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</B

Details

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

About the Author

R. J. Mihalek

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