Metric Affine Geometry (eBook)

Ernst Snapper, Robert J. Troyer (Autoren)

eBook Download: PDF

2014 | 1. Auflage
456 Seiten
Elsevier Science (Verlag)
978-1-4832-6933-7 (ISBN)

Infos

Metric Affine Geometry focuses on linear algebra, which is the source for the axiom systems of all affine and projective geometries, both metric and nonmetric. This book is organized into three chapters. Chapter 1 discusses nonmetric affine geometry, while Chapter 2 reviews inner products of vector spaces. The metric affine geometry is treated in Chapter 3. This text specifically discusses the concrete model for affine space, dilations in terms of coordinates, parallelograms, and theorem of Desargues. The inner products in terms of coordinates and similarities of affine spaces are also elaborated. The prerequisites for this publication are a course in linear algebra and an elementary course in modern algebra that includes the concepts of group, normal subgroup, and quotient group. This monograph is suitable for students and aspiring geometry high school teachers.

Inhalt

Front Cover 1
Metric Affine Geometry 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 16
Symbols 20
Chapter 1. AFFINE GEOMETRY 22
1 Intuitive Affine Geometry 22
2 Axioms for Affine Geometry 27
3 A Concrete Model for Affine Space 29
4 Translations 30
5 Affine Subspaces 32
6 Intersection of Affine Subspaces 36
7 Coordinates for Affine Subspaces 37
8 Analytic Geometry 41
9 Parallelism 47
10 Affine Subspaces Spanned by Points 54
11 The Group of Dilations 57
12 The Ratio of a Dilation 67
13 Dilations in Terms of Coordinates 76
14 The Tangent Space X(c) 80
15 Affine and Semiaffine Transformations 86
16 From Semilinear to Semiaffine 94
17 Parallelograms 99
18 From Semiaffine to Semilinear 102
19 Semiaffine Transformations of Lines 109
20 Interrelation among the Groups Acting on X and on V 111
21 Determination of Affine Transformations by Independent Points and by Coordinates 114
22 The Theorem of Desargues 119
23 The Theorem of Pappus 124
Chapter 2. METRIC VECTOR SPACES 134
24 Inner Products 135
25 Inner Products in Terms of Coordinates 140
26 Change of Coordinate System 144
27 Isometries 152
28 Subspaces 158
29 The Radical 162
30 Orthogonality 168
31 Rectangular Coordinate Systems 174
32 Classification of Spaces over Fields Whose Elements have Square Roots 178
33 Classification of Spaces over Ordered Fields Whose Positive Elements have Square Roots 180
34 Sylvester's Theory 188
35 Artinian Spaces 204
36 Nonsingular Completions 213
37 The Witt Theorem 220
38 Maximal Null Spaces 228
39 Maximal Artinian Spaces 229
40 The Orthogonal Group and the Rotation Group 235
41 Computation of Determinants 245
42 Refinement of the Witt Theorem 252
43 Rotations of Artinian Space around Maximal Null Spaces 259
44 Rotations of Artinian Space with a Maximal Null Space as Axis 265
45 The Cartan-Dieudonné Theorem 272
46 Refinement of the Cartan-Dieudonné Theorem 278
47 Involutions of the General Linear Group 283
48 Involutions of the Orthogonal Group 287
49 Rotations and Reflections in the Plane 290
50 The Plane Rotation Group 292
51 The Plane Orthogonal Group 298
52 Rational Points on Conics 303
53 Plane Trigonometry 309
54 Lorentz Transformations 332
55 Rotations and Reflections in Three-Space 336
56 Null Axes in Three-Space 339
57 Reflections in Three-Space 349
58 Cartan-Dieudonné Theorem for Rotations 352
59 The Commutator Subgroup of a Group 354
60 The Commutator Subgroup of the Orthogonal Group 358
61 The Commutator Subgroup of the Rotation Group 361
62 The Isometries ±1 v 363
63 Centers of O(v), O +(V), and O(v) 368
64 Linear Representations of the Groups 373
65 Definition 379
Chapter 3. METRIC AFFINE SPACES 392
66 Square Distance 393
67 Rigid Motions 400
68 Interrelation among the Groups Mo, Tr, and O(V) 408
69 The Cartan-Dieudonné Theorem for Affine Spaces 421
70 Similarities of Affine Spaces 433
Epilogue 448
Bibliography 450
Index 452

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6933-7 / 1483269337
• ISBN-13: 978-1-4832-6933-7 / 9781483269337