Article

Computing geodesic spectra of surfaces

Authors:

Shing-Tung Yau,

Xianfeng GuAuthors Info & Claims

SPM '07: Proceedings of the 2007 ACM symposium on Solid and physical modeling

Pages 387 - 393

https://doi.org/10.1145/1236246.1236302

Published: 04 June 2007 Publication History

Abstract

Surface classification is one of the most fundamental problems in geometric modeling. Surfaces can be classified according to their conformal structures. In general, each topological equivalent class has infinite conformally equivalent classes.

This paper introduces a novel method to classify surfaces by their conformal structures. Surfaces in the same conformal class share the same uniformization metric, which induces constant Gaussian curvature everywhere on the surface. Under the uniformization metric, each homotopy class of a closed curves on the surface has a unique geodesic. The lengths of all closed geodesics form the geodesic spectrum. The map from the fundamental group to the geodesic spectrum completely determines the conformal structure of the surface.

We first compute the uniformization metric using discrete Ricci flow method, then compute the Fuchsian group generators, finally deduce the geodesic spectra from the generators in a closed form.

The method is rigorous and practical. Geodesic spectra is applied as the signature of surfaces for shape comparison and classification.

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Shi-Qing Xin Ying He Chi-Wing Fu (2012)Efficiently Computing Exact Geodesic Loops within Finite StepsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2011.11918:6(879-889)Online publication date: Jun-2012
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Index Terms

Computing geodesic spectra of surfaces
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Other conferences

SPM '07: Proceedings of the 2007 ACM symposium on Solid and physical modeling

June 2007

455 pages

ISBN:9781595936660

DOI:10.1145/1236246

Program Chairs:
Bruno Levy
INRIA
,
Dinesh Manocha
University of North Carolina

Copyright © 2007 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 June 2007

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Tsinghua University

SPM07: Symposium on Solid and Physical Modeling

June 4 - 6, 2007

Beijing, China

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Cited By

Yuan NWang PMeng WChen SXu JXin SHe YWang W(2023)A Variational Framework for Curve Shortening in Various Geometric DomainsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.313502129:4(1951-1963)Online publication date: 1-Apr-2023
https://doi.org/10.1109/TVCG.2021.3135021
Lai RZhao H(2017)Multiscale Nonrigid Point Cloud Registration Using Rotation-Invariant Sliced-Wasserstein Distance via Laplace--Beltrami EigenmapSIAM Journal on Imaging Sciences10.1137/16M106882710:2(449-483)Online publication date: Jan-2017
https://doi.org/10.1137/16M1068827
Shi-Qing Xin Ying He Chi-Wing Fu (2012)Efficiently Computing Exact Geodesic Loops within Finite StepsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2011.11918:6(879-889)Online publication date: Jun-2012
https://doi.org/10.1109/TVCG.2011.119
Miao Jin Wei Zeng Feng Luo Xianfeng Gu (2009)Computing Teichmuller Shape SpaceIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2008.10315:3(504-517)Online publication date: May-2009
https://doi.org/10.1109/TVCG.2008.103
Miao Jin Wei Zeng Ning Ding Xianfeng Gu (2009)Computing Fenchel-Nielsen coordinates in Teichmuller shape Space2009 IEEE International Conference on Shape Modeling and Applications10.1109/SMI.2009.5170148(193-200)Online publication date: Jun-2009
https://doi.org/10.1109/SMI.2009.5170148
Wei Zeng Miao Jin Feng Luo Gu X(2009)Canonical homotopy class representative using hyperbolic structure2009 IEEE International Conference on Shape Modeling and Applications10.1109/SMI.2009.5170145(171-178)Online publication date: Jun-2009
https://doi.org/10.1109/SMI.2009.5170145
Yin XJin MLuo FGu X(2009)Discrete Curvature Flows for Surfaces and 3-ManifoldsEmerging Trends in Visual Computing10.1007/978-3-642-00826-9_3(38-74)Online publication date: 16-Mar-2009
https://dl.acm.org/doi/10.1007/978-3-642-00826-9_3
Ben-Chen MGotsman C(2008)Characterizing shape using conformal factorsProceedings of the 1st Eurographics conference on 3D Object Retrieval10.5555/2381112.2381114(1-8)Online publication date: 15-Apr-2008
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Jin MKim JLuo FGu X(2008)Discrete Surface Ricci FlowIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2008.5714:5(1030-1043)Online publication date: 1-Sep-2008
https://dl.acm.org/doi/10.1109/TVCG.2008.57

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