Discrete & Computational Geometry

, Volume 43, Issue 2, pp 303–320 | Cite as

Oriented Mixed Area and Discrete Minimal Surfaces



Recently a curvature theory for polyhedral surfaces has been established, which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed area of a derived polygon pair which has only the half number of vertices. Thus we are able to recursively characterize vanishing mixed area for hexagons and other n-gons in an incidence-geometric way. We use these geometric results for the construction of discrete minimal surfaces and a study of equilibrium forces in their edges, especially those with the combinatorics of a hexagonal mesh.


Oriented mixed area Discrete curvatures Geometric configurations Discrete minimal surfaces Reciprocal parallelity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bobenko, A., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996) MATHMathSciNetGoogle Scholar
  2. 2.
    Bobenko, A., Suris, Y.: Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics, vol. 98. American Math. Soc., Providence (2008) MATHGoogle Scholar
  3. 3.
    Bobenko, A.I., Suris, Y.: Discrete Koenigs nets and discrete isothermic surfaces. Int. Math. Res. Not. 2009, 1976–2012 (2009) MATHMathSciNetGoogle Scholar
  4. 4.
    Bobenko, A., Hoffmann, T., Springborn, B.: Minimal surfaces from circle patterns: geometry from combinatorics. Ann. Math. 164, 231–264 (2006) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2, 15–36 (1993) MATHMathSciNetGoogle Scholar
  6. 6.
    Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry of multi-layer free form structures for architecture. ACM Trans. Graph. 26(3), #65 (2007) CrossRefGoogle Scholar
  7. 7.
    Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comput. Math. 29, 249–268 (2008) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sauer, R.: Differenzengeometrie. Springer, Berlin (1970) MATHGoogle Scholar
  9. 9.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993) MATHGoogle Scholar
  10. 10.
    Wallner, J., Pottmann, H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatshefte Math. 153, 347–365 (2008) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of GeometryTU GrazGrazAustria

Personalised recommendations