Infinitesimally flexible meshes and discrete minimal surfaces

Johannes Wallner*, Helmut Pottmann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)347-365
Number of pages19
JournalMonatshefte fur Mathematik
Volume153
Issue number4
DOIs
StatePublished - Apr 1 2008

Keywords

  • Conical meshes
  • Discrete minimal surfaces
  • Reciprocal-parallel meshes

ASJC Scopus subject areas

  • Mathematics(all)

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