Real geometries:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Mannheim [u.a.]
BI-Wiss.-Verl.
1994
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 379 S. graph. Darst. |
| ISBN: | 3411169419 |
Staff View
MARC
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| 245 | 1 | 0 | |a Real geometries |c by Walter Benz |
| 264 | 1 | |a Mannheim [u.a.] |b BI-Wiss.-Verl. |c 1994 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 GEOMETRIES — GENERAL THEORY 17
1.1 Geometry of a transformation group 17
1.2 Geometry of a notion or a function 21
1.3 Substructures of geometries (S,G) 27
1.4 Direct products of geometries 31
1.5 Klein s Erlangen programme 38
1.6 Functional equations in geometry 39
2 HYPERBOLIC GROUPS 47
2.1 The hyperbolic line 47
2.2 Transitivity properties and 2 point invariants 49
2.3 Two functional equations 53
2.4 Weierstrass coordinates 56
2.5 A functional equation of distance preservance 58
2.6 Hyperbolic spaces 60
3 EINSTEIN S CYLINDER UNIVERSE 75
3.1 Points and motions 75
3.2 The functional equation of 2 point invariants 79
3.3 Strong, definite 2 point invariants 80
3.4 Lines in Einstein s cylinder universe 85
3.5 The notion of distance 87
3.6 The affine structure 91
14 CONTENTS
3.7 Null lines 94
3.8 Null line preserving mappings 97
3.9 Distance O preserving mappings 103
3.10 Einstein s cylinder universe over a ring 105
3.11 Laguerre model of Einstein s plane 107
3.12 Laguerre image of a circular helix 112
4 DE SITTER S WORLD 119
4.1 De Sitter s world as a substructure 119
4.2 The group of motions 120
4.3 Lines 121
4.4 Transitivity properties 124
4.5 Pairs of points on lines 129
4.6 The functional equation of 2 point invariants 131
4.7 A functional equation in connection with rings 132
4.8 The notion of distance 133
4.9 Stabilizers 139
4.10 The Minkowski model of de Sitter s plane 144
4.11 Ring coordinates 150
4.12 Fundamental Theorem of de Sitter s plane 156
4.13 A conditional functional equation 160
4.14 De Sitter s world and Lie geometry 162
4.15 Fundamental Theorem of de Sitter s space 165
5 FUNDAMENTAL GEOMETRIES 167
5.1 Euclidean geometry 167
5.1.1 Theorem of Beckman and Quarles 167
5.1.2 A theorem of June Lester 168
5.1.3 All 2 point invariants 173
5.1.4 All 3 point invariants 176
5.2 Similarity transformations 180
CONTENTS 15
5.2.1 Mappings preserving euclidean circles 180
5.2.2 All 2 line invariants 182
5.2.3 The set of angles as invariant notion 187
5.2.4 Angle spaces 190
5.2.5 Additive angle measures 192
5.3 Equiaffine geometry 193
5.3.1 A theorem of G. Martin 193
5.3.2 n dimensional equiaffine geometry 195
5.3.3 Characterization of volumes of simplexes 199
5.4 Affine geometry 203
5.4.1 The set of lines as defining notion 203
5.4.2 An affine invariant 206
5.4.3 The set of parabolas as defining notion 208
5.5 Euclidean geometry of normed spaces 210
5.6 Projective geometry 211
5.6.1 Points, lines, hyperplanes 211
5.6.2 The projective extension of lm 218
5.6.3 The set of lines as defining notion 219
5.6.4 An intrinsic characterization 222
5.6.5 The projective line 225
5.6.6 Collineations on triangles 237
5.7 Non euclidean geometries 243
5.7.1 Hyperbolic geometry as a substructure 243
5.7.2 Angles in hyperbolic geometry 248
5.7.3 Volumes in hyperbolic geometry 253
5.7.4 A definition of elliptic geometry 258
5.8 A definition of spherical geometry 259
5.9 Lorentz Minkowski geometry 260
16 CONTENTS
6 HIGHER GEOMETRIES 263
6.1 Cremona geometries 263
6.1.1 Birational transformations 263
6.1.2 Cremona groups 270
6.1.3 Real chain geometries 271
6.2 Circle geometries 288
6.2.1 2 dimensional chain geometries 288
6.2.2 The real Laguerre plane 291
6.2.3 The other two geometries 301
6.3 Sphere geometries 307
6.3.1 The group of automorphisms 307
6.3.2 Lie geometry 309
6.3.3 Theorem of Liouville for arbitrary signature 310
6.4 Line geometries 312
6.4.1 Introduction 312
6.4.2 A theorem of Wen ling Huang 313
6.5 Proportion geometries 333
6.5.1 Proportion relations, 1 point invariants 333
6.5.2 Proportion functions in two variables 336
6.5.3 A generalization 340
6.5.4 Continuous proportion functions 342
6.5.5 Proportion polynomials 344
6.5.6 The general C1 solution 346
6.5.7 Rectangle patterns 354
6.5.8 Three and more variables 356
BIBLIOGRAPHY 359
NOTATION AND SYMBOLS 369
INDEX 373
|
| any_adam_object | 1 |
| author | Benz, Walter 1931-2017 |
| author_GND | (DE-588)117708046 |
| author_facet | Benz, Walter 1931-2017 |
| author_role | aut |
| author_sort | Benz, Walter 1931-2017 |
| author_variant | w b wb |
| building | Verbundindex |
| bvnumber | BV008958901 |
| classification_rvk | SK 380 |
| classification_tum | MAT 001f MAT 500f |
| ctrlnum | (OCoLC)243776424 (DE-599)BVBBV008958901 |
| dewey-full | 516 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV008958901 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:27:28Z |
| institution | BVB |
| isbn | 3411169419 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005913149 |
| oclc_num | 243776424 |
| open_access_boolean | |
| owner | DE-29T DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-703 DE-739 DE-12 DE-20 DE-210 DE-634 DE-11 |
| owner_facet | DE-29T DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-703 DE-739 DE-12 DE-20 DE-210 DE-634 DE-11 |
| physical | 379 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | BI-Wiss.-Verl. |
| record_format | marc |
| spelling | Benz, Walter 1931-2017 Verfasser (DE-588)117708046 aut Real geometries by Walter Benz Mannheim [u.a.] BI-Wiss.-Verl. 1994 379 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometrie (DE-588)4020236-7 gnd rswk-swf Reelle Geometrie (DE-588)4343628-6 gnd rswk-swf Geometrie (DE-588)4020236-7 s DE-604 Reelle Geometrie (DE-588)4343628-6 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005913149&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Benz, Walter 1931-2017 Real geometries Geometrie (DE-588)4020236-7 gnd Reelle Geometrie (DE-588)4343628-6 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4343628-6 |
| title | Real geometries |
| title_auth | Real geometries |
| title_exact_search | Real geometries |
| title_full | Real geometries by Walter Benz |
| title_fullStr | Real geometries by Walter Benz |
| title_full_unstemmed | Real geometries by Walter Benz |
| title_short | Real geometries |
| title_sort | real geometries |
| topic | Geometrie (DE-588)4020236-7 gnd Reelle Geometrie (DE-588)4343628-6 gnd |
| topic_facet | Geometrie Reelle Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005913149&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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