Projective Geometry and Projective Metrics (eBook)

PrefaceI. 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 ExercisesII. 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 ExercisesIII. 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 ExercisesIV. 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 ExercisesV. 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 ExercisesVI. 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 SpaceBibliographyIndex