Metric Affine Geometry

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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