§7 Isometric transformations

Kapitel in diesem Buch

1. Frontmatter I
2. Chapter I. Möbius transformations and non-euclidean geometry.
3. §1 Pencils of circles - inversive geometry 1
4. §2 Cross-ratio 4
5. §3 Möbius transformations, direct and reversed 6
6. §4 Invariant points and classification of Möbius transformations 8
7. §5 Complex distance of two pairs of points 14
8. §6 Non-euclidean metric 18
9. §7 Isometric transformations 23
10. §8 Non-euclidean trigonometry 27
11. §9 Products and commutators of motions 43
12. Chapter II. Discontinuous groups of motions and reversions.
13. §10 The concept of discontinuity 58
14. §11 Groups with invariant points or lines 70
15. §12 A discontinuity theorem 78
16. §13 ℱ-groups. Fundamental set and limit set 82
17. §14 The convex domain of an ℱ-group. Characteristic and isometric neighbourhood 95
18. §15 Quasi-compactness modulo ℱ and finite generation of ℱ 115
19. Chapter III. Surfaces associated with discontinuous groups.
20. §16 The surfaces D modulo ℭ and K(ℱ) modulo ℱ 127
21. §17 Area and type numbers 135
22. Chapter IV. Decompositions of groups.
23. §18 Composition of groups 153
24. §19 Decomposition of groups 174
25. §20 Decompositions of ℱ-groups containing reflections 196
26. §21 Elementary groups and elementary surfaces 213
27. §22 Complete decomposition and normal form in the case of quasi-compactness 242
28. §23 Exhaustion in the case of non-quasi-compactness 270
29. Chapter V. Isomorphism and homeomorphism.
30. §24 Topological and geometrical isomorphism 283
31. §25 Topological and geometrical homeomorphism 308
32. §26 Construction of g-mappings. Metric parameters. Congruent groups 318
33. Symbols and definitions 349
34. Alphabets 353
35. Bibliography 355
36. Index 361