Variational Method on Discrete Ricci Flow
International Workshop on Combinatorial Image Analysis (IWCIA08)
Miao Jin, Junho Kim, Feng Luo and Xianfeng Gu
Conformal geometry is in the core of pure mathematics. It 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
Riemannain 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 Ricci flow can be defined as the deformation of edge lengths
driven by the discrete curvature.We invented numerical algorithms to compute Riemannian
metrics with prescribed Gaussian curvatures using discrete Ricci flow.
We also showed broad applications using discrete Ricci flow in graphics, geometric modeling,
and medical imaging, such as surface parameterization, surface matching, manifold
splines, and construction of geometric structures on general surfaces.