Preface |

1. Parallel Lines | Push the infinite into the finite |

2. Triangle | Move it to infinity |

3. Triangle and Inscribed Angle | (07/06/25) Not constant, Absolute unit length (1) |

4. Model | Make a disk model, Open and closed |

5. Disk and Hyperboloid | Interchangeable relations |

6. Equivalent Deformation, | |

Comparison with Elliptic Geometry (1) | (07/04/14) Locus of vertex of a triangle |

7. Parallel Angle | What is it? |

8. Lobachevsky's Postulate | (07/07/19) How was it found? Can we draw infinity? |

9. Distance and Angle, Hyperbolic Trigonomery | (07/06/22) Dependent relations, Absolute unit length (2) |

10. Equidistant Lines (1) | How does it look? Parallel curves |

11. Ultraparallel Lines | Common perpendicular |

12. Tiling (tessellation) | (07/05/01) Reflection, Ideal triangle |

13. Parallel Translation (1) | Get locus by reflection |

14. Circle | Cutting hemisphere |

15. Move Circle | Toward infinity |

16. Extending Disk Model | (07/07/02) To get real hyperbolic plane |

17. Horocycle (1) | Hemisphere, Translation, Equidistant lines (2) |

18. Pseudo-sphere (1) | Is hyperbolic surface realized? |

19. Upper Half-plane (1) | Relation with Poincare's disk, Digitized model |

20. Horocycle (2) | Poincare's disk, Upper Half-plane (2) |

21. Metric, Break Pythagorean Theorem | (07/06/13) Riemann, Mercator, Pseudo-sphere (2) |

22. Mutual Relations among Models | Ray and chain 5 models |

23. Curvature (1) | Osculating circle, Curvature of pseudosphere (1) |

24. Curvature (2) | What happens if curvature changes? |

25. Gauss-Bonnet Theorem | Curvature of pseudosphere (2) |

26. Touchable Model | (07/06/20) Paper craft of true hyperbolic plane |

27. H(+) Plane and Euclidean Plane | Visual comparison under unchanged length |

28. Analytic Curve | Lambert's quardrangle |

29. Hyperbolic Function | Source, Hyperbolic angle, Definition of logarithm |

30. Inversion and Infinity (under construction) | What is the end of hyperbola? |

31. Upper Half-plane (4) (under construction) | Can we come to Eucledean plane without jump? |

32. Right Angled Polygon | (07/03/29) Infinite polygon |

33. Dodecahedron | 3-D tiling |

34. Ideal Tetrahedron | 3-D tiling |

35. Projective Geometry (1) | Camera shot, Mobius strip |

36. Mobius Strip (1) | If twisted? |

37. Mobius Strip (2) | |

38. Projective Geometry (2) | Quadratic curve |

39. Crosscap (1) | Can Mobius strip be covered with a disk? |

40. Crosscap (2) | Mobius strip and a disk as projective planes |

41. Crosscap (3) | (06/06/08) Correspondence between Cross-cap |

and Mobius strip | |

42. Klein Bottle & Projective plane | (06/06/27) We cut and see their true colors. |

43. Paper craft | (06/08/13) Cross-cap and twist |

44. Projective Geometry (3) | Cross ratio (1), the Origin of model, Make Klein's disk |

45. Projective Geometry (4) | Real and Imaginary sphere, Hyperboloid |

46. Projective/Mobius Trans (under construction) | How do they change? |

47. Projective Geometry (5) (under construction) | Principle of continuity, Erlangen Programme, Cross ratio (2) |

48. Conclusion for Part 2 (under construction) |

49. Figure-8 knot (1) (under construction) | topology |

50. Figure-8 knot (2) (under construction) | Ultra sphere |

51. Figure-8 knot (3) (under construction) | Formula of complement |

52. Figure-8 knot (4) (under construction) | tetrahedron |

53. Figure-8 knot (5) (under construction) | Hyperbolization (Gometrization) |

54. Figure-8 knot (6) (under construction) | Where has the knot gone? |

55. Figure-8 knot (7) (under construction) | horosphere |

56. Summary (under construction) |

Post Comment / Free Software (07/07/09) |

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Aappendix / Practical Calculation (under construction) |

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