Mobius maps to {0, 1, ∞}

Belyi functions are unique up to Mobius transformations, which are conformal maps from the Riemann sphere to the Riemann sphere that preserve, not merely infinitesimal circles, but all circles. The fate of any three distinct points determines a Mobius function. A particularly simple case is when we know which three points will map to 0, 1, and ∞. Namely, we seek a Mobius transformation S, such that:

S(z₀) = 0

S(z₁) = 1

S(z_∞) = ∞

Then S = ((z-z₀)*(z₁-z_∞))/((z-z_∞)*(z₁-z₀)).

For example, the vertex-color permuting Belyi functions can be derived this way.