Edge offset meshes in Laguerre geometry

Helmut Pottmann*, Philipp Grohs, Bernhard Blaschitz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

A mesh M with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh Md such that corresponding edges of M and Md lie on parallel lines of constant distance d. The edges emanating from a vertex of M lie on a right circular cone. Viewing M as set of these vertex cones, we show that the image of M under any Laguerre transformation is again an EO mesh. As a generalization of this result, it is proved that the cyclographic mapping transforms any EO mesh in a hyperplane of Minkowksi 4-space into a pair of Euclidean EO meshes. This result leads to a derivation of EO meshes which are discrete versions of Laguerre minimal surfaces. Laguerre minimal EO meshes can also be constructed directly from certain pairs of Koebe meshes with help of a discrete Laguerre geometric counterpart of the classical Christoffel duality.

Original languageEnglish (US)
Pages (from-to)45-73
Number of pages29
JournalAdvances in Computational Mathematics
Volume33
Issue number1
DOIs
StatePublished - Jul 1 2010

Keywords

  • Discrete differential geometry
  • Edge offset mesh
  • Koebe polyhedron
  • Laguerre geometry
  • Laguerre minimal surface
  • Minimal surface

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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