The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications.

This updated second edition also features:

• An expanded discussion of planar models of the hyperbolic plane arising from complex analysis
• The hyperboloid model of the hyperbolic plane;
• A brief discussion of generalizations to higher dimensions
• Many new exercises

Table of Contents

1 - The Basic Spaces

2 - The General Mobius Group

3 - Length and Distance in H

4 - Planar Models of the Hyperbolic Plane

5 - Convexity Area, and Trigonometry

6 - Nonplanar models

Solutions to Exercises