Projective Geometry and Algebraic Structures - 1st Edition - ISBN: 9780124955509, 9781483265209

Projective Geometry and Algebraic Structures

1st Edition

Authors: R. J. Mihalek
eBook ISBN: 9781483265209
Imprint: Academic Press
Published Date: 1st January 1972
Page Count: 232
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Description

Projective Geometry and Algebraic Structures focuses on the relationship of geometry and algebra, including affine and projective planes, isomorphism, and system of real numbers.

The book first elaborates on euclidean, projective, and affine planes, including axioms for a projective plane, algebraic incidence bases, and self-dual axioms. The text then ponders on affine and projective planes, theorems of Desargues and Pappus, and coordination. Topics include algebraic systems and incidence bases, coordinatization theorem, finite projective planes, coordinates, deletion subgeometries, imbedding theorem, and isomorphism.

The publication examines projectivities, harmonic quadruples, real projective plane, and projective spaces. Discussions focus on subspaces and dimension, intervals and complements, dual spaces, axioms for a projective space, ordered fields, completeness and the real numbers, real projective plane, and harmonic quadruples.

The manuscript is a dependable reference for students and researchers interested in projective planes, system of real numbers, isomorphism, and subspaces and dimensions.

Table of Contents


Preface

Acknowledgments

Chapter 1 Introduction

1.1 Euclidean Planes

1.2 Incidence Bases

1.3 Set Theory

Chapter 2 Affine Planes

2.1 Axioms for an Affine Plane

2.2 Examples

Chapter 3 Projective Planes

3.1 Axioms for a Projective Plane

3.2 Examples

3.3 Algebraic Incidence Bases

3.4 Self-Dual Axioms

Chapter 4 Affine And Projective Planes

4.1 Isomorphism

4.2 Deletion Subgeometries

4.3 The Imbedding Theorem

Chapter 5 Theorems of Desargues and Pappus

5.1 Configurations

5.2 Theorem of Desargues

5.3 Theorem of Pappus

Chapter 6 Coordinatization

6.1 Coordinates

6.2 Addition

6.3 Multiplication

6.4 Algebraic Systems and Incidence Bases

6.5 The Coordinatization Theorem

6.6 Finite Projective Planes

Chapter 7 Projectivities

7.1 Perspectivities and Projectivities

7.2 Some Classical Theorems

7.3 A Nonpappian Example

Chapter 8 Harmonic Quadruples

8.1 Fano Axiom

8.2 Harmonic Quadruples

Chapter 9 The Real Projective Plane

9.1 Separation

9.2 Ordered Fields

9.3 Completeness and the Real Numbers

9.4 Separation for Basis 3.5

9.5 The Real Projective Plane

9.6 Euclidean Planes

Chapter 10 Projective Spaces—Part 1

10.1 Axioms for a Projective Space

10.2 Examples

Chapter 11 Projective Spaces—Part 2

11.1 Subspaces and Dimension

11.2 Intervals and Complements

11.3 Dual Spaces

Appendix A Hilbert's Axioms for a Euclidean Plane

Group I. Axioms of Connection

Group II. Axioms of Order

Group III. Axiom of Parallels

Group IV. Axioms of Congruence

Group V. Axiom of Continuity

Appendix B Division Rings

Appendix C Quaternions

References

Index of Special Symbols

Subject Index

Details

No. of pages:
232
Language:
English
Copyright:
© Academic Press 1972
Published:
Imprint:
Academic Press
eBook ISBN:
9781483265209

About the Author

R. J. Mihalek