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Related Experiment Videos

Discrete surface Ricci flow.

Miao Jin1, Junho Kim, Feng Luo

  • 1Department of Computer Science, Stony Brook University, Stony Brook, NY 11794-4400, USA. mjin@cs.sunysb.edu

IEEE Transactions on Visualization and Computer Graphics
|July 5, 2008
PubMed
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This study presents a unified framework for discrete surface Ricci flow algorithms, enabling the design of Riemannian metrics on surfaces with user-defined Gaussian curvatures. These efficient and accurate algorithms offer broad applications in graphics and geometric modeling.

Area of Science:

  • Differential Geometry
  • Computational Geometry
  • Computer Graphics

Background:

  • Ricci flow is a powerful tool for understanding the geometry of surfaces.
  • Designing Riemannian metrics with prescribed Gaussian curvatures is a fundamental problem.
  • Existing discrete Ricci flow algorithms often lack generality or efficiency.

Purpose of the Study:

  • To introduce a unified framework for discrete surface Ricci flow algorithms.
  • To enable the design of Riemannian metrics on surfaces with arbitrary topologies and user-defined Gaussian curvatures.
  • To demonstrate the efficiency, accuracy, and flexibility of the proposed algorithms.

Main Methods:

  • A unified framework for discrete surface Ricci flow algorithms (spherical, Euclidean, hyperbolic).

Related Experiment Videos

  • Variational framework defining Ricci energy on the metric space.
  • Optimization of Ricci energy using Newton's method.

    • Main Results:

    • Algorithms design Riemannian metrics conformal to original metrics with user-defined Gaussian curvatures.
    • The curvature evolves like a heat diffusion process, reaching the target curvature.
    • Experimental results demonstrate efficiency, accuracy, and flexibility.

    Conclusions:

    • The proposed discrete Ricci flow algorithms provide a rigorous and efficient solution for metric design.
    • The framework has potential applications in graphics, geometric modeling, and medical imaging.
    • Demonstrated practical value through global surface parameterizations.