The Unified Discrete Surface Ricci flow

Min Zhang Stony Brook University Ren Guo Oregon State Univeristy Wei Zeng School of Computing and Information Sciences, Florida International University Feng Luo Rutgers University Shing Tung Yau Harvard University Xianfeng Gu Stony Brook Univerisity

Computational Geometry Differential Geometry Geometric Modeling and Processing Convex and Discrete Geometry mathscidoc:1612.01001

Distinguished Paper Award in 2017

Graphical Models/Geometric Modeling and Processing 2014, 76, (5), 321-339, 2014.9
Ricci flow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Experimental results show the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.
Unified, Ricci flow, circle packing, discrete Ricci energy. Hessian matrix
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  title={The Unified Discrete Surface Ricci flow},
  author={Min Zhang, Ren Guo, Wei Zeng, Feng Luo, Shing Tung Yau, and Xianfeng Gu},
  booktitle={Graphical Models/Geometric Modeling and Processing 2014},
Min Zhang, Ren Guo, Wei Zeng, Feng Luo, Shing Tung Yau, and Xianfeng Gu. The Unified Discrete Surface Ricci flow. 2014. Vol. 76. In Graphical Models/Geometric Modeling and Processing 2014. pp.321-339.
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