Möbius Transformations

Abstract

In this thesis the Möbius transformations are studied along with two of its applications. Firstly, Möbius transformations are characterized as transformations of the extended complex plane to itself M : C1 7! C1. Some of their geometric properties, as preservation of generalized circles and angles are studied. Their fixed points are then reviewed, and with the help of the concept of cross-ratio, a way to find Möbius transformations with specific behaviour is found. Lastly, the set of all Möbius transformations M is studied as a group isomorphic to PSL(2; C), and a classification into four categories is made. In the second part, their role in the Poincaré half-plane model of hyperbolic geometry is studied. Such model is briefly introduced, defining the concepts of hyperbolic length and distance in the half-plane, as well as lines and geodesics. Using the geometric properties of Möbius transformations, the lines of the model are derived. Lastly, the connection of the Möbius group and the Lorentz transformations is reviewed. The foundamentals of special relativity are briefly introduced. Then, the isomorphism between the Möbius group and a subset of the Lorentz gorup, the restricted Lorentz group, is fully reviewed. This will allow the study of transformations of the celestial sphere using Möbius transformations.