# Matematical: fractional-linear transformations

Let's think about fractional-linear transformations. These are analytical functions on the complex plane C of the form f(z)=(az+b)/(cz+d) where a,b,c,d are complex, and ad-bc=1. This set of functions forms a group with respect to the composition which might be identified with SL(2,C) - special linear group of 2x2 matrices over C of determinant 1.

Each fractional linear transformation is actually is an automorphism of extended complex plane {C union with {infinity}} which via stereographic projection is nothing but S^2 - two dimensional sphere.

The following property of this group is remarkable:
given three points of C: z1,z2,z3 and other three points w1,w2,w3. Then there is a unique fractional -linear transformation f:C->C s.t. f(z1)=w1,f(z2)=w2,f(z3)=w3.
You can write this transformation explicitly:
((z-z1)(z3-z2))/((z-z2)(z3-z1)) =
((w-w1)(w3-w2))/((w-w2)(w3-w1))
and then solve for w=f(z).

Now the natural question is coming what if we have n points Z1,Z2,...Zn and W1,W2,...Wn.
Any examples of the families of analytical functions which form a group under the compositon such that we can find a unique function which transforms Z's to W's.

Of course, it would be nice to find as simple family as possible.
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