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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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 0-85358-064-2
Bobenko, A.I., Hoffmann, T., Springborn, B.A.: Minimal surfaces from circle patterns: Geometry from combinatorics. Annals of Mathematics 164, 231–264 (2006)
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)
Desbrun, M., Grinspun, E., Schroder, P.: Discrete differential geometry: an applied introduction. In: SIGGRAPH Course Notes (2005)
Do Carmo, M.: Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs (1976)
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)
Gu, X., Yau, S.: Surface classification using conformal structures. ICCV, 701-708 (2003)
Gu, X., Yau, S.: Computing conformal structure of surfaces CoRR cs.GR/0212043 (2002)
Hoscheck, J., Schneider, F.: Spline conversion for trimmed rational Bezier and B-spline surfaces. Computer Aided Design 9, 580–590 (1990)
Jin, M., Luo, F., Gu, X.F.: Computing general geometric structures on surfaces using Ricci flow. Computer-Aided Design 8, 663–675 (2007)
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)
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)
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)
Man, J.J., Wang, G.Z.: Approximating to nonparameterzied minimal surface with B-spline surface. Journal of Software 4, 824–829 (2003)
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)
Monterde, J.: Bézier surfaces of minimal area: The Dirichlet approach. Computer Aided Geometric Design 1, 117–136 (2004)
Monterde, J., Ugail., H.: On harmonic and biharmonic Bézier surfaces. Computer Aided Geometric Design 7, 697–715 (2004)
Monterde, J., Ugail, H.: A general 4th-order PDE method to generate Bézier surfaces from boundary. Computer Aided Geometric Design 2, 208–225 (2006)
Nitsche, J.C.C.: Lectures on minimal surfaces, vol. 1. Cambridge Univ. Press, Cambridge (1989)
Osserman, R.: A survey of minimal surfaces, 2nd edn. Dover publ., New York( (1986)
Pinkall, U., Polthier, K.: Computing discrete minimal surface and their conjugates. Experiment Mathematics 1, 15–36 (1993)
Polthier, K.: Polyhedral surface of constant mean curvature. Habilitationsschrift TU Berlin (2002)
Pottmann, H., Liu, Y.: Discrete surfaces in isotropic geometry. In: Mathematics of Surfaces, vol. XII, pp. 341-363 (2007)
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)
Séquin, C.H.: CAD Tools for Aesthetic Engineering. Computer Aided Design 7, 737–750 (2005)
Sullivan, J.: The aesthetic value of optimal geometry. In: Emmer, M. (ed.) The Visual Mind II, pp. 547–563. MIT Press, Cambridge (2005)
Wallner, J., Pottmann, H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatsh. Math (to appear, 2006)
Wang, Y.: Periodic surface modeling for computer aided nano design. Computer Aided Design 3, 179–189 (2007)
Xu, G., Wang, G.Z.: Harmonic B-B surfaces over triangular domain. Journal of Computers 12, 2180–2185 (2006)
Xu, G.L., Zhang, Q.: G 2 surface modeling using minimal mean-curvature-variation flow. Computer-Aided Design 5, 342–351 (2007)
Xu, G.L.: Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design 10, 767–784 (2004)
Xu, G.L.: Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces. Computer Aided Geometric Design 2, 193–207 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xu, G., Wang, G. (2008). Parametric Polynomial Minimal Surfaces of Degree Six with Isothermal Parameter. In: Chen, F., Jüttler, B. (eds) Advances in Geometric Modeling and Processing. GMP 2008. Lecture Notes in Computer Science, vol 4975. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79246-8_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-79246-8_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79245-1
Online ISBN: 978-3-540-79246-8
eBook Packages: Computer ScienceComputer Science (R0)