Descriptive Geometry
This Descriptive Geometry is addressed to foreign students at Warsaw University of
Technology, especially to the first year students of Civil Engineering Faculty and
Environmental Engineering Faculty. It is based on courses that the author has taught for a
number of years at the school. Thus, this textbook is intended to meet the requirements of
the syllabuses of these courses that contain a presentation of principal geometric methods
to give a one-to-one (then reversible) representation of 3D-space in a plane. The aim of
the courses has been to give a working knowledge of the engineer’s language: how to make
and how to read drawings; to become familiar with presented methods and acquire the
ability to specify their use with assurance as well. Skills to be achieved are:
proficiency in operating 3-dimensional systems and drawing them using some advanced
projection methods.
PREFACE . 9
NOTATION . 11
INTRODUCTION 13
Chapter 1. PRELIMINARY TO DESCRIPTIVE GEOMETRY 17
1.1. VOCABULARY AND DEFINITIONS . 17
1.1.1. Plane Geometry (Planimetry) . 18
1.1.2. Space Geometry (Stereometry) 24
1.2. SOME GEOMETRIC CONSTRUCTIONS 27
1.3. PROJECTION. INFINITY. IDEAL POINTS 32
1.4. POINT TRANSFORMATIONS OF PLANE 34
1.4.1. Plane Isometries 34
1.4.2. Plane Similarities 37
1.4.3. Affi nity . 38
EXERCISES . 42
Chapter 2. PARALLEL PROJECTION 44
2.1. FUNDAMENTAL CONCEPTS 44
2.2. INVARIANTS . 45
2.3. AXONOMETRIC PROJECTION . 48
2.3.1. Defi nitions and Basic Properties 48
2.3.2. Associated Axonometric Systems 62
EXERCISES . 63
Chapter 3. ORTHOGONAL PROJECTION, MONGE’S PROJECTIONS . 64
3.1. PROPERTIES OF THE ORTHOGONAL PROJECTION 64
3.2. REPRESENTATIONS OF POINTS, LINES AND PLANES . 67
3.2.1. Representation of Point 68
3.2.2. Representation of Line . 69
3.2.3. Representation of Plane 70
3.2.4. Incidence Relation 71
3.2.5. Determination of Visibility . 73
3.2.6. Multi-view Projections . 77
3.3. SPECIAL POSITIONS OF LINES AND PLANES 78
3.3.1. Special Positions of Straight Lines . 78
3.3.2. Special Positions of Planes – Projecting Planes . 80
3.4. COMMON ELEMENTS – INTERSECTIONS 81
3.4.1. Intersection of Projecting Planes 82
3.4.2. Intersection of Line and Plane in General Position 86
3.4.3. Intersection of Planes in General Position 93
3.4.4. Intersection of Polyhedra 95
EXERCISES . 97
3.5. TRANSFORMATION OF THE PROJECTION SYSTEM (?1, ?2) . 98
3.5.1. Principle of Transformation 98
3.5.2. Application of Transformation . 99
EXERCISES . 106
3.6. REVOLUTIONS AND RABATMENTS 106
3.6.1. Rabatment of Line 107
3.6.2. Rabatment of Vertical Plane (side rabatment) 108
3.6.3. Rabatment of Plane in General Position . 109
3.6.4. Dihedral Angle 113
EXERCISES . 114
3.7. ROOFS . 115
3.7.1. Roof Terminology . 115
3.7.2. Geometric Solution of Roof . 116
3.7.3. Roofs on Not Detached Buildings 119
3.8. ORTHOGONAL AXONOMETRIC PROJECTION . 122
3.8.1. Basic Properties 122
3.8.2. Isometric Projection 124
3.8.3. Isometric Drawing 124
3.9. PICTORIAL DRAWING . 126
EXERCISES . 128
Chapter 4. SURFACES . 129
4.1. SURFACES OF REVOLUTION . 129
4.1.1. Defi nitions 129
4.1.2. Representation in Orthographic Projections . 130
4.1.3. Location of Point in Surface of Revolution 131
4.2. INTERSECTION OF SURFACES OF REVOLUTION BY PLANES AND LINES IN
GENERAL POSITION 135
4.2.1. Intersection of Right Circular Cones and Planes – Conic Sections 135
4.2.2. Representation of Smooth Conics by Orthographic Projections 137
4.2.3. Construction of Smooth Conics 141
4.2.4. Intersection of Spheres and Planes . 144
4.2.5. Intersection of Cylinders and Planes 146
4.2.6. Piercing of Surface of Revolution by Line in General Position . 148
4.3. DEVELOPMENTS OF SURFACES OF REVOLUTION 152
4.3.1. Some Methods of Approximation of Length of Circle Arc 152
4.3.2. Development of Truncated Right Circular Cylinder 153
4.3.3. Development of Truncated Right Circular Cone 154
4.3.4. Geodesics 155
4.4. INTERSECTION OF SURFACES OF REVOLUTIONS . 157
4.4.1. Methods of Intersection of Surfaces of Revolution 157
4.5. REDUCIBILITY OF INTERSECTION LINE OF TWO SECON-ORDER SURFACES .165
4.5.1. Properties 165
4.5.2. Reducibility of Intersection Line of Two Second-order Surfaces 166
4.5.3. Applications. Cylindrical vaults. Conical connection 168
4.6. TANGENCY 173
4.6.1. Plane Tangent to Surface of Revolution . 173
4.5.2. Mutual Tangency of Surfaces of Revolution 175
EXERCISES . 176
4.7. RULED SURFACES . 177
4.7.1. Basic Notions 177
4.7.2. Warped Surfaces . 180
4.7.3. Construction of Elements of Warped Surface 182
4.7.4. Applications 188
EXERCISES . 188
Chapter 5. MAP (TOPOGRAPHIC) PROJECTION 189
5.1. REPRESENTATIONS OF POINTS, LINES AND PLANES . 189
5.1.1. Basic Notions 189
5.1.2. Mutual Position of Lines and Planes. Parallelism and Perpendicularity 199
5.1.3. Remarks Concerning Specifi cation of Plane by Strike and Dip . 201
5.2. COMMON ELEMENTS . 202
5.2.1. Intersections of Planes and Lines . 202
5.2.2. Intersections of Topographic Surfaces and Planes . 206
5.3. NORMAL VIEW OF A PLANE . 207
5.4. APPLICATIONS . 209
EXERCISES . 211
Chapter 6. CENTRAL PROJECTION – PERSPECTIVE 214
6.1. FUNDAMENTAL CONCEPTS 214
6.1.1. Defi nitions and Terminology 214
6.1.2. Representation of Straight Line 216
6.1.3. Representation of Plane 219
6.1.4. Angle of Inclination to Projection Plane 221
6.1.5. Relations between Points Lines and Planes 223
6.1.6. Common Elements – Intersection of Lines and Planes 227
EXERCISES . 229
6.2. PERSPECTIVE DRAWING – VERTICAL PERSPECTIVE . 230
6.2.1. Ground Plane 230
6.2.2. Angle between Two Lines in ? . 232
6.2.3. Measurement of Lines in Ground Plane . 233
6.2.4. Measurements of Lines in Vertical Planes 240
EXERCISES . 247
TABLES 249
BIBLIOGRAPHY . 255
INDEX 256
258 pages , Paperback
