Projective Geometry and Algebraic Structures (eBook)

R. J. Mihalek (Autor)

eBook Download: PDF

2014 | 1. Auflage
232 Seiten
Elsevier Science (Verlag)
978-1-4832-6520-9 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Projective Geometry and Algebraic Structures 4
Copyright Page 5
Table of Contents 6
PREFACE 10
ACKNOWLEDGMENTS 12
CHAPTER 1. 
14 
1.1 Euclidean Planes 14
1.2 Incidence Bases 19
1.3 Set Theory 21
CHAPTER 
25 
2.1 Axioms for an Affine Plane 25
2.2 Examples 30
CHAPTER 
41 
3.1 Axioms for a Projective Plane 41
3.2 Examples 46
3.3 Algebraic Incidence Bases 50
3.4 Self-Dual Axioms 53
CHAPTER 4. 
57 
4.1 Isomorphism 58
4.2 Deletion Subgeometries 62
4.3 The Imbedding Theorem 67
CHAPTER 5. 
74 
5.1 Configurations 74
5.2 Theorem of Desargues 77
5.3 Theorem of Pappus 86
CHAPTER 6. 
92 
6.1 Coordinates 93
6.2 Addition 95
6.3 Multiplication 102
6.4 Algebraic Systems and Incidence Bases 111
6.5 The Coordinatization Theorem 113
6.6 Finite Projective Planes 117
CHAPTER 7. 
119 
7.1 Perspectivities and Projectivities 119
7.2 Some Classical Theorems 127
7.3 A Nonpappian Example 131
CHAPTER 8. 
134 
8.1 Fano Axiom 135
8.2 Harmonic Quadruples 137
CHAPTER 9. 
144 
9.1 Separation 146
9.2 Ordered Fields 151
9.3 Completeness and the Real Numbers 155
9.4 Separation for Basis 3.5 160
9.5 The Real Projective Plane 167
9.6 Euclidean Planes 170
CHAPTER 10. 
173 
10.1 Axioms for a Projective Space 175
10.2 Examples 181
CHAPTER 11. 
185 
11.1 Subspaces and Dimension 185
11.2 Intervals and Complements 199
11.3 Dual Spaces 204
APPENDIX 
208 
Group I. Axioms of Connection 208
Group II. Axioms of Order 209
Group III. Axiom of Parallels 210
Group IV. Axioms of Congruence 210
Group V. Axiom of Continuity 211
APPENDIX 
212 
APPENDIX 
218 
REFERENCES 224
Index of Special Symbols 226
Subject Index 230

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6520-X / 148326520X
• ISBN-13: 978-1-4832-6520-9 / 9781483265209