The famous Finite Riemann Mapping Theorem (FRMT) that was proved independently by Cannon-Floyd-Parry and Schramm (mid 90s) asserts that given a triangulated planar topological quad. there exists a realization of it by a straight rectangle which is tilled by squares. Each square correspond to a vertex of the given triangulation. While (in general) degeneracies will occur the combinatorics of the triangulation is roughly preserved in the tilled rectangle. We will present the following.

Theorem (Her 2006). LetWbe a bounded planar triangulated domain whose boundary consist of a finite number of curves. Let E1 and E2 be disjoint and each a finite union of closed arcs or closed curves contained in the boundary of W. Then there exists a unique (up to scaling) geometric realization of W, E1, E2.