Geometriae Dedicata

, Volume 154, Issue 1, pp 27–46 | Cite as

Conformal hexagonal meshes

Original Paper


We explore discrete conformal and discrete minimal surfaces whose faces are planar hexagons throughout. Discrete conformal meshes are built of conformal hexagons for which we establish a dual construction. We apply this dual construction to conformal hexagonal meshes covering the sphere and get discrete hexagonal minimal surfaces via a discrete analogue to the Christoffel dual construction. We compare the smooth and the discrete settings by means of limit considerations and also by a discussion of Möbius invariants.


Discrete differential geometry Hexagonal mesh Discrete conformal map Discrete Christoffel duality Discrete minimal surface Elementary geometry 

Mathematics Subject Classification (2000)

51M04 52C26 52C99 53A40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bobenko A.I., Hoffmann T., Springborn B.: Minimal surfaces from circle patterns: geometry from combinatorics. Ann. Math. 164, 231–264 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bobenko A.I., Hoffmann T., Suris Y.B.: Hexagonal circle patterns and integrable systems: patterns with the multi-ratio property and Lax equations on the regular triangular lattice. Int. Math. Res. Not. 2002(3), 111–164 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bobenko A.I., Pinkall U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bobenko A.I., Pottmann H., Wallner J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348, 1–24 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry: Integrable Structure, Graduate Studies in Mathematics, vol. 98. American Math. Soc. (2008)Google Scholar
  6. 6.
    Bobenko A.I., Suris Y.B.: Discrete Koenigs nets and discrete isothermic surfaces. Int. Math. Res. Notices 11, 1976–2012 (2009)MathSciNetGoogle Scholar
  7. 7.
    Christoffel E.B.: Ueber einige allgemeine Eigenschaften der Minimumsflächen. J. Reine Angew. Math. 67, 218–228 (1867)MATHCrossRefGoogle Scholar
  8. 8.
    Eschenburg J.H., Jost J.: Differentialgeometrie und Minimalflächen. Springer-Verlag, Berlin (2007)MATHGoogle Scholar
  9. 9.
    Liu, Y., Wang, W.: On vertex offsets of polyhedral surfaces. In: Proceeding of Advances in Architectural Geometry. TU Wien, Vienna, pp. 61–64 (2008)Google Scholar
  10. 10.
    Müller C.: Hexagonal meshes as discrete minimal surfaces. PhD Thesis TU Graz (2010)Google Scholar
  11. 11.
    Müller C., Wallner J.: Oriented mixed area and discrete minimal surfaces. Discret. Comput. Geom. 43, 303–320 (2010)MATHCrossRefGoogle Scholar
  12. 12.
    Pinkall U., Polthier K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)MathSciNetMATHGoogle Scholar
  13. 13.
    Pottmann, H., Liu, Y., Wallner, J., Bobenko, A.I., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), #65,1–11 (2007)Google Scholar
  14. 14.
    Pottmann H., Wallner J.: The focal geometry of circular and conical meshes. Adv. Comp. Math. 29, 249–268 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sauer R.: Differenzengeometrie. Springer-Verlag, Berlin (1970)MATHGoogle Scholar
  16. 16.
    Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), #77,1–11 (2008)Google Scholar
  17. 17.
    Wallner J., Pottmann H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatshefte Math. 153, 347–365 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and Geometry, TU WienViennaAustria

Personalised recommendations