, the Loxodromic Möbius transformations forms subgroup isomorphic to the group of matrices , , the elliptic Möbius transformations forms subgroup isomorphic to the group of matrices , the hyperbolic Möbius transformations forms subgroup isomorphic to the group of matrices Tr 2(A) = 4, the parabolic Möbius transformations forms subgroup isomorphic to the group of matricesĪnd this transformation is orientation preserving. There are Parabolic, elliptic, hyperbolic and loxodromic which are distinguished by looking at the trace tr(A) = a + b. This mean, then the fixed points are given byįor non parabolic transformation, there are two fixed points 0, ¥ but for parabolic transformation, there is only fixed points ¥ because the fixed points are coincide. invariant points) is defined by f(z) = z, then The Fixed Points in Mobius Transformation Then matrix A and matrix B must be conjugate.Ģ. Since matrix A and B are Möbius transformations, then Since the trace of matrix A is tr(A) = a + b and this trace is invariant under conjugation, this is mean,Įvery Möbius transformation can be represented by normalized matrix A such that its determinant equal one which mean ad − bc = 1. If any one of z i = 0 for example z 3 = 0, then Since translation, rotation and dilation preserve cross ratio and Möbius transformation consists of them so Möbius transformation preserves cross ratio. The cross ratio is invariant of the group of all Möbius transformation so if we transform the four points z i intoīy an inversion, the cross ratio of these points are taken into its conjugate value, and the cross ratio is invariant under a product of two or any even number of inversions and exchanging any two pairs of coordinates preserves the cross-ratio. Given four distinct points z 1, z 2, z 3, z 4, their cross ratio is defined by Möbius transformations also preserve cross ratio. The inverse Möbius transformation is evaluated from the inverse of the metric We can write Möbius transformations as follows Then the plane inside turn out and the lines on the plane are lines or circles and right angles stay true and also the circles are circles Since Möbius transformation takes the formĪ Möbius transformation consists of four composition functions.Ģ) inversion and reflection with respect to real axis The angles are Euclidean angles.Ī Möbius transformations form a group which is denoted by. The lines (geodesics) are vertical rays and semicircles orthogonal to ¶H. The upper half plane model is defined by the set Which sending each point to a corresponding point, where z is the complex variable and the coefficients a, b, c, d are complex numbers. Möbius transformations are formed a group action PSL (2,Â) on the upper half plane model.Ī Möbius transformation of the plane is a map f: The basic properties of these transformations are introduced and classified according to the invariant points. The purpose of this paper is studied the properties of Möbius transformations in detail, and some definitions and theorems are given. Möbius transformations are also called homographic transformations, linear fractional transformations, or fractional linear transformations and it is a bijective holomorphic function (conformal map). Furthermore, the conformal mapping is represented as bilinear translation, linear fractional transformation and Mobius transformation. Möbius transformations have applications to problems in physics, engineering and mathematics. The Upper Half-Plane Model, Möbius Transformation, Hyperbolic Distance, Fixed Points, The Group PSL (2,Â) Moreover, we can see that Möbius transformation is hyperbolic isometries that form a group action PSL (2,Â) on the upper half plane model. For instance, Möbius transformation is classified according to the invariant points. In this paper, I have provided a brief introduction on Möbius transformation and explored some basic properties of this kind of transformation. Received revised 2 July 2014 accepted 14 July 2014 This work is licensed under the Creative Commons Attribution International License (CC BY). Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, EgyptĮmail: © 2014 by author and Scientific Research Publishing Inc.
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