If z1, z2, z3, z4 are complex numbers, then the quantity:
(z3-z1)   (z4-z2)  
------- * -------
(z3-z2)   (z4-z1)
is called the cross ratio of the 4-tuple (z1,z2,z3,z4).
The cross ratio is conserved under translation (z+a), inversion (1/z), dilation and rotation (a*z). For such a transformation s, the cross ratio of (s(z1),s(z2),s(z3),s(z4)).

In the complex numbers, any transformation s that is a composition of these four simple transformations is of the form:

       a*z + b
s(z) = -------
       c*z + d
where a,b,c,d,z are complex numbers.
From this it is simple calculation to verify that the cross ratio is conserved.

Log in or register to write something here or to contact authors.