The Poincaré Half-plane: A Gateway to Modern GeometryJones & Bartlett Learning, 1993 - 298 sider The Poincare Half-Planeprovides an elementary and constructive development of this geometry that brings the undergraduate major closer to current geometric research. At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor. |
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Side iv
... storage or retrieval system , without written permission from the copyright owner . Printed in the United States of America 07 06 05 04 03 10 9 8 7 6 5 4 To Susan Table of Contents Preface xi Chapter Dependencies xiii.
... storage or retrieval system , without written permission from the copyright owner . Printed in the United States of America 07 06 05 04 03 10 9 8 7 6 5 4 To Susan Table of Contents Preface xi Chapter Dependencies xiii.
Side v
Beklager, innholdet på denne siden er tilgangsbegrenset..
Beklager, innholdet på denne siden er tilgangsbegrenset..
Side vii
... 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 ...
... 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 ...
Side viii
... 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 ...
... 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 ...
Side ix
... CHAPTER 12. DIFFERENTIAL GEOMETRY AND GAUSSIAN CURVATURE 183 1. Differential geometry 183 2. A review of lengths and areas on surfaces 3 . 190 Gauss's formula for the curvature at a point 196 4. Riemannian geometry revisited 5 ...
... CHAPTER 12. DIFFERENTIAL GEOMETRY AND GAUSSIAN CURVATURE 183 1. Differential geometry 183 2. A review of lengths and areas on surfaces 3 . 190 Gauss's formula for the curvature at a point 196 4. Riemannian geometry revisited 5 ...
Innhold
EUCLIDEAN RIGID MOTIONS | 35 |
INVERSIONS | 51 |
THE HYPERBOLIC PLANE | 63 |
EUCLIDEAN VERSUS HYPERBOLIC GEOMETRY | 79 |
THE ANGLES OF THE HYPERBOLIC TRIANGLE | 93 |
HYPERBOLIC AREA | 109 |
THE TRIGONOMETRY OF THE HYPERBOLIC | 119 |
The general hyperbolic triangle | 125 |
SPHERICAL TRIGONOMETRY AND ELLIPTIC | 167 |
DIFFERENTIAL GEOMETRY AND GAUSSIAN | 183 |
THE CROSS RATIO AND THE UNIT DISK | 207 |
Explicit rigid motions of the unit disk model | 221 |
Regular tesselations of the unit disk model | 228 |
A BRIEF HISTORY OF NONEUCLIDEAN | 247 |
Exercises | 254 |
APPPENDIX | 277 |
COMPLEX NUMBERS AND RIGID MOTIONS | 131 |
ABSOLUTE GEOMETRY AND THE ANGLES | 161 |
BIBLIOGRAPHY | 293 |
Vanlige uttrykk og setninger
absolute geometry arbitrary arcs axioms axis Beltrami-Klein Beltrami-Klein model bowed geodesic Chapter Common Notions complex numbers composition congruent containing cos² cosh cross ratio curvature curve defined denote dx² dy² elliptic elliptic geometry equation Euclidean center Euclidean circle Euclidean geometry Euclidean plane Euclidean straight lines Example Exercise FIGURE fixed point flow diagram flow lines follows function Gauss Gaussian curvature geodesic segment given glide-reflections Hence hyperbolic area hyperbolic distance hyperbolic geometry hyperbolic length hyperbolic plane hyperbolic reflection hyperbolic rigid motion hyperbolic rotation hyperbolic triangle interior intersect inversion Lemma mathematicians Moebius rigid motion Moebius transformation non-Euclidean geometry parallel Parallel Postulate Poincaré metric proof of Proposition Prove reader Riemann metric right angles sides sin² sinh sphere straight geodesic surface tangent Theorem translation triangle ABC unit disk model upper half-plane vertices x-axis xy plane მუ
Referanser til denne boken
Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Begrenset visning - 2000 |