Metric Affine Geometry - 1st Edition - ISBN: 9780126536508, 9781483269337

Metric Affine Geometry

1st Edition

Authors: Ernst Snapper Robert J. Troyer
eBook ISBN: 9781483269337
Imprint: Academic Press
Published Date: 1st January 1971
Page Count: 456
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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



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





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© Academic Press 1971
Academic Press
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About the Author

Ernst Snapper

Robert J. Troyer

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