Introduction to hyperbolic geometry (2^nd Ed.)

This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical, the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane, and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Changes to this second edition include: new chapters on Euclidean geometry, as a means of comparing Euclidean and hyperbolic geometry, a new chapter on Euclidean planes, a new proof of the uniqueness of the hyperbolic plane (up to isomorphism and choice of unit length), a proof that the upper half-plane can be made a hyperbolic plane with a new way to measure distance, a new appendix on analysis, the addition of new theorems and the improvement of existing ones.

Introduction.- Axioms for Plane Geometry.- Euclidean Geometry.- Euclidean Planes.- Some Neutral Theorems of Plane Geometry.- Qualitative Description of the Hyperbolic Plane.- H3 and Euclidean Approximations in H2.- Differential Geometry of Surface.- Quantitative Considerations.- Consistency and Categoricalness of the Hyperbolic Axioms- the Classical Models.- Matrix Representation of the Isometry Group.- Differential and Hyperbolic Geometry in More Dimensions.- Connections with the Lorentz Group of Special Relativity.- Constructions by Straightedge and Compass in the Hyperbolic Plane.- Index.