Projective geometry

Cover......Page 1
Title: Projective Geometry, 2nd Ed.......Page 3
ISBN 3-540-96532-7......Page 4
Preface to the First Edition......Page 5
Preface to the Second Edition......Page 7
Contents......Page 9
1.1 What is Projective Geometry?......Page 13
1.2 Historical Remarks......Page 14
EXERCISES......Page 16
1.3 Definitions......Page 17
1.4 The Simplest Geometric Objects......Page 18
1.5 Projectivities......Page 20
1.6 Perspectivities......Page 22
EXERCISES......Page 25
2.1 Axioms......Page 26
EXERCISES......Page 27
2.2 Simple Consequences of the Axioms......Page 28
EXERCISES......Page 29
2.3 Perspective Triangles......Page 30
2.4 Quadrangular Sets......Page 32
2.5 Harmonic Sets......Page 34
EXERCISES......Page 35
3.1 The Axiomatic Basis of the Principle of Duality......Page 36
3.2 The Desargues Configuration......Page 38
EXERCISES......Page 39
3.3 The Invariance of the Harmonic Relation......Page 40
3.4 Trilinear Polarity......Page 41
3.5 Harmonic Nets......Page 42
EXERCISES......Page 44
4.1 How Three Pairs Determine a Projectivity......Page 45
4.2 Some Special Projectivities......Page 47
4.3 The Axis of a Projectivity......Page 48
4.4 Pappus and Desargues......Page 50
EXERCISES......Page 52
5.1 Superposed Ranges......Page 53
5.2 Parabolic Projectivities......Page 55
EXERCISE......Page 56
5.3 Involutions......Page 57
5.4 Hyperbolic Involutions......Page 59
EXERCISES......Page 60
6.1 Projective Collineations......Page 61
6.2 Perspective Collineations......Page 64
EXERCISES......Page 66
6.3 Involutory Collineations......Page 67
EXERCISES......Page 68
6.4 Projective Correlations......Page 69
EXERCISE......Page 71
7.1 Conjugate Points and Conjugate Lines......Page 72
7.2 The Use of a Self-Polar Triangle......Page 74
7.3 Polar Triangles......Page 76
7.4 A Construction for the Polar of a Point......Page 77
EXERCISES......Page 78
EXERCISES......Page 79
7.7 The Product of Two Polarities......Page 80
7.8 The self-polarity of the Desargues configuration......Page 82
8.1 How a Hyperbolic Polarity Determines a Conic......Page 83
EXERCISES......Page 86
8.2 The Polarity Induced by a Conic......Page 87
8.3 Projectively Related Pencils......Page 88
EXERCISES......Page 89
8.4 Conics Touching two Lines at Given Points......Page 90
EXERCISES......Page 91
EXERCISES......Page 92
9.1 The Conic Touching Five Given Lines......Page 93
EXERCISES......Page 96
9.2 The Conic Through Five Given Points......Page 97
EXERCISES......Page 98
9.3 Conics Through Four Given Points......Page 99
9.4 Two Self-Polar Triangles......Page 100
9.5 Degenerate Conics......Page 101
EXERCISES......Page 102
10.1 The Idea of a Finite Geometry......Page 103
10.2 A Combinatorial Scheme for PG(2, 5)......Page 104
EXERCISES......Page 105
10.3 Verifying the Axioms......Page 107
10.4 Involutions......Page 108
10.5 Collineations and Correlations......Page 109
10.6 Conics......Page 110
EXERCISES......Page 112
11.1 Is the Circle a Conic?......Page 114
11.2 Affine Space......Page 115
11.3 How Two Coplanar Lines Determine a Flat Pencil and a Bundle......Page 117
11.4 How Two Planes Determine an Axial Pencil......Page 118
11.5 The Language of Pencils and Bundles......Page 119
11.6 The Plane at Infinity......Page 120
11.7 Euclidean Space......Page 121
EXERCISE......Page 122
12.1 The Idea of Analytic Geometry......Page 123
12.2 Definitions......Page 124
EXERCISES......Page 127
12.3 Verifying the Axioms for the Projective Plane......Page 128
EXERCISES......Page 130
12.4 Projective Collineations......Page 131
12.5 Polarities......Page 134
12.6 Conics......Page 136
12.7 The Analytic Geometry of PG(2, 5)......Page 138
EXERCISES......Page 140
12.8 Cartesian Coordinates......Page 141
EXERCISES......Page 143
12.9 Planes of characteristic two......Page 144
Answers To Exercises......Page 145
References......Page 169
Index......Page 171
List of Publications BY THE SAME AUTHOR......Page 175