skip to main content
10.1145/1185657.1185663acmconferencesArticle/Chapter ViewAbstractPublication PagessiggraphConference Proceedingsconference-collections
Article

Discrete quadratic curvature energies

Published:30 July 2006Publication 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.

Skip Supplemental Material Section

Supplemental Material

a20-bergou-high.mov

mov

97 MB

a20-bergou-low.mov

mov

34.4 MB

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.Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle Scholar
  3. Baraff, D., and Witkin, A. 1998. Large steps in cloth simulation. In Proceedings of SIGGRAPH, 43--54. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Baraff, D., Witkin, A., and Kass, M. 2003. Untangling cloth. ACM Transactions on Graphics 22, 3, 862--870. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Blascke, W. 1929. Vorlesungen über Differentialgeometrie III. Springer.Google ScholarGoogle Scholar
  6. Bobenko, A. I., , and Schröder, P. 2005. Discrete Willmore flow. SGP, 101--110. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Bobenko, A. I. 2005. A conformal energy for simplicial surfaces. Combinatorial and Computational Geometry, 133--143.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital 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. Google ScholarGoogle ScholarDigital LibraryDigital 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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Bridson, R., Marino, S., and Fedkiw, R. 2003. Simulation of clothing with folds and wrinkles. SCA '03, 28--36. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle ScholarCross RefCross Ref
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Choi, K.-j., and Ko, H.-S. 2002. Stable but responsive cloth. ACM Transactions on Graphics 21, 3 (July), 604--611. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Choi, K.-j., and Ko, H.-S. 2005. Research problems in clothing simulation. CAD 37, 6, 585--592. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Ciarlet, P. 2000. Mathematical Elasticity, Vol III. North-Holland.Google ScholarGoogle Scholar
  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.Google ScholarGoogle ScholarCross RefCross Ref
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Cohen-steiner, D., and Morvan, J.-M. 2003. Restricted Delaunay triangulations and normal cycle. SoCG 2003, 312--321. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Desbrun, M., Schroder, P., and Barr, A. 1999. Interactive animation of structured deformable objects. Proceedings. Graphics Interface '99, 1--8. Google ScholarGoogle ScholarDigital LibraryDigital 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. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Feynman, C. 1986. Modeling the Appearance of Cloth. MSc thesis, MIT.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Grinspun, E., Hirani, A. N., Desbrun, M., and Schröder, P. 2003. Discrete shells. SCA '03, 62--67. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle Scholar
  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.Google ScholarGoogle ScholarCross RefCross Ref
  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.Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Hauth, M. 2004. Visual Simulation of Deformable Models. PhD thesis, University of Tübingen.Google ScholarGoogle Scholar
  33. Helfrich, W. 1973. Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift fr Naturforschung Teil C 28, 693--703.Google ScholarGoogle ScholarCross RefCross Ref
  34. Hildebrandt, K., and Polthier, K. 2004. Anisotropic filtering of non-linear surface features. CGF 23, 3, 391--400.Google ScholarGoogle ScholarCross RefCross Ref
  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.Google ScholarGoogle Scholar
  36. House, D. H., and Breen, D. E., Eds. 2000. Cloth modeling and animation. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle ScholarCross RefCross Ref
  38. Hughes, T. J. R. 1987. Finite Element Method - Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs.Google ScholarGoogle Scholar
  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.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle ScholarCross RefCross Ref
  42. Mercat, C. 2001. Discrete Riemann surfaces and the Ising model. Communications in Mathematical Physics 218, 1, 177--216.Google ScholarGoogle ScholarCross RefCross Ref
  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.Google ScholarGoogle Scholar
  44. Ng, H. N., and Grimsdale, R. L. 1996. Computer graphics techniques for modeling cloth. IEEE CG&A 16, 5 (Sept.), 28--41. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experim. Math. 2, 15--36.Google ScholarGoogle ScholarCross RefCross Ref
  46. Polthier, K., 2002. Polyhedral surfaces of constant mean curvature. Habilitationsschrift, TU Berlin.Google ScholarGoogle Scholar
  47. Provot, X. 1995. Deformation constraints in a mass-spring model to describe rigid cloth behavior. In Graphics Interface '95, 147--154.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Schneider, R., and Kobbelt, L. 2001. Geometric fairing of irregular meshes for free-from surface design. CAGD 18, 359--379. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Sorkine, O. 2005. Laplacian mesh processing. Eurographics STAR - State of The Art Report, 53--70.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. 1987. Elastically deformable models. In Proceedings of SIGGRAPH, 205--214. Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. Thomaszewski, B., and Wacker, M. 2006. Bending Models for Thin Flexible Objects. In WSCG Short Communication proceedings.Google ScholarGoogle Scholar
  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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  55. White, J. H. 2000. A global invariant of conformal mappings in space. Proceedings of the American Mathematical Society 38, 162--164.Google ScholarGoogle ScholarCross RefCross Ref
  56. Willmore, T. J. 2000. Surfaces in conformal geometry. Annals of Global Analysis and Geometry 18, 255--264.Google ScholarGoogle ScholarCross RefCross Ref
  57. Yoshizawa, S., and Belyaev, A. 2002. Fair triangle mesh generation via discrete elastica. In GMP2002, IEEE, 119--123. Google ScholarGoogle ScholarDigital LibraryDigital 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.Google ScholarGoogle Scholar
  59. Zienkiewicz, O. C., and Taylor, R. L. 2000. The finite element method: The basis, 5th ed., vol. 1. Butterworth and Heinemann.Google ScholarGoogle Scholar

Index Terms

  1. Discrete quadratic curvature energies

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        SIGGRAPH '06: ACM SIGGRAPH 2006 Courses
        July 2006
        83 pages
        ISBN:1595933646
        DOI:10.1145/1185657

        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]

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 30 July 2006

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • Article

        Acceptance Rates

        Overall Acceptance Rate1,822of8,601submissions,21%

        Upcoming Conference

        SIGGRAPH '24

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader