Article

Computing surface hyperbolic structure and real projective structure

Authors:

Xianfeng GuAuthors Info & Claims

SPM '06: Proceedings of the 2006 ACM symposium on Solid and physical modeling

Pages 105 - 116

https://doi.org/10.1145/1128888.1128904

Published: 06 June 2006 Publication History

Abstract

Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces. Algorithms designed for planar domains based on a specific geometry can be systematically generalized to surface domains via the corresponding geometric structure. For example, polar form splines with planar domains are based on affine invariants. Polar form splines can be generalized to manifold splines on the surfaces which admit affine structures and are equipped with affine geometries.Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. All surfaces admit real projective structures and are equipped with real projective geometry. Because of their general existence, both hyperbolic structures and real projective structures have the potential to replace the role of affine structures in defining manifold splines.This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The method is based on a novel geometric tool - discrete variational Ricci flow. Any metric surface admits a special uniformization metric, which is conformal to its original metric and induces constant curvature. Ricci flow is an efficient method to calculate the uniformization metric, which determines the hyperbolic structure and real projective structure.The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure.More importantly, this work introduces the framework of general geometric structures, which enable different geometries to be defined on manifolds and lay down the theoretical foundation for many important applications in geometric modeling.

References

[1]

Benzècri, J. P. 1959. Varits localement affines. Sem. Topologie et Gom. Diff., Ch. Ehresmann 7, 229--332.

[2]

Bobenko, A., and Schroder, P. 2005. Discrete willmore flow. In Eurographics Symposium on Geometry Processing.

Digital Library

[3]

Bobenko, A. I., and Springborn, B. A. 2004. Variational principles for circle patterns and koebe's theorem. Transactions of the American Mathematical Society 356, 659.

[4]

Carner, C., Jin, M., Gu, X., and Qin, H. 2005. Topology-driven surface mappings with robust feature alignment. In IEEE Visualization, 543--550.

[5]

Chow, B., and Luo, F. 2003. Combinatorial ricci flows on surfaces. Journal Differential Geometry 63, 1, 97--129.

[6]

Colin De Verdiére, E., and Lazarus, F. 2002. Optimal system of loops on an orientable surface. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 627--636.

Digital Library

[7]

Dai, J., Luo, W., Yau, S.-T., and GU, X. 2006. Geometric accuracy analysis for discrete surface approximation. In Geometric Modeling and Processing. submitted.

Digital Library

[8]

De Verdiére, C. 1991. Yves un principe variationnel pour les empilements de cercles. (french) "a variational principle for circle packings". Invent. Math. 104 3, 655--669.

[9]

Erickson, J. G., and Whittlesey, K. 2005. Greedy optimal homotopy and homology generators. In Proc. 16th Symp. Discrete Algorithms, ACM and SIAM, 1038--1046.

Digital Library

[10]

Ferguson, H., Rockwood, A. P., and Cox, J. 1992. Topological design of sculptured surfaces. In SIGGRAPH, 149--156.

Digital Library

[11]

Goldman, R. 2003. Computer graphics in its fifth decade: Ferment at the foundations. In 11th Pacific Conference on Computer Graphics and Applications (PG'03), 4--21.

Digital Library

[12]

Gotsman, C., Gu, X., and Sheffer, A. 2003. Fundamentals of spherical parameterization for 3d meshes. ACM Trans. Graph. 22, 3, 358--363.

Digital Library

[13]

Grimm, C., and Hughes, J. F. 2003. Parameterizing n-holed tori. In IMA Conference on the Mathematics of Surfaces, 14--29.

[14]

Gu, X., and Yau, S.-T. 2003. Global conformal parameterization. In Symposium on Geometry Processing, 127--137.

Digital Library

[15]

Gu, X., Wang, Y., Chan, T. F., Thompson, P. M., and Yau, S.-T. 2004. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23, 8, 949--958.

[16]

Gu, X., He, Y., and Qin, H. 2005. Manifold splines. In Symposium on Solid and Physical Modeling, 27--38.

Digital Library

[17]

Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., and Halle, M. 2000. Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph. 6, 2, 181--189.

Digital Library

[18]

Hamilton, R. S. 1988. The ricci flow on surfaces. Mathematics and general relativity 71, 237--262.

[19]

Jin, M., Wang, Y., Yau, S.-T., and Gu, X. 2004. Optimal global conformal surface parameterization. In IEEE Visualization, 267--274.

Digital Library

[20]

J. W. Milnor. 1958. On the existence of a connection with curvature zero. Comm. Math. Helv. 32, 215--223.

[21]

Kharevych, L., Springborn, B., and Schroder, P. 2005. Discrete conformal mappings via circle patterns.

[22]

Luo, W., and Gu, X. 2006. Discrete curvature flow and derivative cosine law. In preprint.

[23]

Luo, W. 2006. Error estimates for discrete harmonic 1-forms over riemann surfaces. In Communications in Analysis and Geometry. submitted.

[24]

Munkres, J. 1984. Elements of Algebraic Topology. Addison-Wesley Co.

[25]

Petersen, P. 1997. Riemannian Geometry. Springer Verlag.

[26]

Praun, E., and Hoppe, H. 2003. Spherical parametrization and remeshing. ACM Trans. Graph. 22, 3, 340--349.

Digital Library

[27]

Ratcliffe, J. G. 1994. Foundations of Hyperbolic Manifolds. Springer Verlag.

[28]

Rodin, B., and Sullivan, D. 1987. The convergence of circle packings to the riemann mapping. Journal of Differential Geometry 26, 2, 349--360.

[29]

Sovakar, A., and Kobbelt, L. 2004. Api design for adaptive subdivision schemes. Computers & Graphics 28, 1, 67--72.

[30]

Stephenson, K. 2005. Introduction To Circle Packing. Cambridge University Press.

[31]

Thurston, W. 1976. Geometry and Topology of 3-manifolds. Princeton lecture notes.

[32]

Thurston, W. P. 1997. Three-Dimensional Geometry and Topology. Princeton University Presss.

[33]

Wallner, J., and Pottmann, H. 1997. Spline orbifolds. Curves and Surfaces with Applications in CAGD, 445--464.

Cited By

Leng ZWu SSaleh MMontanaro AYu HWang YNavab NLiang XTombari F(2023)Dynamic Hyperbolic Attention Network for Fine Hand-object Reconstruction2023 IEEE/CVF International Conference on Computer Vision (ICCV)10.1109/ICCV51070.2023.01368(14848-14858)Online publication date: 1-Oct-2023
https://doi.org/10.1109/ICCV51070.2023.01368
Behera ASingh JVerma SKumar M(2022)Data visualization through non linear dimensionality reduction using feature based Ricci flow embeddingMultimedia Tools and Applications10.1007/s11042-021-11479-781:11(14831-14850)Online publication date: 3-Jan-2022
https://doi.org/10.1007/s11042-021-11479-7
Jiang LGuo YChen SWei PLei NGu X(2019)Concurrent optimization of structural topology and infill properties with a CBF-based level set methodFrontiers of Mechanical Engineering10.1007/s11465-019-0530-5Online publication date: 5-Feb-2019
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Index Terms

Computing surface hyperbolic structure and real projective structure
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 Conferences

SPM '06: Proceedings of the 2006 ACM symposium on Solid and physical modeling

June 2006

235 pages

ISBN:1595933581

DOI:10.1145/1128888

Conference Chairs:
Ralph Martin
Cardiff University
,
Shi-Min Hu
Tsinghua University

Copyright © 2006 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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Published: 06 June 2006

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SPM06: 2006 ACM Symposium on Solid and Physical Modeling

June 6 - 8, 2006

Cardiff, Wales, United Kingdom

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Cited By

Leng ZWu SSaleh MMontanaro AYu HWang YNavab NLiang XTombari F(2023)Dynamic Hyperbolic Attention Network for Fine Hand-object Reconstruction2023 IEEE/CVF International Conference on Computer Vision (ICCV)10.1109/ICCV51070.2023.01368(14848-14858)Online publication date: 1-Oct-2023
https://doi.org/10.1109/ICCV51070.2023.01368
Behera ASingh JVerma SKumar M(2022)Data visualization through non linear dimensionality reduction using feature based Ricci flow embeddingMultimedia Tools and Applications10.1007/s11042-021-11479-781:11(14831-14850)Online publication date: 3-Jan-2022
https://doi.org/10.1007/s11042-021-11479-7
Jiang LGuo YChen SWei PLei NGu X(2019)Concurrent optimization of structural topology and infill properties with a CBF-based level set methodFrontiers of Mechanical Engineering10.1007/s11465-019-0530-5Online publication date: 5-Feb-2019
https://doi.org/10.1007/s11465-019-0530-5
Xu WHancock EWilson R(2014)Ricci flow embedding for rectifying non-Euclidean dissimilarity dataPattern Recognition10.1016/j.patcog.2014.04.02147:11(3709-3725)Online publication date: 1-Nov-2014
https://dl.acm.org/doi/10.1016/j.patcog.2014.04.021
Hancock EXu EWilson R(2014)Pattern Recognition with Non-Euclidean SimilaritiesMan-Machine Interactions 310.1007/978-3-319-02309-0_1(3-15)Online publication date: 2014
https://doi.org/10.1007/978-3-319-02309-0_1
Gao MShi RZhang SZeng WQian ZGu XMetaxas DAxel L(2013)High resolution cardiac shape registration using Ricci flow2013 IEEE 10th International Symposium on Biomedical Imaging10.1109/ISBI.2013.6556518(488-491)Online publication date: Apr-2013
https://doi.org/10.1109/ISBI.2013.6556518
Shuai LGuo XJin M(2013)GPU-based computation of discrete periodic centroidal Voronoi tessellation in hyperbolic spaceComputer-Aided Design10.1016/j.cad.2012.10.02945:2(463-472)Online publication date: 1-Feb-2013
https://dl.acm.org/doi/10.1016/j.cad.2012.10.029
Zeng WSun JGuo RLuo FGu X(2012)Metric and Heat KernelManifold Learning Theory and Applications10.1201/b11431-8(145-166)Online publication date: 17-Jan-2012
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Zeng WGuo RLuo FGu X(2012)Discrete heat kernel determines discrete Riemannian metricGraphical Models10.1016/j.gmod.2012.03.00974:4(121-129)Online publication date: 1-Jul-2012
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Rong GJin MShuai LGuo X(2011)Centroidal Voronoi tessellation in universal covering space of manifold surfacesComputer Aided Geometric Design10.1016/j.cagd.2011.06.00528:8(475-496)Online publication date: 1-Nov-2011
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