Article

Discrete quadratic curvature energies

Authors:

Eitan GrinspunAuthors Info & Claims

SIGGRAPH '06: ACM SIGGRAPH 2006 Courses

Pages 20 - 29

https://doi.org/10.1145/1185657.1185663

Published: 30 July 2006 Publication History

Abstract

Efficient computation of curvature-based energies is important for practical implementations of geometric modeling and physical simulation applications. Building on a simple geometric observation, we provide a version of a curvature-based energy expressed in terms of the Laplace operator acting on the embedding of the surface. The corresponding energy--being quadratic in positions--gives rise to a constant Hessian in the context of isometric deformations. The resulting isometric bending model is shown to significantly speed up common cloth solvers, and when applied to geometric modeling situations built onWillmore flow to provide runtimes which are close to interactive rates.

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References

[1]

Ascher, U. M., and Boxerman, E. 2003. On the modified conjugate gradient method in cloth simulation. Visual Computer 19, 7-8, 526--531.

Digital Library

[2]

Balay, S., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M., Mcinnes, L. C., Smith, B. F., and Zhang, H. 2001. PETSc homepage. http://www.mcs.anl.gov/petsc.

[3]

Baraff, D., and Witkin, A. 1998. Large steps in cloth simulation. In Proceedings of SIGGRAPH, 43--54.

Digital Library

[4]

Baraff, D., Witkin, A., and Kass, M. 2003. Untangling cloth. ACM Transactions on Graphics 22, 3, 862--870.

Digital Library

[5]

Blascke, W. 1929. Vorlesungen über Differentialgeometrie III. Springer.

[6]

Bobenko, A. I., and Schröder, P. 2005. Discrete Willmore flow. SGP, 101--110.

Digital Library

[7]

Bobenko, A. I. 2005. A conformal energy for simplicial surfaces. Combinatorial and Computational Geometry, 133--143.

[8]

Boxerman, E., and Ascher, U. 2004. Decomposing cloth. In SCA '04: Proceedings of the 2004 ACM SIGGRAPH/Eurographics symposium on Computer animation, ACM Press, New York, NY, USA, 153--161.

Digital Library

[9]

Breen, D. E., House, D. H., and Wozny, M. J. 1994. Predicting the drape of woven cloth using interacting particles. In Proceedings of SIGGRAPH 94, Computer Graphics Proceedings, Annual Conference Series, 365--372.

Digital Library

[10]

Bridson, R., Fedkiw, R., and Anderson, J. 2002. Robust treatment of collisions, contact and friction for cloth animation. ACM TOG 21, 3, 594--603.

Digital Library

[11]

Bridson, R., Marino, S., and Fedkiw, R. 2003. Simulation of clothing with folds and wrinkles. SCA '03, 28--36.

Digital Library

[12]

CANHAM, P. 1970. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology 26, 61--81.

[13]

Carignan, M., Yang, Y., Thalmann, N. M., and Thalmann, D. 1992. Dressing animated synthetic actors with complex deformable clothes. In Proceedings of SIGGRAPH, 99--104.

Digital Library

[14]

Choi, K.-j., and Ko, H.-S. 2002. Stable but responsive cloth. ACM Transactions on Graphics 21, 3 (July), 604--611.

Digital Library

[15]

Choi, K.-j., and Ko, H.-S. 2005. Research problems in clothing simulation. CAD 37, 6, 585--592.

Digital Library

[16]

Ciarlet, P. 2000. Mathematical Elasticity, Vol III. North-Holland.

[17]

Cirak, F., Ortiz, M., and Schröder, P. 2000. Subdivision surfaces: A new paradigm for thin-shell finite-element analysis. Internat. J. Numer. Methods Engrg. 47, 12, 2039--2072.

[18]

Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., and Rusu, R. 2004. A finite element method for surface restoration with smooth boundary conditions. CAGD, 427--445.

Digital Library

[19]

Cohen-steiner, D., and Morvan, J.-M. 2003. Restricted Delaunay triangulations and normal cycle. SoCG 2003, 312--321.

Digital Library

[20]

Crouzeix, M., and Raviart, P. A. 1973. Conforming and nonconforming finite elements for solving stationary Stokes equations. RAIRO Anal. Numer. 7, 33--76.

[21]

Deckelnick, K., Dziuk, G., and Elliott, C. M. 2005. Fully discrete semi-implicit second order splitting for anisotropic surface diffusion of graphs. SINUM 43, 1112--1138.

Digital Library

[22]

Desbrun, M., Schroder, P., and Barr, A. 1999. Interactive animation of structured deformable objects. Proceedings. Graphics Interface '99, 1--8.

Digital Library

[23]

Desbrun, M., Meyer, M., Schröder, P., and Barr, A. H. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. SIGGRAPH'99 Conference Proceedings, 317--324.

Digital Library

[24]

Etzmuss, O., Eberhardt, B., and Hauth, M. 2000. Implicitexplicit schemes for fast animation with particle systems. In Computer Animation and Simulation 2000, 138--151.

[25]

Etzmuss, O., Keckeisen, M., and Strasser, W. 2003. A fast finite element solution for cloth modelling. Proceedings 11th Pacific Conference on Computer Graphics and Applications, 244--251.

Digital Library

[26]

Feynman, C. 1986. Modeling the Appearance of Cloth. MSc thesis, MIT.

[27]

Govindaraju, N. K., Knott, D., Jain, N., Kabul, I., Tamstorf, R., Gayle, R., Lin, M. C., and Manocha, D. 2005. Interactive collision detection between deformable models using chromatic decomposition. ACM Transactions on Graphics 24, 3 (Aug.), 991--999.

Digital Library

[28]

Grinspun, E., Hirani, A. N., Desbrun, M., and Schröder, P. 2003. Discrete shells. SCA '03, 62--67.

Digital Library

[29]

Haumann, R. 1987. Modeling the physical behavior of flexible objects. In Topics in Physically-based Modeling, Eds. Barr, Barrel, Haumann, Kass, Platt, Terzopoulos, andWitkin, SIGGRAPH Course Notes.

[30]

Hauth, M., and Etzmuss, O. 2001. A high performance solver for the animation of deformable objects using advanced numerical methods. Computer Graphics Forum 20, 3, 319--328.

[31]

Hauth, M., Etzmuss, O., and Strasser, W. 2003. Analysis of numerical methods for the simulation of deformable models. Visual Computer 19, 7-8, 581--600.

Digital Library

[32]

Hauth, M. 2004. Visual Simulation of Deformable Models. PhD thesis, University of Tübingen.

[33]

Helfrich, W. 1973. Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift fr Naturforschung Teil C 28, 693--703.

[34]

Hildebrandt, K., and Polthier, K. 2004. Anisotropic filtering of non-linear surface features. CGF 23, 3, 391--400.

[35]

Hildebrandt, K., Polthier, K., and Wardetzky, M. 2005. On the convergence of metric and geometric properties of polyhedral surfaces. ZIB-Report ZR-05-24.

[36]

House, D. H., and Breen, D. E., Eds. 2000. Cloth modeling and animation.

Digital Library

[37]

Hsu, L., Kusner, R., and Sullivan, J. 1992. Minimizing the squared mean curvature integral for surfaces in space forms. Experiment. Math. 1, 3, 191--207.

[38]

Hughes, T. J. R. 1987. Finite Element Method - Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs.

[39]

Keckeisen, M., Kimmerle, S., Thomaszewski, B., and Wacker, M. 2004. Modelling Effects of Wind Fields in Cloth Animations. In Journal of WSCG, vol. Vol. 12, 205--212.

[40]

Klosowski, J. T., Held, M., Mitchell, J. S. B., Sowizral, H., and Zikan, K. 1998. Efficient collision detection using bounding volume hierarchies of k-dops. IEEE TVCG 4, 1 (January-March), 21--36.

Digital Library

[41]

Magnenat-Thalmann, N., and Volino, P. 2005. From early draping to haute couture models: 20 years of research. The Visual Computer 21, 8-10, 506--519.

[42]

Mercat, C. 2001. Discrete Riemann surfaces and the Ising model. Communications in Mathematical Physics 218, 1, 177--216.

[43]

Meyer, M., Desbrun, M., Schröder, P., and Barr, A. H. 2003. Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III, H.-C. Hege and K. Polthier, Eds. Springer-Verlag, Heidelberg, 113--134.

[44]

Ng, H. N., and Grimsdale, R. L. 1996. Computer graphics techniques for modeling cloth. IEEE CG&A 16, 5 (Sept.), 28--41.

Digital Library

[45]

Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experim. Math. 2, 15--36.

[46]

Polthier, K., 2002. Polyhedral surfaces of constant mean curvature. Habilitationsschrift, TU Berlin.

[47]

Provot, X. 1995. Deformation constraints in a mass-spring model to describe rigid cloth behavior. In Graphics Interface '95, 147--154.

[48]

Schenk, O., and Gärtner, K. 2004. Solving unsymmetric sparse systems of linear equations with PARDISO. Journal of Future Generation Computer Systems 20, 3, 475--487.

Digital Library

[49]

Schneider, R., and Kobbelt, L. 2001. Geometric fairing of irregular meshes for free-from surface design. CAGD 18, 359--379.

Digital Library

[50]

Sorkine, O. 2005. Laplacian mesh processing. Eurographics STAR - State of The Art Report, 53--70.

[51]

Teran, J., Sifakis, E., Irving, G., and Fedkiw, R. 2005. Robust quasistatic finite elements and flesh simulation. In 2005 ACM SIGGRAPH / Eurographics Symposium on Computer Animation, 181--190.

Digital Library

[52]

Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. 1987. Elastically deformable models. In Proceedings of SIGGRAPH, 205--214.

Digital Library

[53]

Thomaszewski, B., and Wacker, M. 2006. Bending Models for Thin Flexible Objects. In WSCG Short Communication proceedings.

[54]

Volino, P., Courchesne, M., and Thalmann, N. M. 1995. Versatile and efficient techniques for simulating cloth and other deformable objects. In SIGGRAPH '95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 137--144.

Digital Library

[55]

White, J. H. 2000. A global invariant of conformal mappings in space. Proceedings of the American Mathematical Society 38, 162--164.

[56]

Willmore, T. J. 2000. Surfaces in conformal geometry. Annals of Global Analysis and Geometry 18, 255--264.

[57]

Yoshizawa, S., and Belyaev, A. 2002. Fair triangle mesh generation via discrete elastica. In GMP2002, IEEE, 119--123.

Digital Library

[58]

Zhu, H., Jin, X., Feng, J., and Peng, Q. 2004. Survey on cloth animation. Journal of Computer Aided Design& Computer Graphics 16, 5, 613--618.

[59]

Zienkiewicz, O. C., and Taylor, R. L. 2000. The finite element method: The basis, 5th ed., vol. 1. Butterworth and Heinemann.

Cited By

Calatroni LHuska MMorigi SRecupero G(2022)A Unified Surface Geometric Framework for Feature-Aware Denoising, Hole Filling and Context-Aware CompletionJournal of Mathematical Imaging and Vision10.1007/s10851-022-01107-w65:1(82-98)Online publication date: 22-Jun-2022
https://doi.org/10.1007/s10851-022-01107-w
Nabaei SBaverel OWeinand Y(2014)Form Finding of Twisted Interlaced Structures: A Hybrid ApproachAdvances in Architectural Geometry 201410.1007/978-3-319-11418-7_9(127-143)Online publication date: 3-Dec-2014
https://doi.org/10.1007/978-3-319-11418-7_9

Index Terms

Discrete quadratic curvature energies
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

cover image ACM Conferences

SIGGRAPH '06: ACM SIGGRAPH 2006 Courses

July 2006

83 pages

ISBN:1595933646

DOI:10.1145/1185657

Conference Chair:
John Finnegan
Purdue University
,
Program Chair:
Dave Shreiner

Copyright © 2006 ACM.

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Published: 30 July 2006

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Cited By

Calatroni LHuska MMorigi SRecupero G(2022)A Unified Surface Geometric Framework for Feature-Aware Denoising, Hole Filling and Context-Aware CompletionJournal of Mathematical Imaging and Vision10.1007/s10851-022-01107-w65:1(82-98)Online publication date: 22-Jun-2022
https://doi.org/10.1007/s10851-022-01107-w
Nabaei SBaverel OWeinand Y(2014)Form Finding of Twisted Interlaced Structures: A Hybrid ApproachAdvances in Architectural Geometry 201410.1007/978-3-319-11418-7_9(127-143)Online publication date: 3-Dec-2014
https://doi.org/10.1007/978-3-319-11418-7_9

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