Discrete surface Ricci flow

Miao Jin, Junho Kim, Feng Luo, Xianfeng Gu

Research output: Contribution to journalArticlepeer-review

215 Scopus citations


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.

Original languageEnglish (US)
Article number4483509
Pages (from-to)1030-1043
Number of pages14
JournalIEEE Transactions on Visualization and Computer Graphics
Issue number5
StatePublished - Sep 2008

All Science Journal Classification (ASJC) codes

  • Software
  • Signal Processing
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design


  • Discrete ricci flow
  • Global conformal parameterizations
  • Manifold
  • Riemannian metric
  • Uniformization


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