Canonical conformal mapping for high genus surfaces with boundaries

Abstract

Conformal mapping plays an important role in Computer Graphics and Shape Modeling. According to Poincaré's uniformization theorem, all closed metric surfaces can be conformally mapped to one of the three canonical spaces, the sphere, the plane or the hyperbolic disk. This work generalizes the uniformization from closed high genus surfaces to high genus surfaces with boundaries, to map them to the canonical spaces with circular holes. The method combines discrete surface Ricci flow and Koebe's iteration with zero holonomy condition. Theoretic proof for the convergence is given. Experimental results show that the method is general, stable and practical. It is fundamental and has great potential to geometric analysis in various fields of engineering and medicine.