Möbius transformation

A Möbius transformationMathworldPlanetmath is a bijectionMathworldPlanetmath on the extended complex plane {} given by

f(z)={acif z=if z=-dcaz+bcz+dotherwise

where a,b,c,d and ad-bc0

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

The geometric interpretationMathworldPlanetmathPlanetmath of the Möbius group is that it is the group of automorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the Riemann sphere.

Any Möbius map can be composed from the elementary transformations - dilationsMathworldPlanetmath, translations and inversionsMathworldPlanetmathPlanetmath. If we define a line to be a circle passing through 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