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Generalized Discrete Ricci Flow
Author(s) -
Yang YongLiang,
Guo Ren,
Luo Feng,
Hu ShiMin,
Gu Xianfeng
Publication year - 2009
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2009.01579.x
Subject(s) - ricci flow , yamabe flow , mathematics , ricci curvature , euclidean geometry , conformal map , geometric flow , robustness (evolution) , topology (electrical circuits) , polygon mesh , flow (mathematics) , uniqueness , mathematical optimization , computer science , curvature , mathematical analysis , geometry , scalar curvature , sectional curvature , combinatorics , biochemistry , chemistry , gene
Surface Ricci flow is a powerful tool to design Riemannian metrics by user defined curvatures. Discrete surface Ricci flow has been broadly applied for surface parameterization, shape analysis, and computational topology. Conventional discrete Ricci flow has limitations. For meshes with low quality triangulations, if high conformality is required, the flow may get stuck at the local optimum of the Ricci energy. If convergence to the global optimum is enforced, the conformality may be sacrificed. This work introduces a novel method to generalize the traditional discrete Ricci flow. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. Conventional method is based on circle packing, which requires two circles on an edge intersect each other at an acute angle. Generalized method allows the two circles either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. Furthermore, the generalized Ricci flow preserves the convexity of the Ricci energy, this ensures the uniqueness of the global optimum. Therefore the algorithm won't get stuck at the local optimum. Generalized discrete Ricci flow algorithms are explained in details for triangle meshes with both Euclidean and hyperbolic background geometries. Its advantages are demonstrated by theoretic proofs and practical applications in graphics, especially surface parameterization.