Construction of Subdivision Surfaces by Fourth-Order Geometric Flows with G1 Boundary Conditions

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


In this paper, we present a method for constructing Loop’s subdivision surface patches with given G 1 boundary conditions and a given topology of control polygon, using several fourth-order geometric partial differential equations. These equations are solved by a mixed finite element method in a function space defined by the extended Loop’s subdivision scheme. The method is flexible to the shape of the boundaries, and there is no limitation on the number of boundary curves and on the topology of the control polygon. Several properties for the basis functions of the finite element space are developed.


Subdivision Surface Geometric Partial Differential Equations G1 continuity 


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  1. 1.
    Bajaj, C., Xu, G.: Anisotropic diffusion of subdivision surfaces and functions on surfaces. ACM Transactions on Graphics 22(1), 4–32 (2003)CrossRefGoogle 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.
    Bryant, R.: A duality theorem for Willmore surfaces. J. Diff. Geom. 20, 23–53 (1984)zbMATHGoogle Scholar
  4. 4.
    Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., Rusu, R.: A finite element method for surface restoration with boundary conditions. Computer Aided Geometric Design 21(5), 427–445 (2004)zbMATHMathSciNetGoogle Scholar
  5. 5.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992)zbMATHGoogle Scholar
  6. 6.
    Epstein, C.L., Gage, M.: The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modeling, and Computation, pp. 15–59. Springer, New York (1987)Google Scholar
  7. 7.
    Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proceedings of the American Mathematical Society 126(9), 2789–2796 (1998)zbMATHCrossRefMathSciNetGoogle 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.
    Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Diff. Geom. 57(3), 409–441 (2001)zbMATHGoogle Scholar
  10. 10.
    Kuwert, E., Schätzle, R.: Gradient flow for the Willmore functional. Comm. Anal. Geom. 10(5), 1228–1245 (2002)Google Scholar
  11. 11.
    Man, J., Wang, G.: Approximating to Nonparameterzied Minimal Surface with B-Spline Surface. Journal of Software 14(4), 824–829 (2003)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Man, J., Wang, G.: Minimal Surface Modeling Using Finite Element Method. Chinese Journal of Computers 26(4), 507–510 (2003)MathSciNetGoogle Scholar
  13. 13.
    Mullins, W.W.: Two-dimensional motion of idealised grain boundaries. J. Appl. Phys. 27, 900–904 (1956)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957)CrossRefGoogle Scholar
  15. 15.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  16. 16.
    Sapiro, G., Tannenbaum, A.: Area and length preserving geometric invariant scale–spaces. IEEE Trans. Pattern Anal. Mach. Intell. 17, 67–72 (1995)CrossRefGoogle Scholar
  17. 17.
    Schneider, R., Kobbelt, L.: Generating fair meshes with G 1 boundary conditions. In: Geometric Modeling and Processing, Hong Kong, China, pp. 251–261 (2000)Google Scholar
  18. 18.
    Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design 18(4), 359–379 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Simon, L.: Existence of surfaces minimizing the Willmore functional. Commun. Analysis Geom. 1(2), 281–326 (1993)zbMATHGoogle Scholar
  20. 20.
    Xu, G.: Interpolation by Loop’s Subdivision Functions. Journal of Computational Mathematics 23(3), 247–260 (2005)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Xu, G.: Geometric Partial Differential Equation Methods in Computational Geometry. Science Press, Beijing (2008)Google Scholar
  22. 22.
    Xu, G., Pan, Q., Bajaj, C.: Discrete surface modelling using partial differential equations. Computer Aided Geometric Design 23(2), 125–145 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Xu, G., Zhang, Q.: G2 surface modeling using minimal mean–curvature–variation flow. Computer - Aided Design 39(5), 342–351 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

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