Hyperbolic Geometry

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Pages 1-21
  3. Pages 189-215
  4. Back Matter
    Pages 217-276

About this book


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.

Topics covered include the upper half-plane model of the hyperbolic plane, Möbius transformations, the general Möbius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincaré disc model, convex subsets of the hyperbolic plane, hyperbolic area, the Gauss-Bonnet formula 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;

brief discussion of generalizations to higher dimensions;

many new exercises.

The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject and provides the reader with a firm grasp of the concepts and techniques of this beautiful part of the mathematical landscape.





Area Geometry Hyperbolic geometry Hyperbolic plane Hyperbolicity Polygon calculus mathematics

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag London Limited 2005
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-85233-934-0
  • Online ISBN 978-1-84628-220-1
  • Series Print ISSN 1615-2085
  • Buy this book on publisher's site