Geometry in a modern setting:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Paris
Hermann
1969
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 142 Seiten |
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MARC
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| 245 | 1 | 0 | |a Geometry in a modern setting |c Gustave Choquet |
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Record in the Search Index
| _version_ | 1804117098392190976 |
|---|---|
| adam_text | Contents
Foreword ............ 5
Introduction 13
Chapter I. Axioms of incidence and order
§ 1. Lines and parallels
1. Definitions . . . . . . . . . . 17
2. Axioms of incidence . . . . . . . . . 17
3. Oblique projection .......... 19
4. Systems of axes........... 20
§ 2. Axioms of order
5. The ordering on a line ......... 21
6. Linking axiom ........... 22
7. Division of the plane by a line ........ 23
Exercises ............ 23
Chapter II. Axioms for affine structure
§ 1. Affine structure of lines
8. First affine axiom .......... 26
g. Isomorphism of R and pointed lines of II . . . . . 27
§ 2. Additive group structure of (II, 0)
10. Linking axiom ........... 28
11. Oblique parallel projections and parallelograms ..... 29
12. Addition in (II, 0) and its group structure ...... 29
§ 3. Translations of FI
13. Characterization of translations . . . . . . . 31
14. Isomorphism of the groups (II, 0) ....... 32
15. Free vectors and Chasles formula 32
16. Action of translations on orientated lines ...... 33
1*
8 CONTENTS
§ 4. (II, 0) as a vector space
17. Definitions and scalar multiplication ....... 35
18. Linearity of oblique parallel projections ...... 35
19. The vector space theorem ......... 36
20. Bases and coordinates. Equation of a line ...... 37
21. Characterization of homothetic transformations . . . . 37
22. Isomorphism of the vector spaces (II, 0) ...... 39
23. Translations as a vector space ........ 39
§5. Dilations of the plane
24. Characterization of dilations ........ 40
25. The dilation group .......... 40
26. Subgroups of the dilation group . . . . . . . 41
27. Dilations of subsets of II 41
§ 6. Further results
28. A few topics for consideration ........ 42
29. Oblique symmetry 43
Exercises ............ 44
Chapter III. Axioms for metric structure
§ 1. Perpendiculars
30. Perpendicularity axiom ......... 46
31. Perpendicular directions ......... 46
32. Affine properties which are apparently metric ..... 47
33. Projection ratio of a pair of half lines originating at the same point . . 48
§ 2. Inner product
34. Symmetry axiom 48
35. Norm and inner product ......... 48
36. Identities and inequalities ......... 50
37. Invariance of distance and inner product under translation . . • 51
38. An inner product on the vector space of translations .... 52
CONTENTS g
§3. Elementary metric properties
39. Metric relations in parallelograms and triangles 53
40. Orthogonal projection ......... 55
41. Perpendicular bisector ......... 56
42. Moments of inertia .......... 57
43. Inner product and distance with respect to an arbitrary basis ... 58
Chapter IV. Isometries. Similarity transformations. Symmetries of a set
§ 1. Isometries
44. Axial symmetries and central symmetries ...... 59
45. Isometries ....... . . . . 61
46. Group of isometries about a point ....... 62
47. Even and odd isometries ......... 64
48. Analysing isometries .......... 66
§ 2. Similitudes
49. Characteristic properties ......... 67
50. Even and odd similitudes ......... 68
51. Group of similitudes about a point . . . . . . 69
52. Analysing similitudes .......... 70
53. Classification of closed groups of similitudes . . . . . 71
§ 3. Sets stable under a group of transformations
54. Regularity in a set 72
55. Construction of regular pairs (E, 8) 73
56. Symmetries of a given set . . . . . . . . 74
Exercises ............ 75
Chapter V. Angles
§ 1. The group of angles
57. Difficulties in defining an angle 78
58. Definition and notation 79
59. The sum of the angles of a closed plane polygon . . . . . 81
IO CONTENTS
§ 2. Angles and similitudes
60. Angles under symmetries ......... 81
61. Angles under similitudes ......... 82
62. Characterization of rotations ........ 82
63. Characterization of similitudes 83
64. Dividing an angle by 2 . . . . . . . 84
65. Angles formed by a pair of lines ........ 84
Chapter VI. Orientation
66. Difficulties about orientation 86
67. Orientation of subsets of II ........ 86
68. Orientation of other geometrical entities associated with II ... 87
69. Elementary approach to the orientation of pairs of non collinear half lines . 89
70. Relation between orientation and continuous deformations ... 90
71. Continuous deformations ......... 91
Exercises ............ 92
Chapter VII. Trigonometry
§ 1. Elementary trigonometry
72. The cosine and sine of an angle relative to a given basis. • • • 94
73. Matrix of a rotation relative to a positive orthonormal basis ... 95
74. Addition formulas 96
§ 2. Measuring angles
75. In search of a definition ......... 97
76. Definition and immediate consequences ...... 98
77. Sketch proof of the existence of continuous homomorphisms from R onto T . 99
78. Arithmetical measure of an angle . . . . . . .101
Exercises ............ 101
Chapter VIII. The Circle
79. Definition and symmetries of a circle ....... 102
80. Image of a circle under a similitude . . . . . . .103
81. Convexity of discs 104
82. Intersection of a circle and a line ....... i°4
CONTENTS II
83. Tangent to a circle .......... 104
84. Intersection of two circles . . . . . . . . .105
85. Equation of a circle .......... 105
86. Some characteristic properties of the circle . . . . . .106
87. The power of a point with respect to a circle . . . . .108
Exercises ............ 109
Chapter IX. Space
§ 1. Axioms
88. Choosing a method . . . . . . . . . .110
89. Axioms for three dimensional space . . . . . . .111
90. Elementary consequences . . . . . . . . .112
§ 2. Affine structure of space
91. The pointed space [E, 0] 113
92. Translations . . . . . . . . . . .114
93. Parallelism 114
94. Consequences of the dimension axiom . . . . . . 115
§ 3. Metric structure of space
95. Translations and perpendicularity . . . . . . 117
96. Inner product . . . . . . . . . . 117
97. Application to two classical theorems 119
98. Further topics 120
Exercises 120
Appendix 1. A metrically based axiomatization
99. First axioms ........... 122
100. Folding axiom (or symmetry axiom) 123
101. Symmetry about a line 123
102. Perpendiculars and projections 124
103. Symmetry about a point and products of symmetries .... 126
104. A pointer to subsequent development . . . . . . .128
Appendix 2. Axiomatization of non euclidean geometry . . . . .129
Appendix 3. Axiomatization for the little geometry 131
Appendix 4. Alternative scheme for defining angles 133
List of Symbols X35
Index of Terms r39
Bibliography 41
|
| any_adam_object | 1 |
| author | Choquet, Gustave 1915-2006 |
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| dewey-full | 513 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 513 - Arithmetic |
| dewey-raw | 513 |
| dewey-search | 513 |
| dewey-sort | 3513 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T15:49:00Z |
| institution | BVB |
| language | English French |
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| physical | 142 Seiten |
| publishDate | 1969 |
| publishDateSearch | 1969 |
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| publisher | Hermann |
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| spelling | Choquet, Gustave 1915-2006 Verfasser (DE-588)107837897 aut L' enseignement de la géométrie Geometry in a modern setting Gustave Choquet Paris Hermann 1969 142 Seiten txt rdacontent n rdamedia nc rdacarrier Geometry, Plane Elementargeometrie (DE-588)4137488-5 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Axiomatik (DE-588)4004038-0 gnd rswk-swf Geometrie (DE-588)4020236-7 s Mathematikunterricht (DE-588)4037949-8 s 1\p DE-604 Elementargeometrie (DE-588)4137488-5 s 2\p DE-604 Axiomatik (DE-588)4004038-0 s 3\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001755231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Choquet, Gustave 1915-2006 Geometry in a modern setting Geometry, Plane Elementargeometrie (DE-588)4137488-5 gnd Geometrie (DE-588)4020236-7 gnd Mathematikunterricht (DE-588)4037949-8 gnd Axiomatik (DE-588)4004038-0 gnd |
| subject_GND | (DE-588)4137488-5 (DE-588)4020236-7 (DE-588)4037949-8 (DE-588)4004038-0 |
| title | Geometry in a modern setting |
| title_alt | L' enseignement de la géométrie |
| title_auth | Geometry in a modern setting |
| title_exact_search | Geometry in a modern setting |
| title_full | Geometry in a modern setting Gustave Choquet |
| title_fullStr | Geometry in a modern setting Gustave Choquet |
| title_full_unstemmed | Geometry in a modern setting Gustave Choquet |
| title_short | Geometry in a modern setting |
| title_sort | geometry in a modern setting |
| topic | Geometry, Plane Elementargeometrie (DE-588)4137488-5 gnd Geometrie (DE-588)4020236-7 gnd Mathematikunterricht (DE-588)4037949-8 gnd Axiomatik (DE-588)4004038-0 gnd |
| topic_facet | Geometry, Plane Elementargeometrie Geometrie Mathematikunterricht Axiomatik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001755231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT choquetgustave lenseignementdelageometrie AT choquetgustave geometryinamodernsetting |