Discrete Surface Ricci Flow
Mathematics of Surfaces
Miao Jin, Junho Kim, and Xianfeng David Gu
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¡äe 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.