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Parametric Polynomial Minimal Surfaces of Degree Six with Isothermal Parameter

  • Gang Xu
  • Guozhao Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4975)

Abstract

In this paper, parametric polynomial minimal surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a minimal surface. Then we obtain two kinds of new minimal surfaces from the condition. The new minimal surfaces have similar properties as Enneper’s minimal surface, such as symmetry, self-intersection and containing straight lines. A new pair of conjugate minimal surfaces is also discovered in this paper. The new minimal surfaces can be represented by tensor product Bézier surface and triangular Bézier surface, and have several shape parameters. We also employ the new minimal surfaces for form-finding problem in membrane structure and present several modeling examples.

Keywords

minimal surface harmonic surfaces isothermal parametric surface parametric polynomial minimal surface of degree six membrane structure 

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References

  1. 1.
    Bletzinger, K.-W.: Form finding of membrane structures and minimal surfaces by numerical continuation. In: Proceeding of the IASS Congress on Structural Morphology: Towards the New Millennium, Nottingham, pp. 68–75 (1997) ISBN 0-85358-064-2Google Scholar
  2. 2.
    Bobenko, A.I., Hoffmann, T., Springborn, B.A.: Minimal surfaces from circle patterns: Geometry from combinatorics. Annals of Mathematics 164, 231–264 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cosín, 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)Google Scholar
  4. 4.
    Desbrun, M., Grinspun, E., Schroder, P.: Discrete differential geometry: an applied introduction. In: SIGGRAPH Course Notes (2005)Google Scholar
  5. 5.
    Do Carmo, M.: Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  6. 6.
    Grandine, S., Del Valle, T., Moeller, S.K., Natarajan, G., Pencheva, J., Sherman, S.: Wise. Designing airplane struts using minimal surfaces IMA Preprint 1866 (2002)Google Scholar
  7. 7.
    Gu, X., Yau, S.: Surface classification using conformal structures. ICCV, 701-708 (2003)Google Scholar
  8. 8.
    Gu, X., Yau, S.: Computing conformal structure of surfaces CoRR cs.GR/0212043 (2002)Google Scholar
  9. 9.
    Hoscheck, J., Schneider, F.: Spline conversion for trimmed rational Bezier and B-spline surfaces. Computer Aided Design 9, 580–590 (1990)CrossRefGoogle Scholar
  10. 10.
    Jin, M., Luo, F., Gu, X.F.: Computing general geometric structures on surfaces using Ricci flow. Computer-Aided Design 8, 663–675 (2007)CrossRefGoogle Scholar
  11. 11.
    Jung, K., Chu, K.T., Torquato, S.: A variational level set approach for surface area minimization of triply-periodic surfaces. Journal of Computational Physics 2, 711–730 (2007)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Li, X., Guo, X.H., Wang, H.Y., He, Y., Gu, X.F., Qin, H.: Harmonic volumetric mapping for solid modeling applications. In: Symposium on Solid and Physical Modeling, pp. 109–120 (2007)Google Scholar
  13. 13.
    Liu, Y., Pottmann, H., Wallner, J., Yang, Y.L., Wang, W.P.: Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics 3, 681–689 (2006)CrossRefGoogle Scholar
  14. 14.
    Man, J.J., Wang, G.Z.: Approximating to nonparameterzied minimal surface with B-spline surface. Journal of Software 4, 824–829 (2003)Google Scholar
  15. 15.
    Monterde, J.: The Plateau-Bézier Problem. In: Wilson, M.J., Martin, R.R. (eds.) Mathematics of Surfaces. LNCS, vol. 2768, pp. 262–273. Springer, Heidelberg (2003)Google Scholar
  16. 16.
    Monterde, J.: Bézier surfaces of minimal area: The Dirichlet approach. Computer Aided Geometric Design 1, 117–136 (2004)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Monterde, J., Ugail., H.: On harmonic and biharmonic Bézier surfaces. Computer Aided Geometric Design 7, 697–715 (2004)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Monterde, J., Ugail, H.: A general 4th-order PDE method to generate Bézier surfaces from boundary. Computer Aided Geometric Design 2, 208–225 (2006)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Nitsche, J.C.C.: Lectures on minimal surfaces, vol. 1. Cambridge Univ. Press, Cambridge (1989)zbMATHGoogle Scholar
  20. 20.
    Osserman, R.: A survey of minimal surfaces, 2nd edn. Dover publ., New York( (1986)Google Scholar
  21. 21.
    Pinkall, U., Polthier, K.: Computing discrete minimal surface and their conjugates. Experiment Mathematics 1, 15–36 (1993)MathSciNetGoogle Scholar
  22. 22.
    Polthier, K.: Polyhedral surface of constant mean curvature. Habilitationsschrift TU Berlin (2002)Google Scholar
  23. 23.
    Pottmann, H., Liu, Y.: Discrete surfaces in isotropic geometry. In: Mathematics of Surfaces, vol. XII, pp. 341-363 (2007)Google Scholar
  24. 24.
    Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.P.: Geometry of multi-layer freeform structures for architecture. ACM Transactions on Graphics 3, 1–11 (2007)Google Scholar
  25. 25.
    Séquin, C.H.: CAD Tools for Aesthetic Engineering. Computer Aided Design 7, 737–750 (2005)Google Scholar
  26. 26.
    Sullivan, J.: The aesthetic value of optimal geometry. In: Emmer, M. (ed.) The Visual Mind II, pp. 547–563. MIT Press, Cambridge (2005)Google Scholar
  27. 27.
    Wallner, J., Pottmann, H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatsh. Math (to appear, 2006)Google Scholar
  28. 28.
    Wang, Y.: Periodic surface modeling for computer aided nano design. Computer Aided Design 3, 179–189 (2007)CrossRefGoogle Scholar
  29. 29.
    Xu, G., Wang, G.Z.: Harmonic B-B surfaces over triangular domain. Journal of Computers 12, 2180–2185 (2006)Google Scholar
  30. 30.
    Xu, G.L., Zhang, Q.: G 2 surface modeling using minimal mean-curvature-variation flow. Computer-Aided Design 5, 342–351 (2007)CrossRefGoogle Scholar
  31. 31.
    Xu, G.L.: Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design 10, 767–784 (2004)CrossRefGoogle Scholar
  32. 32.
    Xu, G.L.: Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces. Computer Aided Geometric Design 2, 193–207 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Gang Xu
    • 1
  • Guozhao Wang
    • 1
  1. 1.Institute of Computer Graphics and Image Processing, Department of MathematicsZhejiang UniversityHangzhouP.R. China

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