Discrete Surface Ricci Flow
IEEE TVCG 2008
Miao Jin, Junho Kim, Feng Luo and Xianfeng Gu
This work introduces a unified framework for
discrete surface Ricci flow algorithms, including spherical, Euclidean,
and hyperbolic Ricci flows, which can design Riemannian
metrics on surfaces with arbitrary topologies by user-defined
Gaussian curvatures. Furthermore, the target metrics are conformal
(angle-preserving) to the original metrics.
Ricci flow conformally deforms the Riemannian metric on
a surface according to its induced curvature, such that the
curvature evolves like a heat diffusion process. Eventually, the
curvature becomes the user defined curvature.
Discrete Ricci flow algorithms are based on a variational
framework. Given a mesh, all possible metrics form a linear
space, and all possible curvatures form a convex polytope. The
Ricci energy is defined on the metric space, which reaches its
minimum at the desired metric. The Ricci flow is the negative
gradient flow of the Ricci energy. Furthermore, the Ricci energy
can be optimized using Newton’s method more efficiently.
Discrete Ricci flow algorithms are rigorous and efficient. Our
experimental results demonstrate the efficiency, accuracy and
flexibility of the algorithms. They have the potential for a
wide range of applications in graphics, geometric modeling, and
medical imaging. We demonstrate their practical values by global
surface parameterizations.