Introduction to Classical Geometries
by Galarza, Ana Irene Ramirez; Seade, JoseFormat: Paperback
Pub. Date: 2007-03-06
Publisher(s): Birkhauser
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Summary
This book follows Kleins proposal of studying geometry by looking at the symmetries (or rigid motions) of the space in question. In this way the classical geometries are studied: Euclidean, affine, elliptic, projective and hyperbolic. For simplicity the focus is on the two-dimensional case, which is already rich enough, though some aspects of the 3 or $n$-dimensional geometries are included. Once plane geometry is well understood, it is much easier to go into higher dimensions. The book appeals to, and develops, the geometric intuition of the reader. Some basic notions of algebra and analysis are also used to get better understandings of various concepts and results.
Table of Contents
| Preface | p. vii |
| List of Symbols | p. x |
| Euclidean geometry | p. 1 |
| Symmetries | p. 2 |
| Rigid transformations | p. 15 |
| Invariants under rigid transformations | p. 28 |
| Cylinders and tori | p. 37 |
| Finite subgroups of E(2) and E(3) | p. 46 |
| Frieze patterns and tessellations | p. 58 |
| Affine geometry | p. 75 |
| The line at infinity | p. 76 |
| Affine transformations and their invariants | p. 83 |
| Projective geometry | p. 91 |
| The real projective plane | p. 92 |
| The Duality Principle | p. 99 |
| The shape of P[superscript 2](R) | p. 103 |
| Coordinate charts for P[superscript 2](R) (and for P[superscript 1] (C)) | p. 109 |
| The projective group | p. 113 |
| Invariance of the cross ratio | p. 121 |
| The space of conics | p. 126 |
| Projective properties of the conics | p. 129 |
| Poles and polars | p. 134 |
| Elliptic geometry | p. 141 |
| Hyperbolic geometry | p. 149 |
| Models of the hyperbolic plane | p. 149 |
| Transformations of the hyperbolic plane | p. 157 |
| Steiner network | p. 164 |
| The hyperbolic metric | p. 168 |
| First results in hyperbolic geometry | p. 177 |
| Surfaces with hyperbolic structure | p. 183 |
| Tessellations | p. 192 |
| Appendices | p. 199 |
| Differentiable functions | p. 199 |
| Equivalence relations | p. 201 |
| The symmetric group in four symbols: S[subscript 4] | p. 202 |
| Euclidean postulates | p. 205 |
| Topology | p. 206 |
| Some results on the circle | p. 208 |
| Bibliography | p. 211 |
| Index | p. 215 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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