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Springer Undergraduate Mathematics

By: James Anderson

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Paperback | 2 August 2005 | Edition Number 2

At a Glance

Paperback
292 Pages

Edition Type
Revised

Dimensions(cm)
22.86 x 17.78 x 1.91

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The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This second edition of Hyperbolic Geometry has been thoroughly rewritten and updated. Chapter 4 focuses on planar models of hyperbolic plane that arise from complex analysis and looks at the connections between planar hyperbolic geometry and complex analysis.However most of the new material will appear in Chapter 6 and concentrates on an introduction to the hyperboloid model of the hyperbolic plane. The chapter concludes with a discussion of hyperbolic geometry in higher dimensions, and generalizations of hyperbolicity (this, in particular, is an important topic that allows for an in-depth development of the fundamental concepts). This book is written primarily for third or fourth year undergraduate students with some calculus knowledge. It contains new exercises with solutions and is ideal for self-study or as a classroom text.

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Non-Fiction Mathematics Geometry Non-Euclidean Geometry

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Non-Fiction Mathematics Geometry Non-Euclidean Geometry

Preamble to the Second Edition	p. vii
Preamble to the First Edition	p. ix
The Basic Spaces	p. 1
A Model for the Hyperbolic Plane	p. 1
The Riemann Sphere [characters not reproducible]	p. 8
The Boundary at Infinity of H	p. 18
The General Mobius Group	p. 23
The Group of Mobius Transformations	p. 23
Transitivity Properties of Mob[superscript +]	p. 30
The Cross Ratio	p. 36
Classification of Mobius Transformations	p. 39
A Matrix Representation	p. 42
Reflections	p. 48
The Conformality of Elements of Mob	p. 53
Preserving H	p. 56
Transitivity Properties of Mob(H)	p. 62
The Geometry of the Action of Mob(H)	p. 65
Length and Distance in H	p. 73
Paths and Elements of Arc-length	p. 73
The Element of Arc-length on H	p. 80
Path Metric Spaces	p. 88
From Arc-length to Metric	p. 92
Formulae for Hyperbolic Distance in H	p. 99
Isometries	p. 103
Metric Properties of (H,d[subscript H])	p. 108
Planar Models of the Hyperbolic Plane	p. 117
The Poincare Disc Model	p. 117
A General Construction	p. 130
Convexity, Area, and Trigonometry	p. 145
Convexity	p. 145
Hyperbolic Polygons	p. 154
The Definition of Hyperbolic Area	p. 164
Area and the Gauss-Bonnet Formula	p. 169
Applications of the Gauss-Bonnet Formula	p. 174
Trigonometry in the Hyperbolic Plane	p. 181
Nonplanar models	p. 189
The Hyperboloid Model of the Hyperbolic Plane	p. 189
Higher Dimensional Hyperbolic Spaces	p. 209
Solutions to Exercises	p. 217
Referfences	p. 265
List of Notation	p. 269
Table of Contents provided by Ingram. All Rights Reserved.

Springer Undergraduate Mathematics

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