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 toand
tothen
sends
to 0 then to
to 1 then toand
to infinity then to
Since 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 gives
and collecting the terms in
on the left hand side and factorising gives
hence