Monatshefte für Mathematik

, Volume 153, Issue 4, pp 347–365 | Cite as

Infinitesimally flexible meshes and discrete minimal surfaces

  • Johannes WallnerEmail author
  • Helmut Pottmann


We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.

2000 Mathematics Subject Classification: 53A40, 52C99, 51B15, 65D18 
Key words: Conical meshes, discrete minimal surfaces, reciprocal-parallel meshes 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Technische Universität GrazGrazAustria

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