Laguerre minimal surfaces, isotropic geometry and linear elasticity

Helmut Pottmann*, Philipp Grohs, Niloy Mitra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

40 Scopus citations

Abstract

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy. They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of, which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.

Original languageEnglish (US)
Pages (from-to)391-419
Number of pages29
JournalAdvances in Computational Mathematics
Volume31
Issue number4
DOIs
StatePublished - Oct 1 2009

Keywords

  • Airy stress function
  • Biharmonic function
  • Differential geometry
  • Geometrical optics
  • Isotropic geometry
  • Laguerre geometry
  • Laguerre minimal surface
  • Linear elasticity
  • Thin plate spline

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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