Computing Surface Uniformization Using Discrete Beltrami Flow

Abstract

In this paper, we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology. According to the celebrated uniformization theorem, all Riemann surfaces can be classified as elliptic, parabolic, or hyperbolic. Our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain, such as a sphere, or a polygon in the plane $\mathbb{R}^2$ or the hyperbolic disk $D$, and then morphing the diffeomorphism based on the discrete Beltrami flow algorithm. For high genus surfaces, both the final mapping and the target domain are unknown, which presents a challenge in general. Each such surface can be conformally mapped onto $D$ modulo a discrete subgroup of all fractional linear transforms on $D$. A conformal copy of the surface, which is also its uniformization domain, can be visualized as a fundamental polygon in $D$ corresponding to the discrete subgroup, where the generators of the discrete subgroup map each side of the fundamental polygon to its corresponding side, giving the conformality information of the surface. The novelty in our method lies in the iterative change of these generators as the diffeomorphism is morphing, which indicates a change of geometry of the target domain to match the geometry of the original surface. Numerical results are presented to show the efficiency and accuracy (in terms of distortion) of our methods as well as to make a comparison to other state-of-the-art algorithms.