Defining a Mobius Transformation With the Standard Triple
A Mobius transformation is defined completely by its effect on three points of the extended complex planeWe can find a Mobius transformation that maps any three pointsof the extended complex plane to any other three pointsFinding this transformation is tedious and it is easier to consider the effect of the Mobius transformation (or its inverse) on a standard triple.
The standard triple is the set of points
It is easily seen thatand
This formula also works if one ofis the point at infinity providing that factors containing infinity are cancelled out, so that for example
Letsendto 0,to 1 andto infinity . Then
sends 0 to1 toandtothensendsto 0 then toto 1 then toandto infinity then toSince a composition of extended Mobius transformations is also and extended Mobius transformation,the extended Mobius transformation that sendstotoandto
We can find this transformation using the 'implicit formula'by solving for
Example: Find the extended Mobius transformation that sends the points -i, -1, i to 4, 3, 2 respectively.
Using the implicit formula we obtain
Finding the constant factors and cross multiplication gives
Expanding the brackets givesand collecting the terms inon the left hand side and factorising giveshence