Measuring Geodesic Distances via the Uniformization Theorem

According to the Uniformization Theorem any surface can be conformally mapped into a flat domain, that is, a domain with zero Gaussian curvature. The conformal factor indicates the local scaling introduced by such a mapping. This process could be used to compute geometric quantities in a simplified flat domain. For example, the computation of geodesic distances on a curved surface can be mapped into solving an eikonal equation in a plane weighted by the conformal factor. Solving an eikonal equation on the weighted plane can then be done with regular sampling of the domain using, for example, the fast marching method. The connection between the conformal factor on the plane and the surface geometry can be justified analytically. Still, in order to construct consistent numerical solvers that exploit this relation one needs to prove that the conformal factor is bounded. In this paper we provide theoretical bounds over the conformal factor and introduce optimization formulations that control its behavior. It is demonstrated that without such a control the numerical results are unboundedly inaccurate. Putting all ingredients in the right order, we introduce a method for computing geodesic distances on a two dimensional manifold by using the fast marching algorithm on a weighed flat domain.