Discrete Surface Ricci Flow: Theory and Applications

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 Poincar´e 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 ...

• Type: Conference
• Year: 2007
• Where: IMAMS
• Authors: Miao Jin, Junho Kim, Xianfeng David Gu
• Keywords: Discrete Ricci flow, IMAMS 2007, Mathematics, Ricci flow, Riemannian Metric