Conformal mapping with as uniform as possible conformal factor

According to the uniformization theorem, any surface can be conformally mapped into a domain of a constant 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 with zero Gaussian curvature. 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 by regular sampling of the domain using, for example, the celebrated fast marching method (FMM). 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. We provide theoretical bounds of the conformal factor and introduce optimization formulations that control its behavior. It is demonstrated that without such restrictions 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 FMM on a weighted flat domain. It is also shown how a metric on a curved domain can be reconstructed by reformulating the nonflat metric restoration problem into a weighted flat domain, again, with bounded weights for consistent results.