Surfaces with Piecewise Linear Support Functions over Spherical Triangulations

  • Henrik Almegaard
  • Anne Bagger
  • Jens Gravesen
  • Bert Jüttler
  • Zbynek Šír
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4647)


Given a smooth surface patch we construct an approximating piecewise linear structure. More precisely, we produce a mesh for which virtually all vertices have valency three. We present two methods for the construction of meshes whose facets are tangent to the original surface. These two methods can deal with elliptic and hyperbolic surfaces, respectively. In order to describe and to derive the construction, which is based on a projective duality, we use the so–called support function representation of the surface and of the mesh, where the latter one has a piecewise linear support function.


Unit Sphere Shell Structure Support Function Tangent Plane Surface Patch 
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  1. 1.
    Almegaard, H.: The Stringer System—a Truss Model of Membrane Shells for Analysis and Design of Boundary Conditions. Int. J. Space Structures 19, 1–10 (2004)CrossRefGoogle Scholar
  2. 2.
    Bonnesen, T., Fenchel, W.: Theory of Convex Bodies. BCS Associates, Moscow, Idaho (1987)Google Scholar
  3. 3.
    Brückner, M.: Vielecke und Vielflache—Theorie und Geschichte. Teubner, Leipzig (1900)Google Scholar
  4. 4.
    Cutler, B., Whiting, E.: Constrained Planar Remeshing for Architecture. In: Symposium on Geometry Processing 2006, Poster proceedings (electronic), p. 5 (2006),
  5. 5.
    Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  6. 6.
    Gruber, P.M., Wills, J.M. (eds.): Handbook of Convex Geometry. North-Holland, Amsterdam (1993)Google Scholar
  7. 7.
    Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. AK Peters, Wellesley, Mass (1996)Google Scholar
  8. 8.
    Hoschek, J.: Dual Bézier Curves and Surfaces. In: Barnhill, R.E., Boehm, W. (eds.) Surfaces in Computer Aided Geometric Design, pp. 147–156. North-Holland, Amsterdam (1983)Google Scholar
  9. 9.
    Kawarahada, H., Sugihara, K.: Dual Subdivision: A New Class of Subdivision Schemes using Projective Duality. In: Jorge, J., Skala, V. (eds.) Proc. WSCG 2006, pp. 9–16. University of West Bohemia, Plzen (2006)Google Scholar
  10. 10.
    Liu, Y., Pottmann, H., Wallner, J., Yang, Y., Wang, W.: Geometric Modeling with Conical Meshes and Developable Surfaces. ACM Trans. Graphics 25, 681–689 (2006)CrossRefGoogle Scholar
  11. 11.
    Patanè, G., Spagnuolo, M.: Triangle Mesh Duality: Reconstruction and Smoothing. In: Wilson, M.J., Martin, R.R. (eds.) Mathematics of Surfaces. LNCS, vol. 2768, pp. 111–128. Springer, Heidelberg (2003)Google Scholar
  12. 12.
    Pottmann, H., Wallner, J.: The Focal Geometry of Circular and Conical Meshes. Adv. Comput. Math. (to appear)Google Scholar
  13. 13.
    Ros, L., Sugihara, K., Thomas, F.: Towards Shape Representation using Trihedral Mesh Projections. The Visual Computer 19, 139–150 (2003)zbMATHGoogle Scholar
  14. 14.
    Sabin, M.: A Class of Surfaces Closed under Five Important Geometric Operations. Technical Report VTO/MS/207, British Aircraft Corporation (1974), Available at
  15. 15.
    Šír, Z., Gravesen, J., Jüttler B.: Curves and surfaces represented by polynomial support functions. SFB report no. 2006-36 (2006), Available at
  16. 16.
    Weisstein, E.W.: Dual Polyhedron. From MathWorld — A Wolfram Web Resource.
  17. 17.
    Wenninger, M.J.: Dual Models. Cambridge University Press, Cambridge (1983)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Henrik Almegaard
    • 1
  • Anne Bagger
    • 1
  • Jens Gravesen
    • 2
  • Bert Jüttler
    • 3
  • Zbynek Šír
    • 4
  1. 1.Technical University of Denmark, Dept. of Civil Engineering 
  2. 2.Technical University of Denmark, Dept. of Mathematics 
  3. 3.Johannes Kepler University, Institute of Applied Geometry, LinzAustria
  4. 4.Charles University, Faculty of Mathematics and Physics, PragueCzech Republic

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