This is not the most precise question but rather a hope that someone has seen something like this.
I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I want to do some processing of that which would benefit from all triangles being about the same size or the vertices being distributed in a somewhat uniform manner. The processing I do afterwards is invariant under rotations so that part does not need to be uniformized but rather only the local scaling of Moebius transforms needs to be addressed.
The question is this: Is there a canonical representation of my triangulation maximizing some metric of the sort that I described? If so, is there an algorithm that finds the corresponding Moebius transform?
Thank you
