Redei / Sneddon / Stark Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein

1. Auflage 2014
ISBN: 978-1-4832-8270-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 97, 410 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein aims to remedy the deficiency in geometry so that the ideas of F. Klein obtain the place they merit in the literature of mathematics. This book discusses the axioms of betweenness, lattice of linear subspaces, generalization of the notion of space, and coplanar Desargues configurations. The central collineations of the plane, fundamental theorem of projective geometry, and lines perpendicular to a proper plane are also elaborated. This text likewise covers the axioms of motion, basic projective configurations, properties of triangles, and theorem of duality in projective space. Other topics include the point-coordinates in an affine space and consistency of the three geometries. This publication is beneficial to mathematicians and students learning geometry.

Autoren/Hrsg.

Redei, L.

Sneddon, I. N.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Foundation of Euclidean and Non-Euclidean Geometries According to F. Klein;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;CHAPTER 1. AXIOMS;12
6.1;§ 1. Axioms of incidence;12
6.2;§ 2. Axioms of betweenness;13
6.3;§ 3. Axiom of continuity;14
6.4;§ 4. Axioms of motion;14
7;CHAPTER 2. CONSEQUENCES OF THE SYSTEM OF AXIOMS I;16
7.1;§ 5. Simple properties of straight lines and planes;16
7.2;§ 6. Desargues configurations;19
7.3;§ 7. Linear subspaces;22
7.4;§ 8. The lattice of linear subspaces;23
7.5;§ 9. Basic projective configurations;24
7.6;§ 10. Projection and intersection;26
8;CHAPTER 3. SIMPLE CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II;28
8.1;§ 11. Segments. Triangles;28
8.2;§ 12. Properties of segments;31
8.3;§ 13. Linear ordering;34
8.4;§ 14. Properties of triangles;40
8.5;§ 15. The tetrahedron;44
8.6;§ 16. Neighbourhoods;47
8.7;§ 17. Validity of the systems of Axioms I, II for the basic domain R';48
8.8;§ 18. Generalization of the notion of space;50
8.9;§ 19. The extension and restriction of spaces;50
9;CHAPTER 4. PROJECTIVE CLOSURE;51
9.1;§ 20. Half-subspaces;51
9.2;§ 21. Half-pencils. Angles;56
9.3;§ 22. Some properties of pencils and bundles;61
9.4;§ 23. Coplanar Desargues configurations;62
9.5;§ 24. Improper pencils of lines;66
9.6;§ 25. Improper bundles of lines;72
9.7;§ 26. The projective closure R of R;76
9.8;§ 27. The projective axioms;95
9.9;§ 28. The general case;104
10;CHAPTER 5. INVESTIGATION OF THE PROJECTIVE SPACE;107
10.1;§ 29. Preliminaries;107
10.2;§ 30. Theorem of duality in projective space;109
10.3;§ 31. Collineations;110
10.4;§ 32. The Erlangen programme;113
10.5;§ 33. Theorem of duality of the plane;114
10.6;§ 34. Perspectivities and projectivities;116
10.7;§ 35. Central collineations of the plane;121
10.8;§ 36. Separation;133
10.9;§ 37. Cyclic ordering;144
10.10;§ 38. Projective segments and angles;149
10.11;§ 39. Complete quadrangles. Harmonic points;155
10.12;§ 40. Preliminaries about coordinate systems;164
10.13;§ 41. Coordinates in projective scales;174
10.14;§ 42. Halving a projective scale;179
10.15;§ 43. Coordinates for dyadic sets of points on a line;181
11;CHAPTER 6. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III;186
11.1;§ 44. Preliminaries;186
11.2;§ 45. Theorem concerning the infinite point;191
11.3;§ 46. Coordinates in an affine line;196
11.4;§ 47. Coordinates on the basic projective configurations of the first degree;202
11.5;§ 48. Point-coordinates in an affine plane;205
11.6;§ 49. The fundamental theorem of projective geometry;214
11.7;§ 50. Point-coordinates in an affine space;223
11.8;§ 51. Vectors;229
11.9;§ 52. Homogeneous point- and plane-coordinates in space. Point- and line-coordinates in a plane;231
11.10;§ 53. Determination of all collineations of the space;241
11.11;§ 54. Determination of the coordinate transformations of space;245
11.12;§ 55. Transformation of projective coordinates;250
11.13;§ 56. Cross ratio;252
11.14;§ 57. Imaginary points;258
11.15;§ 58. Fixed elements of projectivities;259
11.16;§ 59. Involutions;260
11.17;§ 60. Involutory collineations of a plane;263
12;CHAPTER 7. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III, IV;265
12.1;§ 61. Extended motions;265
12.2;§ 62. The comparability of segments;270
12.3;§ 63. Reflections and rotations. Absolute polar plane;275
12.4;§ 64. Metric scales. Infinite and ultra-infinite points. Elliptic, parabolic and hyperbolic geometries;286
12.5;§ 65. Absolute involution of points on a proper line;294
12.6;§ 66. Midpoint and bisector;299
12.7;§ 67. The lines perpendicular to a proper plane;304
12.8;§ 68. Motions as products of reflections;312
12.9;§ 69. Polarities with respect to surfaces and curves of the second order;314
12.10;§ 70. The absolute configuration in the elliptic case;322
12.11;§ 71. The absolute configuration in the hyperbolic case;324
12.12;§ 72. Characterization of motions in the non-parabolic case;333
12.13;§ 73. The absolute configuration and characterization of motions in the parabolic case;336
12.14;§ 74. Formulae of motion of the three geometries;348
12.15;§ 75. The consistency of the three geometries;363
12.16;§ 76. Measuring of segments;377
12.17;§ 77. Measuring of angles;387
12.18;§ 78. Applications to trigonometry;393
13;Bibliography;402
14;Index;404