Metric Affine Geometry

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Author(s): Ernst Snapper and Robert J. Troyer (Auth.)
Edition: First Edition
Publisher: Elsevier Inc, Academic Press Inc
Year: 1971

Language: English
Commentary: with ToC
Pages: 435

Front Cover
Title
Copyright
Dedication
Contents
Preface
Symbols
Chapter 1 Affine Geometry
1. Intuitive Affine Geometry
Vector space of translations
Limited measurement in the affine plane
2. Axioms for Affine Geometry
Division rings and fields
Axiom system for n-dimensional affine space (X, V, k)
Action of V on X
Dimension of the affine space X
Real affine space (X, V, R)
3. A Concrete Model for Affine Space
4. Translations
Definition
Translation group
5. Affine Subspaces
Definition
Dimension of affine subspaces
Lines, planes, and hyperplanes in X
Equality of affine subspaces
Direction space of S(x, U)
6. Intersection of Affine Subspaces
7. Coordinates for Affine Subspaces
Coordinate system for V (ordered basis)
Affine coordinate system
Action of V on X in terms of coordinates
8. Analytic Geometry
Parametric equations of a line
Linear equations for hyperplanes
9. Parallelism
Parallel affine subspaces of the same dimension
The fifth parallel axiom
General definition of parallel affine spaces
10. Affine Subspaces Spanned by Points
Independent (dependent) points of X
The affine space spanned by a set of points
11. The Group of Dilations
Definition
Magnifications
The group of magnifications with center c
Classification of dilations
Trace of a dilation
12. The Ratio of a Dilation
Parallel line segments
Line segments, oriented line segments
Ratio of lengths of parallel line segments
Dilation ratios of translations and magnifications
Direction of a translation
13. Dilations in Terms of Coordinates
Dilation ratio
14. The Tangent Space X(c)
Definition
Isomorphism between X(c) and X(b)
A side remark on high school teaching
15. Affine and Semiaffine Transformations
Semiaffine transformations
The group Sa of semiaffine transformations
Affine transformations
The group Af of affine transformations
16. From Semilinear to Semiaffine
Semilinear mappings
Semilinear automorphisms
17. Parallelograms
18. From Semiaffine to Semilinear
Characterization of semilinear automorphisms of V
19. Semiaffine Transformations of Lines
20. Interrelation among the Groups Acting on X and on V
21. Determination of Affine Transformations by Independent Points and by Coordinates
22. The Theorem of Desargues
The affine part of the theorem of Desargues
Side remark on the projective plane
23. The Theorem of Pappus
Degenerate hexagons
The affine part of the theorem of Pappus
Side remark on associativity
Side remark on the projective plane
Chapter 2 Metric Vector Spaces
24. Inner Products
Definition
Metric Vector Spaces
Orthogonal (perpendicular) vectors
Orthogonal (perpendicular) subspaces
Nonsingular metric vector spaces
25. Inner Products in Terms of Coordinates
Inner products and symmetric bilinear forms
Inner products and quadratic forms
26. Change of Coordinate System
Congruent matrices
Discriminant of V
Euclidean space
The Lorentz plane
Minkowski space
Negative Euclidean space
27. Isometries
Definition
Remark on terminology
Classification of metric vector spaces
28. Subspaces
29. The Radical
Definition
The quotient space V/rad V
Rank of a metric vector space
Orthogonal sum of subspaces
30. Orthogonality
Orthogonal complement of a subspace
Relationships between U and U^*
31. Rectangular Coordinate Systems
Definition
Orthogonal basis
32. Classification of Spaces over Fields Whose Elements have Square Roots
Orthonormal coordinate system
Orthonormal basis
33. Classification of Spaces over Ordered Fields Whose Positive Elements have Square Roots
34. Sylvester’s Theory
Positive semidefinite (definite) spaces
Negative semidefinite (definite) spaces
Maximal positive (negative) definite spaces
Main theorem
Signature of V
Remark about algebraic number fields
Remark about projective geometry
35. Artinian Spaces
Artinian plane
Defense of terminology
Artinian coordinate systems
Properties of Artinian planes
Artinian spaces
36. Nonsingular Completions
Definition
Characterization of Artinian spaces
Orthogonal sum of isometries
37. The Witt Theorem
Fundamental question about isometries
Witt theorem
Witt theorem translated into matrix language
38. Maximal Null Spaces
Definition
Witt index
39. Maximal Artinian Spaces
Definition
Reduction of classification problem to anistropic spaces
A research idea of Artin
40. The Orthogonal Group and the Rotation Group
General linear group GL(m, k)
The orthogonal group O(W)
Rotations and reflections
180° rotations
Symmetries
Rotation group O^+(V)
Remark on teaching high school geometry
41. Computation of Determinants
42. Refinement of the Witt Theorem
43. Rotations of Artinian Space around Maximal Null Spaces
44. Rotations of Artinian Space with a Maximal Null Space as Axis
45. The Cartan–Dieudonné Theorem
Set of generators of a group
Bisector of the vectors A and B
Cartan–Dieudonné theorem
46. Refinement of the Cartan–Dieudonné Theorem
Scherk’s theorem
47. Involutions of the General Linear Group
48. Involutions of the Orthogonal Group
Type of an involution
180° rotation
49. Rotations and Reflections in the Plane
Plane reflections
Plane rotations
50. The Plane Rotation Group
Commutativity of O^+(V)
Extended geometry from V to V′
51. The Plane Orthogonal Group
The exceptional plane
Characterizations of the exceptional plane
52. Rational Points on Coniсs
Circle C_r with radius r
Parametric formulas of the circle C_r
Pythagorean triples
53. Plane Trigonometry
Cosine of a rotation
Matrix of a rotation
Orientation of a vector space
Clockwise and counterclockwise rotations
Sine of a rotation
Sum formulas for the sine and cosine
Circle group
Remark on teaching trigonometry
54. Lorentz Transformations
55. Rotations and Reflections in Three-Space
Axis of a rotation
Four classes of isometries
Rotations with a nonsingular line as axis
56. Null Axes in Three-Space
57. Reflections in Three-Space
Reflections which leave only the origin fixed
Two types of reflections
Remark on high school teaching
58. Cartan–Dieudonné Theorem for Rotations
Fundamental question
Cartan–Dieudonné theorem for rotations
59. The Commutator Subgroup of a Group
60. The Commutator Subgroup of the Orthogonal Group
Birotations
Main theorem
61. The Commutator Subgroup of the Rotation Group
62. The Isometries ±1_V
Center of a group
Magnification with center 0
Magnification and the invariance of lines
Isometries which leave all lines through 0 invariant
63. Centers of O(V), O^+(V), and Ω(V)
Centralizer in O(V) of the set O^+(V)^2
Main theorem
64. Linear Representations of the Groups O(V), O^+(V), and Ω(V)
Definition
Natural representations of (O(V), V), (O^+(V), V), and (Ω(V), V)
Simple representations
Main theorem
65. Similarities
Definition
Main theorem
Factorization of similarities into the product of magnifications and isometries
Chapter 3 Metric Affine Spaces
66. Square distance
Metric affine space (X, V, k)
Orthogonal (perpendicular) affine subspaces
Square distance between points
The metric vector space X(c)
Perpendicular line segments
Perpendicular bisector of a line segment
Remark on high school teaching
67. Rigid Motions
Definition and properties
n-dimensional Euclidean group
Reflections, rotations, symmetries, etc., of X
Isometric affine spaces
68. Interrelation among the Groups Mo, Tr, and O(V)
Diagram of relationships
Glide reflections
69. The Cartan–Dieudonné Theorem for Affine Spaces
Parallel symmetries
Cartan–Dieudonné theorem for anisotropic affine spaces
Null motions
Cartan–Dieudonné theorem for affine spaces
Congruent sets
70. Similarities of Affine Spaces
Definition
A similarity as the product of a magnification and rigid motion
Main theorem
Direct and opposite similarities
Angles
Similar figures
Epilogue
Bibliography
Index