Projective Geometry [Paperback ed.] 0387967524, 9780387967523
The purpose of this book is to revive some of the beautiful results obtained by various geometers of the 19th century, a
314 37 2MB
English Pages 156 [170] Year 1988
Table of contents :
Introduction v
Contents ix
Chapter 1. Projective Spaces 1
1.1 Projective Spaces and Projective Bases 1
1.2 Projective Transformations and the Projective Group 6
1.3 Projective and Affine Spaces 8
1.4 Axiomatic Presentation of Projective and Affine Planes 21
1.5 Projective Spaces of Hyperplanes and Duality 34
1.6 The Projective Space of Circles 36
1.7 The Projective Space of Conics 44
1.8 Projective Spaces of Divisors in Algebraic Geometry 51
Chapter 2. One-Dimensional Projective Geometry 53
2.1 Cross-ratios and Rational Maps 53
2.2 Cross-ratios and permutations 57
2.3 Harmonic Division 58
2.4 Projective Transformations and Involutions on a Projective Line 61
2.5 The Projective Structure of a Conic 66
2.6 Unicursal Curves 71
2.7 The Complex Projective Line and the Circular Group 81
2.8 Topology of Projective Spaces 87
Chapter 3. Classification of Conics and Quadrics 90
3.1 What Is a Quadric? 90
3.2 Classification of Affine and Euclidean Quadrics 92
3.3 Projective Classification of Real Quadrics 97
3.4 Classification of Conics and Quadrics over a Finite Field 99
Chapter 4. Polarity with Respect to a Quadric 102
4.1 Polars and Poles 102
4.2 Polarity with Respect to Conics 104
4.3 Polarity and Tangential Equations 111
4.4 Applications to Conics 116
Appendix: (2, 2)-Correspondences 123
Bibliography 145
Index of Symbols and Notations 148
Index 149
Introduction v
Contents ix
Chapter 1. Projective Spaces 1
1.1 Projective Spaces and Projective Bases 1
1.2 Projective Transformations and the Projective Group 6
1.3 Projective and Affine Spaces 8
1.4 Axiomatic Presentation of Projective and Affine Planes 21
1.5 Projective Spaces of Hyperplanes and Duality 34
1.6 The Projective Space of Circles 36
1.7 The Projective Space of Conics 44
1.8 Projective Spaces of Divisors in Algebraic Geometry 51
Chapter 2. One-Dimensional Projective Geometry 53
2.1 Cross-ratios and Rational Maps 53
2.2 Cross-ratios and permutations 57
2.3 Harmonic Division 58
2.4 Projective Transformations and Involutions on a Projective Line 61
2.5 The Projective Structure of a Conic 66
2.6 Unicursal Curves 71
2.7 The Complex Projective Line and the Circular Group 81
2.8 Topology of Projective Spaces 87
Chapter 3. Classification of Conics and Quadrics 90
3.1 What Is a Quadric? 90
3.2 Classification of Affine and Euclidean Quadrics 92
3.3 Projective Classification of Real Quadrics 97
3.4 Classification of Conics and Quadrics over a Finite Field 99
Chapter 4. Polarity with Respect to a Quadric 102
4.1 Polars and Poles 102
4.2 Polarity with Respect to Conics 104
4.3 Polarity and Tangential Equations 111
4.4 Applications to Conics 116
Appendix: (2, 2)-Correspondences 123
Bibliography 145
Index of Symbols and Notations 148
Index 149
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