Metric Affine Geometry

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