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Applied Mathematics, 2014, 5, 2216-2225
Published Online August 2014 in SciRes. /journal/am
/10.4236/am.2014.515215
Lecture Notes of Möbuis Transformation in
Hyperbolic Plane
Rania B. M. Amer
Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
Email: dr.raniaamer@
Received 26 May 2014; revised 2 July 2014; accepted 14 July 2014
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
/licenses/by/4.0/
Abstract
In this paper, I have provided a brief introduction on Möbius transformation and explored some
basic properties of this kind of transformation. For instance, Möbius transformation is classified
according to the invariant points. Moreover, we can see that Möbius transformation is hyperbolic
isometries that form a group action PSL (2,ℜ) on the upper half plane model.
Keywords
The Upper Half-Plane Model, Möbius Transformation, Hyperbolic Distance, Fixed Points,
The Group PSL (2,ℜ)
1. Introduction
Möbius transformations have applications to problems in physics, engineering and mathematics. Furthermore,
the conformal mapping is represented as bilinear translation, linear fractional transformation and Mobius trans-
formation.
Möbius transformations are also called homographic transformations, linear fractional transformations, or
fractional linear transformations and it is a bijective holomorphic function (conformal map) [1] [2].
The purpose of this paper is studied the properties of Möbius transformations in detail, and some definitions and
theorems are given. The basic properties of these transformations are introduced and classified according to the
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