Hyperbolic Geometry
Description
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.
Reviews / Votes
Praise for the first edition: "... The textbook is a good and useful introduction to hyperbolic geometry, and can be recommended for undergraduate courses."
Newsletter of the EMS, Issue 41, December 2001
From the reviews of the second edition:
"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, suitable for third or fourth year undergraduates. . The basic approach taken is to define hyperbolic lines and develop a natural group of transformations preserving hyperbolic lines, and then study hyperbolic geometry as those quantities invariant under this group of transformations." (L'Enseignement Mathematique, Vol. 51 (3-4), 2005)
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