Mobius Transformations

A Mobius transformation is any rational function of the formwhereandsatisfy


Every Mobius transformation is conformal – preserves angles – and analytic on

Every Mobius transformation is either a linear function or a quotient of linear functions. If


Every Mobius transformationmay be extended toso that ifthenandThe extended function is labelledExtended Mobius transformations mapone to one ontoand generalized circles – circles and lines – onto generalized circles.


The set of extended Mobius transformations has the following group properties:

  1. Closure: Ifandare extended Mobius transformations then so is

  2. Identity: The identity function inis an extended Mobius transformation.

  3. Associativity: Ifandare extended Mobius transformations then

An extended Mobius transformationhas at least one fixed pointsuch that

An extended Mobius transformation is defined by it's effect on three distinct points.

Given three distinct pointsinand any other three distinct pointsin there is a unique extended Mobius transformation that sendstototo In particular ifarerespectively then

Ifare generalized circles then there is an extended Mobius transformation that maps to