p. 157, § 157. 
 First angle, the, p. 16, § 24. 
 Fourth angle, the, p. 16, § 24. 
 Fundamental principles of isometric projection, p. 155, § 153. 
 
 G. 
 
 General considerations and definitions, p. 107, § 119; p. 119, § I31. 
 
 notation, p. 17, § 25. 
 Geometric properties of the line, p. i, § 3. 
 
 plane, p. 2, § 3. 
 
 point, p. I, § 3. 
 
 solid, p. 2, § 3. 
 
INDEX 217 
 
 Geometry, Descriptive, definition of, p. i, § i. 
 drawing for, p. 4, § 5. 
 measurements in, p. 4, § 5. 
 Great circle of a sphere, p. 120, § 131. 
 Ground line, p. 8, § 11; p. 12, § 21. 
 
 convention for, p. 18, § 26. 
 
 location of, in this work, p. 163, § 166. 
 
 notation for, p. 12, § 21. 
 
 H. 
 Helicoid, definition of, p. 105, § 117. ' 
 
 Helix,construction of, p. 104, § 117. 
 definition of, p. 104, § 117. 
 pitch of, p. 105, § 117. 
 Horizontal piercing point of a line, p. 20, § 2T, (Prob. 3) p. 28, § 34. 
 plans' of projection, p. 8, § 9. 
 projection of line of intersection of oblique plane and plane parallel to 
 
 H, etc., (Theo. VII) p. 58, § 69. 
 projection of line, p. 20, § 27. 
 
 notation for, p. 18, § 25. 
 point, p. 2, § s; p. 3, § 5; p. 10, § 14. 
 notation for, p. 18, § 25. 
 trace of plane, p. 49, § 56. 
 Hyperboloid of revolution, p. 116, § 128; (Prob. 59) p. 116, § 129. 
 of two nappes, p. 116, § 128. 
 projection of, (Prob. 59) p. ii6, § 129. 
 
 I. 
 
 Intersecting lines, angle between, p. 29, § 38. 
 
 have one point projected in common, (Prin.) p. 26, § 33. 
 plane passed through two, (Prob. 13) p. 54, § 60. 
 projection of, (Exp. II) p. 20, § 27. 
 projections being at right angles, (Axiom) p. 34, § 45. 
 to determine angle between, (Prob. 9) p. 41, § 51; p. 83, § 94. 
 to pass a plane through two, (Prob. 13) p. 54, § 60. 
 true angle between two, (Cor.) p. 29, §38; (Prob. 9) p. 41, § 51. 
 planes, angle between two, to determine, (Prob. 29) p. 75, § 87. 
 
 perpendicular to third plane, intersect in a Ime perpendicular 
 
 to the third plane, (Prin.) p. 65, § 78. 
 to measure angle between, p. 72, § 84. 
 Intersection, line of, p. 119, § 131; p. 129, § 138. 
 
 between cone and cylinder, to determine, (Prob. 78) p. 148, 
 §150. 
 cone and plane, to determme, (Prob. 70) p. 134, 
 
 § 142; (Prob. 74) p. 138, § 146. 
 cylinder and plane, to determine, p. 131, § 140; 
 
 (Prob. 72) p. 136, § 144. 
 cylinders (two), to determine, (Prob. 76) p. 142, § 148. 
 double curved surfaces (two). of revolution, to de- 
 termine, (Prob. 79) p. 151, § 151. 
 plane and prism, to determine, (Prob'. 67) p. 130, 
 
 plane and pyramid, to determme, p. 132, § 141; p. 137, 
 
 § 145- 
 planes (two), to determine, (Prob. 20) p. 60, § 72; 
 
 (Prob. 21) p. 62, § 74; (Prob. 22) p. 64, § 75. 
 prism and plane, to determine, (Prob. 71) p. 134, 
 
 § 143- 
 pyramid and plane, to determine, (Prob. 69) p. 132, 
 § 141; (Prob. 73) p. 137, § 145. 
 
2i8 INDEX 
 
 Intersection, line of, between pyramid and prism, to determine, (Prob. 77) p. 144, 
 
 § 149- 
 two planes at right angles to a third plane, p. 65, 
 
 §78. 
 two planes, projection of, (Theo. VII) p. 58, § 69; 
 two prisms, to determine, (Prob, 75) p. 138, § 147. 
 to determine, p. 129, § 138. 
 
 cylinder cut by plane, p. 131, § 140. 
 
 double curved surfaces of revolution, p. 151, 
 
 § iSi. , 
 planes (two) when both are perpendicular to a 
 
 third plane, p. 65, § 78. 
 pyramid cut by plane, p. 132, § 141. 
 Invisible line, convention for, p. 19, § 26. 
 Isometric axes, p. 154, § 152. 
 
 drawing, p. 153, § 152. 
 
 of circle, p. 159, § 160. 
 cube, p. 157, § 158. 
 mortise and tenon, p. 159, § 162. 
 piping system, p. 159, § 163. 
 plane curve, p. 159, § 161. 
 pyramid, p. 158, § 159. 
 square, p. 157, § 157. 
 origin, p. 154, § 152. 
 projection, (Chapter IX) p. 153. 
 
 fundamental principles of, p. 155, §■ 153. 
 of any plane curve, (Prob. 87) p. 159, § l6l. 
 circle, Prob. 86) p. 159, § 160. 
 cube, (Prob. 84) p. I57, § 158- 
 hexagonal pyramid, (Prob. 85) p. 158, § 159. 
 line, p. 156, § 156. , , , 
 mortise and tenon, (Prob. 88) p. 159, § 162. 
 piping system, (Prob. 89) p. 159, § 163. 
 plane curve, p. 159, § 161. 
 
 point, (Prob. 80) p. 155, § 154; (Prol>- 81) P- 156, § 155- 
 pyramid to determine, p. 158, § 159. 
 square, (Prob. 83) p. 157, § 157- 
 straight line, (Prob. 82) p. 156, § 156. 
 scale, p. 154, § 152. 
 
 L. 
 
 Layout of plates in this work (see Fig. 154), p. 162, § 166. 
 
 Length, true, notation for, p. 41, § 50. 
 
 Letters, style of , in this work, p. 163, § 166. 
 
 Light, ray of, in projecting a point, p. 5,' §7. 
 
 Line, also intersecting lines, projection of, (Exp. II) p. 20, § 27. 
 
 and plane, angle between, (Cor.) p. 29, § 37; (Prob. 3) p. 78, § 91. 
 
 point, plane passed through a, (Prob. 15) p. 55, § 62. 
 angle it makes with a given plane, (Prob. 31) p. 78, § 91. 
 
 its own projection is the same as angle between the line 
 and the plane, (Prin.) p. 78, § 90. 
 assumed in a given oblique plane, (Prob. 12) p. gi, § 59. 
 construction, convention for,' p. 18, § 26. 
 convention and notation for traces of planes, p. 49, § 56. 
 conventions, p. 12, § 21; p. 18, § 26; p. 164, § l66. 
 
 and notations, p. 49, § 56. 
 determined by its direction and one of its points, or by two of its points, 
 
 (Prin.) p. 25, § 30. 
 drawn through a given point perpendicular to a given plane and to determine 
 distance from the point to tiie plane,' (Prob. 24) p. 68, § 80. 
 
INDEX 2IQ 
 
 Line, drawn through a given point to intersect a given line at a given angle, p. 80, 
 §92. 
 geometric properties of, p. i, § 3. 
 ground, p. 8, § 11. 
 
 convention for, p. 18, § 26. 
 location of, on drawings, p. 163, § 166. 
 notation for, p. 12, § 21. 
 horizontal piercing point of, p. 20, § 2^; (Prob. 3) p. 28, § 34. 
 
 projection of, p. 20, § 27. 
 how specified, p. 15, § 23. 
 in a plane and parallel to V or H, to project, p. 59, § 70. 
 
 whidi makes a given angle with H, ( Prob. 53) p. no, § 122. 
 in space, notation for, p. 18, § 25. 
 invisible, convention for, p. 19, § 26. 
 isometric projection of, p. 156, § 156. 
 
 located by its direction and one of its points, (Prin.) p. 25, § 30. 
 maximum projection of, p. 7, § 7. 
 minimum projection of, p. 7, § 7. 
 normal to a surface, p. 108, § 119. 
 notation for horizontal projection of, p. 18, § 25. 
 
 vertical projection of, p. 18, § 25. 
 of intersection, p. 119, § 131, p. 129, § 138. 
 
 between cone and cyUnder, to determine, (Prob. 78) p. 148, 
 §150- 
 cone and plane, to determine, (Prob. 70) p. 134, S 142; 
 
 (Prob. 74) p. 138, § 146. 
 cylinder and plane, p. 131, § 140; (Prob. 72) p. 136, 
 
 § 144; ^ 
 cylinders (two), to determine, (Prob. 76) p. 142, § 148. 
 double curved surfaces (two) of revolution, (Prob. 
 . 79) P- 151, § 151- 
 mtersecting planes is common to all of them, p. 59, 
 
 §71- 
 plane and prism, (Prob. 67) p. 130, § 139. 
 
 pyramid, p. 132, § 141. 
 planes (two) at right angles to third plane, etc., p. 65, 
 
 §78- 
 planes (two), to determine (Prob. 20) p. 60, § 72; 
 
 (Prob. 21) p. 62, § 74; (Prob. 22) p. 64, § 75. 
 prism and plane, to determine, (Prob. 67) p. 130, 
 
 § 139; (Prob. 71) p. 134, § 143. 
 pyramid and plane, to determine, (Prob. 69) p. 132, 
 
 § 141; (Prob. 73) p. 137, § 145. 
 P3rramid and prism, (Prob. 77) p. 144, § 149. 
 two planes, projection of, (Theo. VII) p. 58, § 69. 
 prisms, to determine, (Prob. 75) p. 138, § 147. 
 definition of, p. 129, § 138. 
 on oblique plane, projection of, (Prob. 25) p. 69, § 82. 
 parallel to another line on a plane is parallel to plane, (Prin.) p. 57, § 65. 
 passed through a given point and parallel to a given line, (Prob. 4) p. 31, 
 
 §43- 
 perpendicular to one of two intersecting planes makes an angle with the 
 
 other plane which is the complement of the angle between the two 
 
 planes, (Prin.) p. 77, § 88. 
 Line, perpendicular to a plane, its projections are respectively perpendicular to 
 
 the traces of the plane, (Theo. VIII) p. 65, § 79. 
 piercing point of, on a given plane, (Prob. 23) p. 64, § 76. 
 
 to determine, (Prob. 3) p. 26, § 34. 
 plane drawn parallel to, and tangent to a given cone, (Prob. 54) p. in, § 123. 
 points of, project on the line, (Theo. II) p. 25, § 32, 
 possible positions of, p. 22, § 28. 
 
220 INDEX 
 
 Line, principles of, p. 20, § 27. 
 
 projected in its true length, p. 6, § 7; fTheo. Ill) p. 28, § 35. 
 projection of a, (Exp. Il) p. 20, § 27; (Theo. VIIl) p. 65, § 79. 
 on a given oblique plane, (Prob. 25) p. 69, § 82. 
 projections of, contain projections of all its points, (Theo. II) p. 25, § 32. 
 
 to determine, (Prin.) p. 25, § 31. 
 revolution of a point about any, (Exp. Ill) p. 32, § 44; p. 38, § 47. 
 straight, isometric projection of, (Prob. 82) p. 156, § 156. 
 
 through which a plane is passed and parallel to any other straight 
 line, (Prin.) p. 57, § 65. 
 tangent to surface, p. 107, § 119. 
 through which a plane is passed, and which is parallel to an"Dther straight 
 
 Une, (Prob. 17) p. 57, § 66. , x <, 
 
 to contain a plane which is tangent to a sphere, (Prob. 58) p. 115, § 127. 
 to measure ofiE a given distance on a, (Prob. 10) p. 44, § 52. 
 to which a plane passed through a given point shaU be perpendicular, (Prob. 
 
 26) p. 69, § 83. 
 trace of, on a plane, to determine, (Prob. 23) p. 64, § 76. 
 true length projection of, p. 6, § 7; (Theo. Ill) p. 28, § 35. 
 two projections of, required to define the position of, (Prin.) p. 25, § 29. 
 vertical piercing point of, p. 20, § 27; (Prob. 3) p. 26, § 34. 
 projection of, p. 20, § 27. 
 
 no tation for, p. 18, § 25. 
 visible, convention for, p. 18, § 26. 
 
 which is parallel to plane passed through another Hne, (Prob. 17) p. 57, § 66. 
 makes a given angle with V and H, to project, (Prob. 1 1) p. 46, § 54. 
 Lines, angle between two intersecting, p. 29, § 38; (Axiom) p. 83, § 94. 
 
 intersecting, angle between, to determine, (Prob. 9) p. 41, § 51; p. 83, §94. 
 at right angles , projection of, etc., (Axiom) p. 34, § 45. 
 have one point projected in common, (Prin.) p. 26, § 33. 
 plane passed through two, (Prob. 13) p. 54, § 60. 
 projection of, (Exp. II) p. 20, § 27. 
 to pass a plane through two, (Prob. 13) p. 54, § 60. 
 true angle between, (Cor. Ill) p. 29, § 38; (Prob. 9^ p. 41, § 51. 
 parallel, have their corresponding projections parallel, (Theo. IV) p. 29, § 40. 
 
 to a plane of projection, etc., p. 59, § 70. 
 revolution of, p. 35, § 46. 
 
 two given, to which a plane is passed parallel and through a given point, 
 (Prob. 18) p. 57, § 67. 
 Location of a line is determined by its direction and one of its points, (Prin.) p. 25, 
 §30. 
 ground line, in this work, p. 163, § 166. 
 points on drawings and plates in diis work, p. 164, § 166. 
 Longitudinal section, p. 120, § 131. 
 
 M. 
 Maximum projection of a line, p. 7, § 7. 
 
 Measurements in Descriptive Geometry, how made in this work, p. 4, § 5. 
 Measuring off a given distance on a given line, (Prob. 10) p. 44, § 52. 
 
 true distance on a line, (Cor. IV) p. 29, § 39. 
 Mechanical penciled line, in this work, p. 163, § 166. 
 Minimum projection of a line, p. 7, § 7. 
 Mortise and tenon, isometric projection of, (Prob. 88) p. 159, § 162. 
 
 N. 
 New position of a point, p. 6, § 7. 
 vertical plane, p. 13, § 22. 
 
 notation for, p. 14, § 22. 
 projection of a point, (Prob. 2) p. 13, § 32. 
 Normal to a surface, p. 108, § 1 19. 
 
INDEX 221 
 
 Notation and line convention, p. 49, § 56. 
 
 for traces of planes, p. 49, § 56. 
 for groimd line, p. 12, § 21. 
 
 horizontal projection of line, p. 18, § 25. 
 
 point, p. 18, § 25. 
 new vertical plane, p. 14, § 22. 
 point, p. 13, §21; p. 18, §25. 
 
 projection of point on a plane other than V or H, p. 69, § 81. 
 true length, p. 41, § 50. 
 vertical projection of line, p. 18, § 25. 
 
 pouats, p. 18, § 25. 
 general, p. 17, § 25. 
 Number of vertical projections of a point possible, p. 3, § 5. 
 Numbering of exercises in this work, p. 163, § 166. 
 
 plates in this work, p. 163, § 166. 
 Numerals, style of, in this work, p. 163, § 166. 
 
 O. 
 Objects of reference in Descriptive Geometry, p. 8, § 8. 
 Oblique circular cone, p. 100, § 1 10. 
 
 prism, p. 100, § 112. 
 
 pyramid, p. 95, § 105. 
 
 section, p. 120, § 131. 
 Opening or unfolding the projection planes, p. 17, § 24. 
 Origin, isometric, p. 154, § 152. 
 Orthographic projection, p. 5, § 5; p. 10, § 16. 
 
 P. 
 
 Parallel planes have their respective traces parallel, (Theo. VI) p. 5S, § 64. 
 
 intersect a third plane in parallel lines, p. 61, § 73. 
 Path traced by a point, convention for, p 19, § 26. 
 Penciled lines, mechanical, in this work, p. 163, § 166. 
 Perpendicular from point to plane, projection of, (Prob. 24) p, 68, § 80. 
 
 line from a given point to a given plane, and to determine the dis- 
 tance frofti point to plane, (Prob. 24) p. 68, § 80. 
 projection, p. 5, § 5- 
 Piercing point of a Une, horizontal, p. 20, § 27- (Prob. 3) p. 28, § 34. 
 vertical, p. 20, § 27; (Prob. 3) p. 26, § 34. 
 (or trace) of given straight line on a given plane, to determine, 
 (Prob. 23) p. 64, § 76. 
 points of a line he in the traces of plane containing the line, (Theo. V) 
 
 P- 51, § 58- 
 to determine, (Prob. 3) p. 26, § 34. 
 Piping system, isometric drawing of, (Prob. 89) p. 159, § 163. 
 Pitch of helix, p. 105, § 117. 
 Plan view, (footnote) p. 83. 
 
 Plane and cone, hne of intersection between, (,Prob. 70) p. 134, § 142; (Prob. 74) 
 p. 138, § 146. . , ^ 
 
 cylinder, line of intersection between, to determine, (Prob. 63) p. 122, 
 
 § 134; (Prob. 68) p. 131, § 140; (Prob. 72) d. 136, § 144. 
 line, angle between, (Cor^ p. 29, §37; (Prob. 31) p. 78, § 91. 
 point, distance between, (Prob. 24) p. 68, § 80. 
 prism, line of intersection between, (Prob. 67) p. 130, § 139; (Prob. 71) 
 
 p. 134, §143. 
 pyramid, line of intersection between, to determine, (Prob. 69) p. 132, 
 § 141; (Prob. 73) p. 137, § 145- ^ 
 angle a given line makes with a, (Prob. 31) p. 78, § 91. 
 
 between traces of, to determine, (Prob. 28) p. 73, § 86. 
 makes with V or H, to determme, (Prob. 27) p. 72, § 85. 
 
222 INDEX 
 
 Plane, containing circle projected in third angle, (Prob. 35) p. 85, § 98. 
 point which is projected, (Cor.) p. 11, § 19. 
 projection of a line, the projection and the plane make the 
 
 same angle with the line, (Prin.) p. 78, § 90. 
 straight line parallel to any other straight line is parallel to that 
 Une, (Prin.) p. 57, § 65. 
 curve, isometric drawing of, p. 159, § 161. 
 cutting, definition of, p. 119, § 131. 
 
 figure, third angle plan and elevation of, to determine, (Prob. 36) p. 88, § 99. 
 projection of, to determine, (Prob. 34) p. 83, §.95. 
 to determine third angle projection of, (Prob. 34) p. 83, § 95; (Prob. 
 36) p. 88, § 99. 
 traces of plane containing it, (Prob. 16) p. 55, § 63. 
 true form of, from its projections, Prob. 33) p. 81, § 93. 
 geometric properties of, p. 2, § 3. 
 
 given, to determine angle it makes with V or H, (Prob. 27) p. 72, § 85. 
 having traces respectively perpendicular to projections of a line is per- 
 pendicular to the Une, (Theo. VTH) p. 65, § 79. 
 horizontal trace of, p. 49, § 56. 
 
 in which a line is parallel to V or H, to project the line, p. 59, § 70. 
 intersection of, with cone, p. 134, § 142; p. 138, § 146. 
 
 cyUnder, p. 131, § 140; p. 136, § 144. 
 prism, p. 134, § 143. 
 pyramid, p. 132, § 141; p. 137, § 145. 
 line drawn through a given point and perpendicular to a, to determine the 
 
 distance between the point and plane, (Prob. 24) p. 68, § 80. 
 line which is projected on, (Prob. 25) p. 69, § 82. 
 new vertical, p. 13, § 22. 
 
 notation for, p. 14, § 22. 
 obUque, on which a Une is assumed, (Prob. 12) p. 51, § 59. 
 
 to project a given straight line on, (Prob. 25) p. 69, § 82. 
 of projection, horizontal, p. 8, § 9. 
 vertical, p. 8, § 10. 
 principal, p. 8, § 8. 
 revolution, p. 35, § 46. 
 rotation, (Exp. Ill) p. 33, § 44. 
 on which a pyramid rests, to determine third angle projection .of pyramid, 
 
 (Prob. 42) p. 97, § 107. 
 other than V or H, notation for projection of point on, p. 69, § 81. 
 paraUel to a given plane and to contain a given point, (Prob. 19) p. 58, 
 
 §68. 
 passed through a given point and paraUel to two given Unes, (Prob. 18) 
 P- 57, § 67. 
 straight Une and parallel to another given straight 
 Une, (Prin.) p. 57, § 65; (Prob. 17) p. 57, § 66. 
 a Une and a point without the line, (Prob. 15) p. 55, § 62. 
 three given points, (Prob. 14) p. 55, § 61. 
 two intersecting lines, (Prob. 13) p. 54, § 60. 
 perpendicular to line of intersection of other planes, p. 65, § 78. 
 piercing point of a line on a, to determine, (Prob. 23) p. 64, § 76. 
 point assumed in an obUque, to determine, p. 53, § 59. 
 positions it may occupy, p. 50, § 57. 
 profile, p. 31, § 42. 
 projection of a line on any obUque, (Prob. 25) p. 69, § 82. 
 
 point on a, (Cor.) p. 11, § 19. 
 section cut from a cone by a, p. 120, § 131; (Prob. 65) p. 124, § 136. 
 
 cyUnder by a, p. 119, § 131; p. 122, § 134. 
 section cut from a pyramid by a, to determine, (Prob. 64) p. 124, § 135. 
 sphere by a, p. 120, § 131. 
 
 torus by a, to determine, (Prob. 66) p. 126, § 137. 
 specifying a, p. 15, § 23. 
 
INDEX 223 
 
 Plane, tangent to a cone and parallel to given straight line, (Prob. 54) p. 11 1, § 123. 
 at a given point on the cone surface, (Prob. 51) p. 108, 
 
 § 120. 
 through a point without the surface of the cone, (Prob. 
 52) p. 109, § 121. 
 cylinder at a given point on the cylinder surface, (Prob. 55) 
 p. 112, § 124. 
 through a given point without the surface of the cylin- 
 der, (Prob. 56) p. 112, § 125. 
 sphere and through any given straight line, (Prob. 58) p. 115, 
 § 127. 
 at a given point on the surface of the sphere, (Prob. 
 57) P- 114, § 126. 
 surface, p. 107, § 119. 
 to assume a line in, (Prob. 12) p. 51, § 59. 
 
 to contain a given Une and make a given angle with H, to detertnine its 
 traces, (Prob. 53) p. no, § 122. 
 point and be parallel to a given plane, (Prob. 19) p. 58, 
 
 § 68. . 
 
 point and be perpendicular to a given straight line, (Prob. 
 
 26) p. 69, § 83. 
 point and make given angles with V and H, to determine 
 its traces, (Prob. 30) p. 77, § 89. 
 a line and a point, (Prob. 15) p. 55, § 62. 
 any plane figure, to determine traces of, (Prob. 16) p. 55, § 63. 
 three given points, (Prob. 14) p. 55, § 61. 
 to intersect a cone, to determine section cut, (Prob. 65) p. 124, § 136. 
 
 prism, and to determine line of intersection, (Prob. 67) p. 130, 
 
 §139- 
 trace of, to determine, p. 49, § 55; p. 55, §60. 
 traces of a line on, to determine, (Prob. 23) p. 64, § 76. 
 
 contain piercing points of all -lines on the plane, (Theo. V) p. 51, 
 §58. 
 true angle between the traces of, to determine, (Prob. 28) p. 73, § 86. 
 vertical trace of, p. 49, § 55. 
 Planes, angle between two intersecting, how measured, p. 72, § 84. 
 coordinate, p. 154, § 152. 
 designation of, p. 49, § 56. 
 
 intersect in a straight line common to all of them, p. 59, § 71. 
 intersecting, angle between two, to determine, (Prob. 29) p. 75, § 87. 
 
 how to measure angle between, p. 72, § 84. 
 line of intersection between, projection of, (Theo. VII) p. 58, § 69. 
 
 two, (Prob. 21) p. 62, § 74; (Prob. 29) p. 75, 
 
 §87. 
 to determine. (Prob. 20) p. 60, § 72; 
 
 (Prob. 21) p. 62, § 74; (Prob. 22) 
 
 P- 64, § 75- 
 notation and line convention for traces of, p. 49, § 56. 
 of projection, principal, p. 8, § 8. 
 
 parallel, have their respective traces parallel, (Theo. VI) p. 55, § 64. 
 revolution of, p. 35, § 46. 
 
 traces of, line convention and notation for, p. 49, § 50- 
 two, at right angles and which intersect a third plane, p. 65, § 78. 
 unfolding or opening the, p. 17, § 24. 
 which are parallel have their respective traces parallel, (Theo. Vi; 
 
 Planes which are parallel intersect a third plane m parallel hnes, (Prin.) p. 61, § 73. 
 intersect are perpendicular to a third plane if line of intersection of 
 
 first two planes is perpendicular to the third plane, (Prin.) p. 65, 
 
 §78. 
 Plates, layout of, in this work (,see Fig. 154), p. 162, § 166. 
 
224 INDEX 
 
 Plates, numbering of, in this work, p. 162, § 166. 
 
 set of drawing, in this work, p. 165, § 167 to § 187 inclusive. {See also 
 
 Chapter X, Table of Contents, p. xv.) 
 size of, in this work {see Fig. 154), p. 162, § 166. 
 Point and direction of a line locate the line, (Prin.) p. 25, § 30. 
 line, plane passed through a, (Prob. 15) p. 55, § 62. 
 plaiie, distance between, (Prob. 24) p. 68, § 80. 
 assumed in an oblique plane, to determine, p. 53, § 59. 
 
 on surface of an hyperboloid of revolution, (Prob. 59) p. 116, § 129. 
 geometric properties of, p. i, § 3. 
 
 horizontal piercing, of a line, p. 20, § 27; (Prob. 3) p. 28, § 34. 
 projection of, p. 2, § 5; p. 3, § 5. 
 in space, how projected, p. 5, § 7. 
 
 isometric projection of, (Prob. 80) p. 155, § 154; (Prob. 81) p. 156, § 155. 
 line drawn through a, and perpendicular to a given plane, also to determine 
 the distance from the point to the plane, (Prob. 24) p. 68. § 80. 
 passed through a, and parallel to a given line, p. 31, § 43. 
 new position of, p. 6, § 7. 
 
 vertical projection of, (Prob. 2) p. 13, § 22. 
 notation for, p. 13, § 21. 
 
 horizontal projection of, p. 18, § 25. 
 projection of, on plane other than V or H, p. 69, § 81. 
 vertical projection of, p. 18, § 25. 
 number of vertical projections possible, p. 3, § 5. 
 of tangency, p. 107, § 119. 
 
 on cone, and through which a plane is passed tangent to the cone, (Prob. 
 51) p. 108, § 120. 
 cylinder, and through which a plane is passed tangent to a cylinder, 
 
 (Prob. 55) p. 112 §124. 
 plane, projections of, (Cor.) p. 11, § 19. 
 
 sphere, and through which a plane is passed tangent to the sphere, 
 (Prob. 57) p. 114, § 126. 
 path traced by a, convention for, p. 19, § 26. 
 piercing, of a given straight line on a given plane, to determine, (Prob. 23) ' 
 
 p. 64, § 76. 
 position of eye when projecting a, p. 16, § 24. 
 
 how determined, (Theo. I) p. 10, § 17; (Prob. 1, p. 12, § 21. 
 possible positions of, p. II, § 20. 
 
 projected on plane other than V or H, notation for projections, p. 69, § 81. 
 projection of, shown in each dihedral angle, p. 17, § 24. 
 projections of, are on same straight line perpendicular to G-L, (Cor.) p. 1 1,§ 18. 
 revolution of, about any line, (Exp. Ill) p. 32, § 44; p. 38, § 47. 
 
 through a given angle, p. 35, § 47. 
 revolved about axis, (Prob. 6) p. 38, § 48; p. 39, § 49. 
 
 position of, p. 6, § 7. 
 specifying a, p. 15, § 23. 
 
 through which a line is drawn to intersect a given line at a given angle, 
 (Prob. 32) p. 80, § 92. 
 plane is passed and parallel to two given straight lines, 
 (Prob. 18) p. 57, § 67. 
 the traces of which plane make given 
 angles with V and H, (Prob. 30) p. 77, 
 §89. 
 vertical piercing, of a line, p. 20, § 27; (Prob. 3) p. 26, § 34, 
 vertical projection of, p. 3, § 5; p. 8, § 13. 
 
 projections of, number possible, p. 3, § 5. 
 which shall be contained by a plane passed parallel to a given plane, (Prob. 
 19), P- 58, § 68. 
 in a plane perpendicular to a given straight line, 
 (Prob. 26) p. 69, § 83. 
 
INDEX 225 
 
 Point, without the surface of a cone and through which a plane is passed tangent to 
 
 the cone, (Prob. 52) p. 109, § 121. 
 cylinder, and through which a plane is passed tan- 
 gent to the surface of the cylinder, (Prob. 56) 
 p. 112, §125. 
 Points, distance between two, to determine, (Prob. 8) p. 40, § 50. 
 in space, p. 5, § 6. 
 
 notation for, p. 18, § 25. 
 of a line project on the projections of the line, (Theo. H) p. 25, § 32. 
 on drawings, location of) in this work, p. 164, § 166. 
 
 piercing, lie in the traces of plane containing the line, (Theo. V) p. 51, § 58. 
 revolution of, p. 33, § 44; p. 35, § 46. 
 three given, to contain a plane, (Prob. 14) p. 55, § 61. 
 to determine distance between two, (Prob. 8) p. 40, § 50. 
 to pass a plane through three given, (Prob. 14) p. 55, § 61. 
 true distance between two, (Cor.) p. 29, § 36; p. 42, § 51. 
 two, are sufficient to locate a line, (Prin.) p. 25, § 30. 
 vertical projection of, p. 8, § 13. 
 
 notation for, p. 18, § 25. 
 Position, definition of, p. 2, § 4. 
 
 new, of a point, p. 6, § 7. 
 of eye when projecting a point, p. 16, § 24. 
 point, how determined, (Theo. I) p. 10, § 17. 
 line, two projections required to define, (Prin.) p. 25, § 29. 
 Positions line may occupy, p. 22, § 28. 
 plane may occupy, p. 50, § 57. 
 point may occupy, p. 11, § 20. 
 Possible positions of point, p. II, § 20. 
 Practical value of Descriptive Geometry, p. I, § 2. 
 Principal planes of projection, p. 8, § 8. 
 
 Principles, p. 25, § 29, § 30 and § 31; p. 26, § 33; p. 57, § 65; p. 61, § 73; p. 65, 
 § 78; p. 72, § 84; p. 77, § 88; p. 78, § 90; p. 85, § 97. {See also 
 statement of principle under proper Index heading, and see Table 
 of Contents.) 
 and definitions, p. 49, § 55. 
 of the line, p. 20, § 27. 
 
 vmderlying isometric projection, p. 155, § 153. 
 Prism, a right, p. 100, § 112. 
 
 altitude of, p. loO; § 112. 
 
 and plane, line of intersection between, (Prob. 67) p. 130, § 139; (Prob. 71) 
 p. 134, § 143. . /„ , N 
 
 pyramid, line of intersection between, to determine, (Prob. 77) p. 144, 
 
 § 149- 
 definition of, p. 100, § 112. 
 development of, p. 129, § 138; (Prob. 67) p. 130, § 139; (Prob. 71) p. 134, 
 
 § 143; (Prob. 75) p. 138, § 147; (Prob. 77) p. 144, § 149- 
 hexagonal, third angle plan and elevation of, (Prob. 46) p. loi, § 113; 
 
 (Prob. 47) p. 102, § 114. 
 oblique, p. 100, § 112. ^ 
 
 projection of, (Prob. 46) p. loi, § 113; (Prob. 47) p. 102, § 114. 
 to determine section of, (Prob. 61) p. 120, § 132; (Prob. 62) p. 122, § 133. 
 Prisms, intersection between two, (Prob. 75) p. 138, § 147. 
 
 Problems {see statement of problem under proper Index heading, also see Table 
 of Contents), 
 in Descriptive Geometry, data for, p. 15, § 23. 
 Problems, to "visualize," p. I, § 2. 
 Profile plane, p. 31, § 42. 
 Projecting a point in space, p. 5, § 7. 
 Projection, cabinet or cavalier, p. 160, § 164. 
 horizontal plane of, p. 8, § 9. 
 
226 INDEX 
 
 Projection, isometric, (Chap. IX) p. 153. 
 
 of any plane curve, (Prob. 87) p. 159, § 161. 
 circle, (Prob. 86) p. 159, § 160. 
 cube, (Prob. 84) p. 157, § 158. 
 mortise and tenon, (Prob. 88) p. 159, § 162. 
 piping system, (Prob. 89) p. 159, § 163. 
 point, (Prob. 80) p. 155, § 154; . (Prob. 81) p. 156, § 155. 
 pyramid, p. 158, § 159. 
 square figure, (Prob. 83) p. 157, § I57. 
 straight Une, to determine, (Prob. 82) p. 156, § 156. 
 principles underlying, p. 155, § 153. 
 of a circle, to determine, (Prob. 35) p. 85, § 97. 
 cone, to determine, (Prob. 45) p. 100, §111. 
 cube, to determuie, (Prob. 37) p. 90, § loi; (Prob. 38) p. 91, § 102; 
 
 (Prob. 39) p. 02, § 103; (Prob. 40) p. 94, § 104. 
 cyUnder, (Prob. 48) p. 103, § 115; (Prob. 49) p. 104, § 116. 
 given straight line on a given obUque plane, (Prob. 25) p. 69, § 82. 
 line, and of intersecting lines, (Exp. II) p. 20, § 27. 
 horizontal, p. 20, § 27. 
 
 notation for, p. 18, § 25. 
 in its true length, p. 6, § 7; (Theo. Ill) p. 28, § 35. 
 in plane parallel to V or H, p. 59, § 70. 
 making a given angle with V and H, to determine, (Prob. 
 
 II) p. 46, § 54. 
 maximum, p. 7, § 7. 
 minimum, p. 7, § 7. 
 of intersection of obUque plane, and plane parallel to H, 
 
 (Theo. VII) p. 58, § 69. 
 on a plane makes same angle with the line that the plane 
 
 makes with the line, (Prin.) p. 78, § 90. 
 perpendicular to plane, (Theo. VIIl) p. 65, § 79. 
 vertical, p. 20, § 27. 
 
 notation for, p. 18, § 25. 
 plane figure, (Prob. 36) p. 88, § 99. 
 
 from which the true form is to be determined, (Prob. 
 33) P- 81, § 93- 
 point, horizontal, p. 2, § 5; p. 3, § 5! P- 10, § 14- 
 in each dihedral angle, p. 17, § 24. 
 new vertical, (Prob. 2) p. 13, § 22. 
 horizontal, notation for, p. 18, § 25. 
 on a plane, (Cor.) p. 11, § 19. 
 
 other than V or H, notation for, p. 69, § 81. 
 to determine, (Prob. l) p. 12, § 21. 
 vertical, p. 3, § 5; p. 8, § 13. 
 
 notation for, p. 18, § 25. 
 prism, (Prob. 46) p. loi, § 113; (Prob. 47) p. 102, § 114. 
 pyramid, (Prob. 41) p. 95, § 106; (Prob. 42) p. 97, § 107; (Prob. 
 
 43) P- 98., § 108; (Prob. 44) p. 98, § 109. 
 screw thread, p. 105, § 117; p. 106, § 118. 
 solid, p. 90, § 100. 
 straight line, (Prin.) p. 25, § 31. 
 
 on a given oblique plane, (Prob. 25) p. 69, § 82. 
 torus, (Prob. 60) p. 117, § 130. 
 V screw thread, (Prob. 50) p. 106, §118. 
 hyperboloid of revolution, (Prob. 59) p. 116, § 129. 
 Projection, of points on a line fall on the projections of tiie line, (Theo. II) p. 25, § 32. 
 pyramid, (Prob. 85) p. 158, § 159. 
 orthographic, p. 5, § 5; p. 10, § 16. 
 perpendicular, p. 5, § 5. 
 principal plane of, p. 8, § 8. 
 
INDEX 227 
 
 Projection, third angle of a circle, (Prob. 35) p. 85, § 98. 
 
 any plane figure, to determine, (Prob. 34) p. 83, § 95. 
 of trace of a plane, p. 49, § 55. 
 vertical, of a point, p. 8, § 13. 
 plane of, p. 8, § 10. 
 Projections, corresponding, of parallel lines are parallel, (Theo. IV) p. 29, § 40. 
 
 of a line perpendicular to a plane are respectively perpendicular to 
 the traces of the plane, (Theo. VIID p. 65, §-79. 
 a point are on the same straight line perpendicvdar to G-L, (Cor.) 
 p. II, § 18. 
 on a plane, (Cor.) p. 11, § 19. 
 vertical, number possible, p. 3, § 5. 
 aU points on a line fall respectively on the projections of the line, 
 
 (Theo. II) p. 25, § 32. 
 lines intersecting at right angles, p. 34, § 45. 
 required to define the position of a line, (Prin.) p. 25, § 29. 
 Projector, nded, p. 10, § 15; p. 12, § 21. 
 
 convention for, p. 18, § 26. 
 space, p. 8, § 12; p. 10, § 15. 
 Properties of data in Descriptive Geometry, p. I, § 3. 
 line, geometric, p. i, § 3. 
 plane, geometric, p. 2, § 3. 
 soHd, geometric, p. 2, § 3. 
 Property, geometric, of a point, p. i, § 3. 
 Pseudoperspective, p. 160, § 165. 
 Pyramid, altitude of, p. 95, § 105. 
 
 and plane, line of intersection between, (Prob. 69) p. 132, § 141; (Prob, 
 
 73) P- 137, § 145- 
 prism, Une of intersection between, (Prob. 77) p. 1 44, § 149. 
 apex of, p. 95, § 105. 
 axis of, p. 95, § 105. 
 definition of, p. 95, § 105. 
 development of, p. 129, § 138; (Prob. 69) p. 132, § 141; (Prob. 73) 
 
 P- 137, § 145; (Prob- 77) P- 144. § 149- 
 isometric projection of, (Prob. 85) p. 158, § 159. 
 inclined to V and H, projection of, (Prob. 41) p. 95, § 106. 
 obUque, p. 95, § 105. 
 
 third angle plan and elevation of, (Prob. 42) p. 97, § 107; (Prob. 
 43) P- 98,^ § 108; (Prob. 44) p.'98, § 109. 
 projection of, (Prob. 42) p. 97, § 107; (Prob. 43) p. 98, § 108; (Prob. 44) 
 
 p. 98, § 109. 
 right, p. 95, § 105. 
 section cut by a -plane from, (Prob. 64) p. 124, § 135. 
 
 R. 
 Radius of revolution, p. 35, § 46. 
 
 rotation, (Exp. Ill) p. 34, § 44. 
 Ray of light, p. 5, § 7. 
 
 Reference, objects of, in Descriptive Geometry, p. 8, § 8. 
 Revolution, arc of, p. 35, § 46. 
 axis of, p. 35, § 46. 
 center of, p. 35, § 46. 
 circle of, p. 35, § 46. 
 cone of, p. 45, § 53. 
 
 double curved surfaces of, intersection between, p. 151, § 151. 
 hyperboloid of, p. 116, § 129. 
 
 of a point about a line in a plane of projection, p. 38, § 47. 
 any axis, fProb. 7} p. 39, § 49. 
 line, (Prob. 6) p. 38, § 48. 
 through a given angle, etc., (Prob. 5) p. 35, § 47. 
 
228 INDEX 
 
 Revolution, of lines, p. 35, § 46. 
 planes, p. 35, § 46. 
 points, p. 35, § 46. 
 solids, p. 35, § 46. 
 plane of, p. 35, §'46. 
 radius of, p. 35, § 46. 
 Revolved position of a point, p. 6, § 7. 
 Right circular cone, p. 45, § 53; p. icx), § no. 
 base angle of, p. 46, § 53. 
 cone and plane, intersection between, (Prob. 65) p. 124, § 136. 
 cylinder and a plane, intersection between, p. 131, § 140. 
 prism, p. 100, § 112. 
 pyramid, p. 95, §105. 
 
 square prism, to determine any section of, (Prob. 61) p. 120, § 132. 
 Rotation, axis of, (Exp. Ill) p. 33, § 44. 
 
 of point about a line, (Exp. Ill) p. 32, § 44. 
 plane of, (Exp. III^ p. 33, § 44. 
 radius of,(Exp. Ill) p. 33, § 44. 
 Ruled projector, p. 10, § 15; p. 12, § 21. 
 
 convention for, p. 18, § 26. 
 
 S. 
 Scale, isometric, p. 154, § 152. 
 Screw thread, projection of, p. 105, § 117. 
 
 V thread, projection of, (Prob. 50) p. 106, § 118. 
 Second angle, the, p. 16, § 24. 
 Section cut from a cone by a plane, p. 120, § 131. 
 
 to determine, (Prob. 65) p. 124, § 136. 
 pyramid by a plane, to determine, (Prob. 64) p. 124, § 135. 
 right cone by a plane, to determine, (Prob. 65) p. 124, § 136. 
 right cylinder by a plane, (Prob. 63) p. 122, § 134. 
 sphere by a plane, p. 120, § 131. 
 
 torus by a plane, to determine, (Prob. 66) p. 127, § 137. 
 annular torus by plane, to determine, (Prob. 66) p. 127, § 137. 
 cylinder by plane, p. 119, § 131. 
 
 to determine, (Prob. 63) p. 122, § 134. 
 definition of, p. 119, § 131. 
 longitudinal, p. ^20, § 131. 
 oblique, p. 120, § 131. 
 
 of prism, to determine, (Prob. 61) p. 120, § 132; (Prob. 62) p. 122, § 133. 
 transverse, p. 120, § 131. 
 Set of drawing exercises in Descriptive Geometry, p. 162, § 166 to § 187 inclusive. 
 {See also Chapter X, Table of Contents, p. xv.) 
 exercises in this work, p. 162, § 166 to § 187 inclusive. (See also Chapter X, 
 Table of Contents, p. xv.) 
 Sight, direction of, convention for, p. 19, § 26. 
 Size of plates in this work {see Fig. 154), p. 162, § 166. 
 Solid, definition of, p. 90, § 100. 
 
 development of, p. 129, § 138. 
 geometric properties of, p. 2, § 3. 
 how specified, p. 15, § 23. 
 properties of, p. 90, § 100. 
 revolution of a, p. 35, § 46. 
 Space projector, p. 8, § 12; p. 10, § 15. 
 Sphere, great circle of a, p. 120, § 131. 
 
 plane passed tangent to a, and through a given straight line, (Prob. 58) 
 
 p. 115, § 127. 
 plane passed tangent to a, and through a point on the sphere surface, 
 
 (Prob. 57) p. 114, § 126. 
 section cut from, by a plane, p. 120, § 131. 
 
INDEX 229 
 
 Square, isometric projection of, p. 157, § 157. 
 
 Straight line, projection of, to determine, (Prin.) p. 25, § 31. 
 
 Style of letters in this work, p. 163, § 166. 
 
 numerals in this work, p. 163, § 166. 
 Surface, conical, p. 100, § no. 
 
 development of, p. 129, § 138. 
 
 double curved, p. 129, § 138. 
 
 line normal to a, p. 108, § 119. 
 tangent to a, p. 107, § 119. 
 
 of a prism, development of, (Prob. 67) p. 130, § 139. 
 
 plane tangent to a, p. 107, § 119. 
 Surfaces of revolution, double curved, intersection between, p. 151, § 151. 
 
 T. 
 Tangency, element of, p. 108, § 119. 
 
 point of, p. 107, § 119. 
 Tangent, line, to a surface, p. 107, § 119. 
 
 plane, to a cone and parallel to a given straight line, (Prob. 54) p. in, 
 §123. 
 cone and through a given point on a cone surface, p. 108, 
 
 § 120. 
 cone and through a given point without the cone surface, 
 
 (Prob. 52) p. 109, § 121. 
 cylinder and through a given point without the cylinder sur- 
 face, (Prob. 56) p. 112, § 125. 
 cylinder and through a given point on the cylinder surface, 
 
 (Prob. 55) p. 112, § 124. 
 sphere and through a given straight line, (Prob. 58) p. 115, 
 
 § 127. 
 sphere at a given point on the surface of the sphere, (Prob. 
 
 57) P- "4, § 126. 
 surface, p. 107, § 119. 
 Theorem I. Position of a point, to determine, p. 10, § 17. 
 
 n. Projections of every point of a line fall on projections of the Une, p. 25 , 
 
 §32. 
 
 III. True length projection of a hne, p. 28, § 35. 
 
 IV. Parallel lines have corresponding projections parallel, p. 29, § 40. 
 
 V. Traces (or piercing points) of a line Ue in traces of a plane containing 
 it, p. 51, § 58. 
 VI. Parallel planes have then:, respective traces parallel, p. 55, § 64. 
 VII. Line of intersection of obUque plane and plane parallel to plane of 
 
 projection — how it projects, p. 58, § 69. 
 VIII. Line perpendicular to a plane projects respectively perpendicular to 
 ihe traces of the plane, p. 65, § 79. 
 Third angle plan and elevation of a cube, to determine, (Prob. 37) p. 90, § loi ; 
 
 (Prob. 38) p. 91, § 102; (Prob. 39) p. 92, 
 § 103; (Prob. 40) p. 94, § 104. ^ 
 a cyhnder, (Prob. 48) p. 103, § 115; (Prob. 49) 
 
 p. 104, § 116. 
 a plane figure included to V and H, to determine, 
 
 (Prob. 36) p. 88, §99- , , ^ 
 
 hexagonal prism, (Prob. 46) p. loi, § 113; (Prob. 
 
 47) p. 102, § 114. 
 hyperboloid of revolution, (Prob. 59) p. 116, 
 
 § 129- 
 obUque hexagonal pyramid, (Prob. 43) p- 98, 
 
 § 108. _ 
 
 right hexagonal pyramid, to determine, (Prob. 
 41) p. 95, § 106. 
 pentagonal pyramid, (Prob. 44) p. 98, § 109. 
 
230 INDEX 
 
 Third angle plan and elevation of right pyramid resting on oblique plane, (Prob. 
 
 42) P-97, §107- 
 torus, (Prob. 60) p. 117, § 136. 
 projection of a circle which lies in a given plane, (Prob. 35) p. 85, 
 §98. 
 plane figure, to determine, (Prob. 34) p. 83, § 95; (Prob. 
 36) p. 88, § 99. 
 the, p. 16, § 24. 
 Thread, screw, projection of, p. 105, § 117. 
 
 V screw, projection of, (Prob. 50) p. 106, § 118. 
 Title form used in this work {see Fig. 155) p. 163, § 166. 
 Torus,- p. 116, § 128. 
 
 annular, cut by a plane, to determine section, (Prob. 66) p. 127, § 137. 
 projection of, (Prob. 60) p. 117, § 130. 
 Trace of plane, horizontal, p. 49, § 56. 
 
 vertical, p. 49, § 56. 
 Traces of a hne, to determine, (Prob. 3) p. 26, § 34. 
 
 on a plane, to determine, (Prob. 23) p. 64, § 76. 
 plane, p. 49, § 55. 
 
 angle between, (Prob. 28) p. 73, § 86. 
 
 contain the piercing points (or traces) of lines on the plane, 
 
 (Theo. V) p. 51, § 58. 
 to determine, p. 55, § 60. 
 
 true angle between, to determine, (Prob. 28) p. 73, § 86. 
 when respectively perpendicular to projections of a line — then 
 plane and line are perpendicular to each other, (Theo. VIII) 
 
 .P-65, §79- 
 which is passed through a given point and to make given angles 
 with V and H, (Prob. 30) p. 77, § 89. 
 is to contain any plane figure, to determine, (Prob. 16) 
 
 P- 55> § 63. 
 , shall contain a given line and make a given angle with 
 
 H, to determine, (Prob. 53) p. no, § 122. 
 of planes, notation and line conventions for, p. 49, § 56. 
 
 when parallel — then the planes are parallel, (Theo. VI) p. 55, 
 
 of parallel planes are respectively parallel, (Theo. VI) p. 55, § 64. 
 (or piercing points) of a line faU in the traces of a plane containing it, 
 (Theo. V) p. 51, § 58. 
 Transverse section, p. 120, § 131. 
 True angle between the traces of a plane, to determine, (Prob. 28) p. 73, § 86. 
 
 two intersecting Enes, (Cor.) p. 29, § 38; (Prob. 9) p. 41, § 51. 
 distance between two points, (Cor.) p. 29, § 36; p. 42, § 51. 
 
 to measure off, on a line, (Cor.) p. 29, § 39. 
 form of intersection between a prism and a plane, to determine; also to de- 
 velop the prism,) Prob. 67) p. 130, § 139. 
 right cylinder and a plane, (Prob. 68) p. 131, 
 § 140. 
 plane figure, determined from its projections, (Prob. 33) p. 81, § 93. 
 length, Ime projected in, (Theo. Ill) p. 28, § 35. 
 notation for, p. 41, § 50. 
 projection of line, p. 6, § 7; (Theo. Ill) p. 28, § 35. 
 
 U. 
 Unfolding or opening out the planes of projection, p. 17, § 24. 
 
 V 
 Vertical piercing point (or trace) of a line, p. 20, § 27; (Prob. 3) p. 26, § 34. 
 plane, new, p. 13, § 22. 
 
 notation for, p. 14, § 22. 
 
INDEX 231 
 
 Vertical plane, of project; .n, p. 8, § 10. 
 projection of a line, p. 20, § 27. 
 
 notation for, p. 18, § 25. 
 point, p. 3, § 5; p. 8, § 13. 
 
 new, (Prob. 2) p. 13, § 22. 
 notation for, p. 18, § 25. 
 projections of the same point, number possible, p. 3, § 5. 
 trace of a plane, p. 49, § 55 and § 56. 
 Visible line, convention for, p. 18, § 26. 
 
 "Visualizing" a Descriptive Geometry problem or exercise, p. i, § 2. 
 V screw thread, projection of, (Prob. 50) p. 106, § 118. 
 
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