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The Poincaré half-plane : a gateway to modern geometry

Author: Saul Stahl
Publisher: Boston : Jones and Bartlett Publishers, ©1993.
Series: Jones and Bartlett books in mathematics.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
In the 1880s, over fifty years after the discovery of the hyperbolic plane, Poincare pointed out that this plane provides a very useful context for describing the properties of the solutions of an important class of differential equations.
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Saul Stahl
ISBN: 086720298X 9780867202984
OCLC Number: 26854265
Description: xiii, 298 pages : illustrations ; 25 cm.
Contents: Euclidean geometry --
Euclidean rigid motions --
Inversions --
The hyperbolic plane --
Euclidean versus hyperbolic geometry --
The angles of the hyperbolic triangle --
Hyperbolic area --
The trigonometry of the hyperbolic triangle --
Complex numbers and rigid motions --
Absolute geometry and the angles of the triangle --
Spherical trigonometry and elliptic geometry --
Differential geometry and Gaussian curvature --
The cross ratio and the unit disk model --
The Beltrami-Klein model --
a brief history of non-Euclidean geometry --
Spheres and horospheres --
Appendix: Proofs of some of Euclid's propositions.
Series Title: Jones and Bartlett books in mathematics.
Responsibility: Saul Stahl.
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Abstract:

In the 1880s, over fifty years after the discovery of the hyperbolic plane, Poincare pointed out that this plane provides a very useful context for describing the properties of the solutions of an  Read more...

Table of Contents:

by WTYILL@KUK (WorldCat user on 2006-06-29)

Preface xi Chapter Dpenedencies xiii Chapter 1. EUCLIDEAN GEOMETERY 1 1. A breif history of Euclidean geometry 1 2. Excerpts from Euclid's Elements 3 3. Hilbert's axiomatization of Euclidean geometry (optional) 22 4. Variations on Euclid's fifth 27 5. Exercises 31 Chapter 2. EUCLIDEAN RIGID MOTIONS 35 1. Introduction 35 2. Rigid motions 36 3. Translations, rotations, and reflections 37 4. Glide-reflections 43 5. The main theorems 46 6. Rigid motions and absolute geometry 48 7. Exercises 49 CHAPTER 3. INVERSIONS 51 1. An interesting non-rigid transformation 51 2. An application of inversions (optional) 59 3. Exercises 61 CHAPTER 4. THE HYPERBOLIC PLANE 63 1. The hyperbolic distance 63 2. Hyperbolic straight lines 66 3. Hyperbolic angles 70 4. Hyperbolic rigid motions 71 5. Riemannian geometry (optional) 73 6. Exercises 76 CHAPTER 5. EUCLIDEAN VERSUS HYPERBOLIC GEOMETRY 79 1. Euclid's postulates revisited 79 2. Absolute geometry 88 3. Hyperbolic geometry 88 4. Hyperbolic rigid motions 88 5. Exercises 90 CHAPTER 6. THE ANGLES OF THE HYPERBOLIC TRIANGLE 93 1. Introduction 93 2. The standard position of a triangle 93 3. The sum of the angeles of the hyperbolic triangle 95 4. A new congruence theorem 103 5. Regular tesselations (optional) 104 6. Exercises 107 CHAPTER 7. HYPERBOLIC AREA 109 1. The general definition of area 109 2. The area of the hyperbolic triangle 114 3. Exercises 116 CHAPTER 8. THE TRIGONOMETRY OF THE HYPERBOLIC TRIANGLE 119 1. The trigonometry of hyperbolic line segments 119 2. Hyperbolic right triangles 122 3. The General hyperbolic triangle 125 4. Exercises 128 CHAPTER 9. COMPLEX NUMBERS AND RIGID MOTIONS 131 1. Complex numbers and Euclidean rigid motions 131 2. Hyperbolic rigid motions 136 3. Euclidean flow diagrams 143 4. Hyperbolic flow diagrams- rotations 145 5. Hyperbolic flow diagrams- translations 145 6. Hyperbolic flow diagrams- the general case 151 7. Hyperbolic rigid motions- constructions 155 8. Exercises 158 CHAPTER 10. ABSOLUTE GEOMETERY AND THE ANGLES OF THE TRIANGLE 161 1. The sum of the angles of the triangle 161 2. Exercises 166 CHAPTER 11. SPHERICAL TRIGONOMETRY AND ELLIPTIC GEOMETRY 167 1. Introduction 167 2. Geodesics on the sphere 168 3. Spherical trigonometry 172 4. Spherical areas 175 5. A digression into geodesy (optional) 177 6. Elliptic geometry 179 7. Exercises 180 CHAPTER 12. DIFFERENTIAL GEOMETRY AND GAUSSIAN CURATURE 183 1. Differential geometry 183 2. A review of lengths and areas on surfaces 190 3. Gauss's formula for the curvature at a point 196 4. Reimannian geometry revisited 198 5. Exercises 205 CHAPTER 13. THE CROSS RATIO AND THE UNIT DISK MODEL 207 1. INTRODUCTION 207 2. CONFORMal transformations 207 3. The cross ratio 209 4. The unit disk model and its flow diagrams 213 5. Explicit rigid motions of the unit disk model 221 6. The Riemann metric of the unit disk model 225 7. Regular tesselations of the unit disk model 228 8. Exercises 230 CHAPTER 14. THE BELTRAMI-KLEIN MODEL 233 1. Introduction 233 2. The Beltrami-Klein Model 234 3. Exercises 245 CHAPTER 15. A BREIF HISTORY OF NON-EUCLIDEAN GEOMETERY 247 1. History 247 2. Exercises 254 CHAPTER 16. SPHERES AND HOROSPHERES 267 1. Introduction 257 2. Hyperbolic space and its rigid motions 258 3. Hyperbolic geodesics 262 4. The sterographic projection of spheres 266 5. The geometry of spheres and horospheres 271 6. Exercises 275 APPENDIX 277 Proofs of some of Euclid's propsoitions 277 BIBLIOGRAPHY 293 INDEX 295

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