Metric Affine Geometry

Metric Affine Geometry

1st Edition - January 1, 1971

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  • Authors: Ernst Snapper, Robert J. Troyer
  • eBook ISBN: 9781483269337

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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.

Table of Contents

  • Preface


    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


    Translation Group

    5 Affine Subspaces


    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



    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)


    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 78 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


    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


    Remark on Terminology

    Classification of Metric Vector Spaces

    28 Subspaces

    29 The Radical


    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


    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


    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


    Witt Index

    39 Maximal Artinian Spaces


    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


    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-Dieudonne Theorem

    Set of Generators of a Group

    Bisector of the Vectors A and B

    Cartan-Dieudonne Theorem

    46 Refinement of the Cartan-Dieudonne 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 Conies

    Circle Cr With Radius r

    Parametric Formulas of the Circle Cr

    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-Dieudionne Theorem for Rotations

    Fundamental Question

    Cartan-Dieudonne Theorem for Rotations

    59 The Commutator Subgroup of a Group

    60 The Commutator Subgroup of the Orthogonal Group


    Main Theorem

    61 The Commutator Subgroup of the Rotation Group

    62 The Isometries ± 1v

    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)


    Natural Representations of (O(K), V), (O+(V), V) and (Ω(v), V)

    Simple Representations

    Main Theorem

    65 Similarities


    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-Dieudonne Theorem for Affine Spaces

    Parallel Symmetries

    Cartan-Dieudonne Theorem for Anisotropic Affine Spaces

    Null Motions

    Cartan-Dieudonne Theorem for Affine Spaces

    Congruent Sets

    70 Similarities of Affine Spaces


    A Similarity as the Product of a Magnification and Rigid Motion

    Main Theorem

    Direct and Opposite Similarities


    Similar Figures




Product details

  • No. of pages: 456
  • Language: English
  • Copyright: © Academic Press 1971
  • Published: January 1, 1971
  • Imprint: Academic Press
  • eBook ISBN: 9781483269337

About the Authors

Ernst Snapper

Robert J. Troyer

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