Projective Geometry and Projective Metrics (eBook)

Front Cover 1
Projective Geometry and Projective Metrics 4
Copyright Page 5
TABLE OF CONTENTS 8
Preface 6
Chapter I. The Projective Plane 10
I. 1. Ideal Points 10
I. 2. The Projective Plane 12
I. 3. Projective Coordinates 15
I. 4. The Content of Projective Geometry. Duality Principle 22
I. 5. Groups of Transformations. Projectivities 29
I. 6. Cross Ratio . Intervals 37
I. 7. Perspectivities 44
I. 8. The Projectivities of Line on Itself 48
I. 9. Collineations 52
Exercises 59
Chapter II. Polarities And Conic Sections 64
II. 10. Polarities 64
II. 11. Conic Sections 70
II. 12. The Theorems of Steiner and Pascal 74
II. 13. Collineations of Conics 77
II. 14. Pencils of Conics 84
Exercises 90
Chapter III. Affine Geometry 92
III. 15. The Content of Affine Geometry . The Affine Group 92
III. 16. The Affine Theory of Conics 97
III. 17. Convex Sets 102
III. 18. The Equiaffine Group . Area 107
Exercises 111
Chapter IV. Projective Metrics 114
IV. 19. Metric Spaces 114
IV. 20. Segment, Straight Lines, Great Circles . Projective- Metric Spaces 121
IV. 21. Perpendiculars in Open Two-Spaces 126
IV. 22. Motions 133
IV. 23. Motions of Open Two-Spaces 136
IV. 24. Minkowskian Geometry 142
IV. 25. Reflections in Minkowskian Geometry . Euclidean Geometry 148
IV. 26. Angles and Motions in Euclidean Geometry 153
IV. 27. The Euclidean Theory of Conics 160
IV. 28. Hilbert's Geometry 165
IV. 29. Motions and Area in Hilbert Geometry . Definition of Hyperbolic Geometry 172
Exercises 177
Chapter V. Non-Euclidean Geometry 183
V. 30. Hyperbolic Trigonometry 183
V. 31. Length and Area 189
V. 32. Equidistant Curves and Limit Circles 195
V. 33. Some Synthetic Properties of Hyperbolic Geometry 203
V. 34. The Group of Hyperbolic Motions 211
V. 35. Weierstrass Coordinates 215
V. 36. Definition of Elliptic Geometry 220
V. 37. Elliptic Trigonometry . Relation of Elliptic and Spherical Geometry 226
V. 38. The Elliptic Line Element . Length and Area 232
V. 39. Cayley's Method 236
Exercises 242
Chapter VI. Spatial Geometry 247
VI. 40. Three-Dimensional Projective Space 247
VI. 41. Collineations of the Projective Space 255
VI. 42. Line Coordinates . Linear Complexes 262
VI. 43. Polarities and Quadrics 268
VI. 44. Linear Congruences and Reguli 275
VI. 45. Affine Geometry in Space 284
VI. 46. Convex Sets in Space 287
VI. 47. Three-Dimensional Projective-Metrics 291
VI. 48. Minkowskian Geometry in Space 295
VI. 49. Euclidean Geometry in Space 301
VI. 50. Hilbert's Geometry in Space . Hyperbolic Geometry 306
VI. 51. Hyperbolic Spheres, Limit Spheres, and Equidistant Surfaces 310
VI. 52. The Hyperbolic Group of Motions 315
VI. 53. Elliptic Geometry in Space 321
VI. 54. The Line Element of Elliptic Space 326
Bibliography 332
Index 334