Advertisement

Construction of Minimal Catmull-Clark’s Subdivision Surfaces with Given Boundaries

  • Qing Pan
  • Guoliang Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)

Abstract

Minimal surface is an important class of surfaces. They are widely used in the areas such as architecture, art and natural science etc.. On the other hand, subdivision technology has always been active in computer aided design since its invention. The flexibility and high quality of the subdivision surface makes them a powerful tool in geometry modeling and surface designing. In this paper, we combine these two ingredients together aiming at constructing minimal subdivision surfaces. We use the mean curvature flow, a second order geometric partial differential equation, to construct minimal Catmull-Clark’s subdivision surfaces with specified B-spline boundary curves. The mean curvature flow is solved by a finite element method where the finite element space is spanned by the limit functions of the modified Catmull-Clark’s subdivision scheme.

Keywords

Minimal Subdivision Surface Catmull-Clark’s Subdivision Mean Curvature Flow 

MR (2000) Classification

65D17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnal, A., Lluch, A., Monterde, J.: Triangular Bézier Surfaces of Minimal Area. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2669. Springer, Heidelberg (2003)Google Scholar
  2. 2.
    Biermann, H., Levin, A., Zorin, D.: Piecewise-smooth Subdivision Surfaces with Normal Control. In: SIGGRAPH, pp. 113–120 (2000)Google Scholar
  3. 3.
    Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10(6), 350–355 (1978)CrossRefGoogle Scholar
  4. 4.
    Cosin, C., Monterde, J.: Bézier surfaces of minimal area. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS-ComputSci 2002. LNCS, vol. 2330, pp. 72–81. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Doo, D.: A subdivison algorithm for smoothing down irregularly shaped polyhedrons. In: proceedings on Interactive Techniques in computer Aided Design, Bologna, pp. 157–165 (1978)Google Scholar
  6. 6.
    Doo, D., Sabin, M.: Behavious of recursive division surfaces near extraordinary points. Computer-Aided Design 10(6), 356–360 (1978)CrossRefGoogle Scholar
  7. 7.
    Hoppe, H., DeRose, T., Duchamp, T., Halstend, M., Jin, H., McDonald, J., Schweitzer, J., Stuetzle, W.: Piecewise smooth surfaces reconstruction. In: Computer Graphics Proceedings, Annual Conference series, ACM SIGGRAPH 1994, pp. 295–302 (1994)Google Scholar
  8. 8.
    Jin, W., Wang, G.: Geometric Modeling Using Minimal Surfaces. Chinese Journal of Computers 22(12), 1276–1280 (1999)MathSciNetGoogle Scholar
  9. 9.
    Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis. Technical report, Department of Mathematices, University of Utah (1978)Google Scholar
  10. 10.
    Man, J., Wang, G.: Approximating to Nonparameterzied Minimal Surface with B-Spline Surface. Journal of Software 14(4), 824–829 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Man, J., Wang, G.: Minimal Surface Modeling Using Finite Element Method. Chinese Journal of Computers 26(4), 507–510 (2003)MathSciNetGoogle Scholar
  12. 12.
    Monterde, J.: Bézier surface of minimal area: The dirichlet approach. Computer Aided Geometric Design 21, 117–136 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nasri, A.H.: Surface interpolation on irregular networks with normal conditions. Computer Aided Geometric Design 8, 89–96 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nasri, A.H.: Polyhedral subdivision methods for free-form surfaces. ACM Transactions on Graphics 6(1), 29–73 (1987)zbMATHCrossRefGoogle Scholar
  15. 15.
    Polthier, K.: Computational aspects of discrete minimal surfaces. In: Hass, J., Hoffman, D., Jaffe, A., Rosenberg, H., Schoen, R., Wolf, M. (eds.) Proc. of the Clay Summer School on Global Theory of Minimal Surfaces (2002), citeseer.ist.psu.edu/polthier02computational.html
  16. 16.
    Xu, G.: Geometric Partial Differential Equation Methods in Computational Geometry. Science Press, Beijing (2008)Google Scholar
  17. 17.
    Xu, G., Shi, Y.: Progressive computation and numerical tables of generalized Gaussian quadrature formulas. Journal on Numerical Methods and the Computer Application 27(1), 9–23 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Qing Pan
    • 1
  • Guoliang Xu
    • 2
  1. 1.College of Mathematics and Computer ScienceHunan Normal UniversityChangshaChina
  2. 2.LSEC, Institute of Computational Mathematics, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations