Abstract. Conformal geometry is in the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces- discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincaré conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one. In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed. Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.

Surface parameterization is a fundamental problem in graphics. Conformal surface parameterization is equivalent to finding a Riemannian metric on the surface, such that the metric is conformal to the original metric and induces zero Gaussian curvature for all interior points. Ricci flow is a theoretic tool to compute such a conformal flat metric. This paper introduces an efficient and versatile parameterization algorithm based on Euclidean Ricci flow. The algorithm can parameterize surfaces with different topological structures in an unified way. In addition, we can obtain a novel class of parameterization, which provides a conformal invariant of a surface that can be used as a surface signature. 1.