Discrete & Computational Geometry

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

Oriented Mixed Area and Discrete Minimal Surfaces

  • Christian Müller
  • Johannes WallnerEmail author


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 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of GeometryTU GrazGrazAustria

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