Transformations and geometries:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Meredith
1969
|
| Series: | The Appleton-Century mathematics series
|
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 402 S. graph. Darst. |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804126527754862592 |
|---|---|
| adam_text | Contents
Chapter I
GENERAL INTRODUCTION
SECTION
1. A bit of history 1
2. Definition of a geometric transformation 2
3. The inverse of a transformation 5
4. The compounding of transformations 6
5. Groups of transformations 7
6. Geometric invariants 9
7. Transformations of the plane into itself 11
8. Linear transformations 13
Chapter II
MOTIONS OF THE EUCLIDEAN PLANE
1. Introduction 16
2. Some general properties of motions 16
3. Motions and congruence 19
4. Translations 22
5. Inverses and resultants 28
6. Equations of translations in vector form 31
7. Rotations 33
8. Equations of rotations about the origin 37
9. Equations of rotations in matrix form 40
10. Equations of the general rotation 46
11. Resultants of translations and rotations 49
12. The group of displacements 51
13. Reflections 54
14. Equations of reflections 56
ix
SECTION
15. Other opposite motions 59
16. Equations of the group of motions 64
17. Further remarks on congruence 66
Chapter III
TRANSFORMATIONS OF SIMILARITY
1. Introduction 71
2. General properties of similarity transformations 71
3. A key similarity transformation 73
4. Equations of the similarity group 75
5. Extension of the notion of similar figures 78
6. Metric geometry 80
Chapter IV
AFFINE TRANSFORMATIONS
1. Introduction 83
2. The affine group 84
3. A key affine transformation 86
4. A key affine transformation (continuation) 91
5. Resolution of the general affine transformation 95
6. Affine properties 97
7. Affine geometry 101
8. Collinearity and concurrence 103
9. Affine equivalence 106
10. Affine properties and 1 1 transformations 110
11. Axioms for affine geometry 113
12. Distance in affine geometry 116
Chapter V
PROJECTIONS, A TRANSITION
1. Introduction 123
2. Parallel projection of a line 123
3. Parallel projection of a plane 125
4. Parallel projections and affine transformations 128
5. Central projections 130
SECTION
6. Central projection of a line on an intersecting line 132
7. Central projection of a plane on an intersecting plane 137
8. Projective properties 144
9. The values of cross ratios 144
10. Harmonic division 148
11. Cross ratio of concurrent lines 149
12. Harmonic division of concurrent lines 155
13. Cross ratio of parallel lines 156
14. Cross ratio and the conic sections 158
15. Applications of projections 160
Chapter VI
PROJECTIVE TRANSFORMATIONS
1. Introduction 165
2. Definition of a projective transformation 165
3. Some implications of the definition 168
4. Equations of projective transformations 170
5. The projective group 176
6. Projective transformations and projections 178
7. Conic sections 180
8. Projective equivalence 181
9. Projective geometry of the Euclidean plane 182
Chapter VII
TOPOLOGICAL TRANSFORMATIONS
1. Introduction 189
2. Topological transformations of the plane 190
3. Example of a nonaffine topological transformation 193
4. Topological properties of curves 196
5. More general topological transformations 199
6. Homeomorphs of lines and circles 201
7. Topological transformations and order 207
8. Homeomorphs of the plane 209
9. Models of the plane 212
10. More on the circular model of the plane 216
11. Surfaces not homeomorphic to the plane 220
12. The projective plane 225
13. A bounded model of Euclidean space 228
Chapter VIII
THE PROJECTIVE PLANE
SECTION
1. Introduction 230
2. Ideal points 230
3. Extended planes 232
4. Model of an extended plane 234
5. The ideal plane; projective planes 235
6. Projective space 237
7. Projections viewed more broadly 238
8. Collinearity, concurrence, duality 241
9. Cross ratio and ideal elements 243
10. Order on a projective line 248
11. Figures in the projective plane 251
12. Harmonic properties of complete figures 254
13. The construction of harmonic conjugates 256
14. The Theorem of Desargues 258
15. Other perspective figures 261
16. The Theorem of Pappus 262
17. Connections with Euclidean geometry 266
18. Projective conies 270
19. Transformations of a projective plane into itself 275
20. Other methods of developing projective geometry 279
Chapter IX
ANALYTIC PROJECTIVE GEOMETRY
1. Introduction 280
2. Homogeneous coordinates of points 280
3. Equations of projective lines 286
4. Linear combination of points 291
5. Linear combination and cross ratio 294
6. Equations of a projectivity in a projective plane 296
7. Geometries of the projective plane 303
8. Equations of projective conies 308
9. Equations of tangents to projective conies 315
10. Projective curves of higher degree 319
11. Homogeneous coordinates of lines 321
12. Equations of points 322
13. Linear combination of lines 325
14. Projective transformations in line coordinates 326
SECTION
15. Line curves 328
16. Nonhomogeneous line coordinates 329
17. Line curves of the second degree 331
18. Line conies as projective figures 337
19. Correlations 339
20. The summation notation 343
21. Vector and matrix notations 347
22. Collineations as topological transformations 353
Chapter X
PROJECTIVE DESCRIPTIONS OF CONICS
1. Introduction 356
2. A projective view of point conies 356
3. Projective correspondences 358
4. A second projective view of point conies 362
5. The Theorem of Pascal 365
6. Line conies and cross ratio 370
7. The Theorem of Brianchon 373
appendix: determinants 376
answers to selected exercises 386
bibliography 395
INDEX 397
|
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| id | DE-604.BV011941191 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:18:53Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008071888 |
| oclc_num | 633076279 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-11 DE-188 |
| physical | XIII, 402 S. graph. Darst. |
| psigel | HUB-ZB011200905 |
| publishDate | 1969 |
| publishDateSearch | 1969 |
| publishDateSort | 1969 |
| publisher | Meredith |
| record_format | marc |
| series2 | The Appleton-Century mathematics series |
| spelling | Gans, David Verfasser aut Transformations and geometries David Gans New York Meredith 1969 XIII, 402 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Appleton-Century mathematics series HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008071888&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gans, David Transformations and geometries |
| title | Transformations and geometries |
| title_auth | Transformations and geometries |
| title_exact_search | Transformations and geometries |
| title_full | Transformations and geometries David Gans |
| title_fullStr | Transformations and geometries David Gans |
| title_full_unstemmed | Transformations and geometries David Gans |
| title_short | Transformations and geometries |
| title_sort | transformations and geometries |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008071888&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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