# Möbius transformation

A Möbius transformation is a bijection on the extended complex plane $\mathbb{C}\cup\{\infty\}$ given by

 $f(z)=\begin{cases}\frac{a}{c}&\text{if }z=\infty\\ \infty&\text{if }z=-{d\over c}\\ \frac{az+b}{cz+d}&\text{otherwise}\end{cases}$

where $a,b,c,d\in\mathbb{C}$ and $ad-bc\neq 0$

It can be shown that the inverse, and composition of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.

The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere.

Any Möbius map can be composed from the elementary transformations - dilations, translations and inversions. If we define a line to be a circle passing through $\infty$ then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

 Title Möbius transformation Canonical name MobiusTransformation Date of creation 2013-03-22 12:23:19 Last modified on 2013-03-22 12:23:19 Owner Koro (127) Last modified by Koro (127) Numerical id 21 Author Koro (127) Entry type Definition Classification msc 30D99 Synonym fractional linear transformation Synonym linear fractional transformation Related topic ProofOfConformalMobiusCircleMapTheorem Related topic AutomorphismsOfUnitDisk Related topic UnitDiskUpperHalfPlaneConformalEquivalenceTheorem Related topic InversionOfPlane