Projective Geometry and Projective Metrics

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ISBN: 9780486445823 | 0486445828
Cover: Paperback
Copyright: 12/10/2005

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The basic results and methods of projective and non-Euclidean geometry are indispensable for the geometer, and this book-different in content, methods, and point of view from traditional texts-attempts to emphasize that fact. Results of special theorems are discussed in detail only when they are needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity.

PREFACE

CHAPTER I. THE PROJECTIVE PLANE

(54)

I.1. Ideal Points

(2)

I.2. The Projective Plane

(3)

1.3. Projective Coordinates

(7)

I.4. The Content of Projective Geometry. Duality Principle

(7)

I.5. Groups of Transformations. Projectivities

(8)

I.6. Cross Ratio. Intervals

(7)

1.7. Perspectivities

(4)

1.8. The Projectivities of a Line on Itself

(4)

I.9. Collineations

(12)

Exercises

(5)

CHAPTER II. POLARITIES AND CONIC SECTIONS

(28)

II.10. Polarities

(6)

II.11. Conic Sections

(4)

II.12. The Theorems of Steiner and Pascal

(3)

II.13. Collineations of Conics

(7)

II.14. Pencils of Conics

(8)

Exercises

(2)

CHAPTER III. AFFINE GEOMETRY

(22)

III.15. The Content of Affine Geometry. The Affine Group

(5)

III.16. The Affine Theory of Conics

(5)

III.17. Convex Sets

(5)

III.18. The Equiaffine Group. Area

(7)

Exercises

102

(3)

CHAPTER IV. PROJECTIVE METRICS

105

(69)

IV.19. Metric Spaces

105

(7)

IV.20. Segment, Straight Lines, Great Circles. Projective-Metric Spaces

112

(5)

IV.21. Perpendiculars in Open Two-Spaces

117

(7)

IV.22. Motions

124

(3)

IV.23. Motions of Open Two-Spaces

127

(6)

IV.24. Minkowskian Geometry

133

(6)

IV.25. Reflections in Minkowskian Geometry. Euclidean Geometry

139

(5)

IV.26. Angles and Motions in Euclidean Geometry

144

(7)

IV.27. The Euclidean Theory of Conics

151

(5)

IV.28. Hilbert's Geometry

156

(7)

IV.29. Motions and Area in Hilbert Geometry. Definition of Hyperbolic Geometry

163

(11)

Exercises

168

(6)

CHAPTER V. NON-EUCLIDEAN GEOMETRY

174

(64)

V.30. Hyperbolic Trigonometry

174

(6)

V.31. Length and Area

180

(6)

V.32. Equidistant Curves and Limit Circles

186

(8)

V.33. Some Synthetic Properties of Hyperbolic Geometry

194

(8)

V.34. The Group of Hyperbolic Motions

202

(4)

V.35. Weierstrass Coordinates

206

(5)

V.36. Definition of Elliptic Geometry

211

(6)

V.37. Elliptic Trigonometry. Relation of Elliptic and Spherical Geometry

217

(6)

V.38. The Elliptic Line Element. Length and Area

223

(4)

V.39. Cayley's Method

227

(11)

Exercises

233

(5)

CHAPTER VI. SPATIAL GEOMETRY

238

(85)

VI.40. Three-Dimensional Projective Space

238

(8)

VI.41. Collineations of the Projective Space

246

(7)

VI.42. Line Coordinates. Linear Complexes

253

(6)

VI.43. Polarities and Quadrics

259

(7)

VI.44. Linear Congruences and Reguli

266

(9)

VI.45. Milne Geometry in Space

275

(3)

VI.46. Convex Sets in Space

278

(4)

VI.47. Three-Dimensional Projective-Metrics

282

(4)

VI.48. Minkowskian Geometry in Space

286

(6)

VI.49. Euclidean Geometry in Space

292

(5)

VI.50. Hilbert's Geometry in Space. Hyperbolic Geometry

297

(4)

VI.51. Hyperbolic Spheres, Limit Spheres, and Equidistant Surfaces

301

(5)

VI.52. The Hyperbolic Group of Motions

306

(6)

VI.53. Elliptic Geometry in Space

312

(5)

VI.54. The Line Element of Elliptic Space

317

(6)

BIBLIOGRAPHY

323

(2)

INDEX

325

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Projective Geometry and Projective Metrics

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