Hyperbolic Geometry

  • James W. Anderson

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

Table of contents

  1. Front Matter
    Pages i-ix
  2. James W. Anderson
    Pages 1-18
  3. James W. Anderson
    Pages 19-56
  4. James W. Anderson
    Pages 57-94
  5. James W. Anderson
    Pages 95-110
  6. James W. Anderson
    Pages 111-151
  7. James W. Anderson
    Pages 153-178
  8. Back Matter
    Pages 179-230

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.





Hyperbolic geometry Hyperbolic plane Hyperbolicity geometry mathematics

Authors and affiliations

  • James W. Anderson
    • 1
  1. 1.Faculty of Mathematical StudiesUniversity of Southampton, HighfieldSouthamptonUK

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag London 1999
  • Publisher Name Springer, London
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-85233-156-6
  • Online ISBN 978-1-4471-3987-4
  • Series Print ISSN 1615-2085
  • Buy this book on publisher's site