Projective Geometry and Projective Metrics: (Dover Books on Mathematics)

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.Topics include the projective plane, polarities and conic sections, affine geometry, projective metrics, and non-Euclidean and spatial geometry. Numerous figures appear throughout the text, which concludes with a bibliography and index.

Table of Contents:
Preface I. The Projective Plane I.1. Ideal Points I.2. The Projective Plane I.3. Projective Coordinates I.4. The Content of Projective Geometry. Duality Principle I.5. Groups of Transformations. Projectivities I.6. Cross Ratio. Intervals I.7. Perspectivities I.8. The Projectivities of a Line on Itself I.9. Collineations Exercises II. Polarities and Conic Sections II.10. Polarities II.11. Conic Sections II.12. The Theorems of Steiner and Pascal II.13. Collineations of Conics II.14. Pencils of Conics Exercises III. Affine Geometry III.15. The Content of Affine Geometry. The Affine Group III.16. The Affine Theory of Conics III.17. Convex Sets III.18. The Equiaffine Group. Area Exercises IV. Projective Metrics IV.19. Metric Spaces IV.20. Segment, Straight Lines, Great Circles. Projective-Metric Spaces IV.21. Perpendiculars in Open Two-Spaces IV.22. Motions IV.23. Motions of Open Two-Spaces IV.24. Minkowskian Geometry IV.25. Reflections in Minkowskian Geometry. Euclidean Geometry IV.26. Angles and Motions in Euclidean Geometry IV.27. The Euclidean Theory of Conics IV.28. Hilbert's Geometry IV.29. Motions and Area in Hilbert Geometry. Definition of Hyperbolic Geometry Exercises V. Non-Euclidean Geometry V.30. Hyperbolic Trigonometry V.31. Length and Area V.32. Equidistant Curves and Limit Circles V.33. Some Synthetic Properties of Hyperbolic Geometry V.34. The Group of Hyperbolic Motions V.35. Weierstrass Coordinates V.36. Definition of Elliptic Geometry V.37. Elliptic Trigonometry. Relation of Elliptic and Spherical Geometry V.38. The Elliptic Line Element. Length and Area V.39. Cayley's Method Exercises VI. Spatial Geometry VI.40. Three-Dimensional Projective Space VI.41. Collineations of the Projective Space VI.42. Line Coordinates. Linear Complexes VI.43. Polarities and Quadrics VI.44. Linear Congruences and Reguli VI.45. Affine Geometry in Space VI.46. Convex Sets in Space VI.47. Three-Dimensional Projective-Metrics VI.48. Minkowskian Geometry in Space VI.49. Euclidean Geometry in Space VI.50. Hilbert's Geometry in Space. Hyperbolic Geometry VI.51. Hyperbolic Spheres, Limit Spheres, and Equidistant Surfaces VI.52. The Hyperbolic Group of Motions VI.53. Elliptic Geometry in Space VI.54. The Line Element of Elliptic Space Bibliography Index