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.