Robust modeling of constant mean curvature surfaces

Abstract

We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT (centroidal Voronoi tessellation) optimization framework. We devise a CVT-CMC energy function defined as a combination of an extended CVT energy and a volume functional. We show that minimizing the CVT-CMC energy is asymptotically equivalent to minimizing mesh surface area with a fixed volume, thus defining a discrete CMC surface. The CVT term in the energy function ensures high mesh quality throughout the evolution of a CMC surface in an interactive design process for form finding. Our method is capable of modeling CMC surfaces with fixed or free boundaries and is robust with respect to input mesh quality and topology changes. Experiments show that the new method generates discrete CMC surfaces of improved mesh quality over existing methods.

References

1. Bobenko, A. I., and Springborn, B. A. 2007. A discrete Laplace-Beltrami operator for simplicial surfaces. Discrete & Computational Geometry 38, 4, 740--756.
2. Brakke, K. 1992. The Surface Evolver. Experimental Mathematics 1, 2, 141--165.
3. Brakke, K., 2012. The Surface Evolver, version 2.30. http://www.susqu.edu/brakke/evolver/evolver.html, May.
4. Brew, J., and Lewis, W. 2003. Computational form-finding of tension membrane structures -- Non-finite element approaches: Part 1. Use of cubic splines in finding minimal surface membranes. International journal for numerical methods in engineering 56, 5, 651--668.
5. Brew, J., and Lewis, W. 2003. Computational form-finding of tension membrane structures -- Non-finite element approaches: Part 2. Triangular mesh discretization and control of mesh distortion in modelling minimal surface membranes. International Journal for Numerical Methods in Engineering 56, 5, 669--684.
6. Chopp, D. 1993. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics 106, 77--77.
7. Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41, 4, 637--676.
8. Du, Q., Gunzburger, M., and Ju, L. 2003. Constrained centroidal Voronoi tessellations for surfaces. SIAM Journal on Scientific Computing 24, 5, 1488--1506.
9. Du, Q., Gunzburger, M., and Ju, L. 2010. Advances in studies and applications of centroidal Voronoi tessellations. Numer. Math. Theor. Meth. Appl. 3, 119--142.
10. Dyer, R., Zhang, H., and Möller, T. 2007. Delaunay mesh construction. In Proceedings of the fifth Eurographics Symposium on Geometry Processing, Eurographics Association, Aire-la-Ville, Switzerland, 273--282.
11. Dziuk, G., and Hutchinson, J. E. 2006. Finite element approximations to surfaces of prescribed variable mean curvature. Numerische Mathematik 102, 611--648. 10.1007/s00211-005-0649-7.
12. Gersho, A. 1979. Asymptotically optimal block quantization. IEEE Transactions on Information Theory 25, 4, 373--380.
13. Goldstein, R., Moffatt, H., Pesci, A., and Ricca, R. 2010. Soap-film Möbius strip changes topology with a twist singularity. Proceedings of the National Academy of Sciences 107, 51, 21979.
14. Gruber, P. 2001. Optimal configurations of finite sets in Riemannian 2-manifolds. Geometriae Dedicata 84, 271--320. 10.1023/A:1010358407868.
15. Hadzhilazova, M., Mladenov, I., and Oprea, J. 2007. Unduloids and their geometry. Archivum Mathematicum 43, 5, 417--429.
16. Kapouleas, N. 1990. Complete constant mean curvature surfaces in Euclidean three-space. The Annals of Mathematics 131, 2, 239--330.
17. Liu, D., and Nocedal, J. 1989. On the limited memory bfgs method for large scale optimization. Mathematical programming 45, 1, 503--528.
18. Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.-M., Lu, L., and Yang, C. 2009. On centroidal Voronoi tessellation---energy smoothness and fast computation. ACM Trans. Graph. 28, 4, 1--17.
19. 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, 35--57.
20. Oberknapp, B., and Polthier, K. 1997. An algorithm for discrete constant mean curvature surfaces. In Visualization and Mathematics, H.-C. Hege and K. Polthier, Eds. Springer Verlag, Heidelberg, 141--161.
21. Osserman, R. 2002. A survey of minimal surfaces. Dover Pubns.
22. Pardalos, P., and Romeijn, H. 2002. Handbook of global optimization. Volume 2. Kluwer.
23. Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experimental mathematics 2, 1, 15--36.
24. Polthier, K., and Rossman, W. 2002. Discrete constant mean curvature surfaces and their index. J. reine angew. Math., 549, 47--77.
25. Smith, J. 2003. Three Applications of Optimization in Computer Graphics. PhD thesis, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA.
26. Taylor, J. E. 1976. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. The Annals of Mathematics 103, 3, 489--539.
27. Xu, G., and Zhang, Q. 2008. A general framework for surface modeling using geometric partial differential equations. Computer Aided Geometric Design 25, 3, 181--202.
28. Xu, G., Pan, Q., and Bajaj, C. 2006. Discrete surface modelling using partial differential equations. Computer Aided Geometric Design 23, 2, 125--145.
29. Yan, D., Lévy, B., Liu, Y., Sun, F., and Wang, W. 2009. Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. In Computer graphics forum, vol. 28, Wiley Online Library, 1445--1454.
30. Yan, D., Wang, W., Lévy, B., and Liu, Y. 2010. Efficient computation of 3D clipped Voronoi diagram. Advances in Geometric Modeling and Processing, 269--282.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures