The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

Every open ball within R∞ has an associated hyperbolic metric and Möbius transformations act as hyperbolic isometries from one ball to another. The Hillam-Thron Theorem is concerned with images of balls under Möbius transformation, yet existing proofs of the theorem do not make use of hyperbolic geometry. We exploit hyperbolic geometry in proving a generalisation of the Hillam-Thron Theorem and...

In this note, we present the hyperbolic Menelaus theorem in the Poincaré disc of hyperbolic geometry. 2000 Mathematical Subject Classi…cation: 30F45, 20N99, 51B10, 51M10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, gyrovector 1. Introduction Hyperbolic Geometry appeared in the …rst half of the 19 century as an attempt to understand Euclid’s axiomatic basis of Geometry. It is ...