## Defining a Mobius Transformation With the Standard Triple

A Mobius transformation is defined completely by its effect on three points of the extended complex plane We can find a Mobius transformation that maps any three points of the extended complex plane to any other three points Finding 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 that and This formula also works if one of is the point at infinity providing that factors containing infinity are cancelled out, so that for example Let send to 0, to 1 and to infinity . Then and  sends 0 to 1 to and to then sends to 0 then to  to 1 then to and to infinity then to Since a composition of extended Mobius transformations is also and extended Mobius transformation, the extended Mobius transformation that sends to  to and to 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 gives and collecting the terms in on the left hand side and factorising gives hence  