Cross-ratio as a complete invariant on ordered 4-tuples:
Let (z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4) be 4-tuples of distinct points in \mathbb{C}_\infty, and define them to be equivalent if there is a Mobius transformation f \in \mathfrak{M} sending one to the other.
Then (z_1, z_2, z_3, z_4) and (w_1, w_2, w_3, w_4) are equivalent if and only if the cross-ratios \left[z_1, z_2; z_3, z_4\right] = \left[w_1, w_2; w_3, w_4\right] are the same.

\Rightarrow. If (z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4) are equivalent then \left[z_1, z_2; z_3, z_4 \right] = \left[ fz_1, fz_2; fz_3, fz_4 \right] = \left[ w_1, w_2; w_3, w_4 \right] .
\Leftarrow. Let Sz = [z, z_2; z_3, z_4], Tw = [w, w_2; w_3, w_4]. Then T^{-1}S \in \mathfrak{M} sends z_2 \rightarrow w_2, z_3 \rightarrow w_3, z_4 \rightarrow w_4 . Furthermore since Sz_1 = Tw_1, we have that T^{-1}Sz_1 = w_1.


Permuting entries of the cross-ratio: If \lambda = \left[ z_1, z_2; z_3, z_4\right] then

  1. \left[ z_1, z_2; z_4, z_3\right] = \frac{1}{\lambda}
  2. \left[ z_1, z_3; z_2, z_4\right] = 1- \lambda
  3. \left[ z_2, z_1; z_3, z_4 \right] = \frac{1}{\lambda}