Discrete surfaces in isotropic geometry

Helmut Pottmann*, Yang Liu

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

24 Scopus citations

Abstract

Meshes with planar quadrilateral faces are desirable discrete surface representations for architecture. The present paper introduces new classes of planar quad meshes, which discretize principal curvature lines of surfaces in so-called isotropic 3-space. Like their Euclidean counterparts, these isotropic principal meshes meshes are visually expressing fundamental shape characteristics and they can satisfy the aesthetical requirements in architecture. The close relation between isotropic geometry and Euclidean Laguerre geometry provides a link between the new types of meshes and the known classes of conical meshes and edge offset meshes. The latter discretize Euclidean principal curvature lines and have recently been realized as particularly suited for freeform structures in architecture, since they allow for a supporting beam layout with optimal node properties. We also present a discrete isotropic curvature theory which applies to all types of meshes including triangle meshes. The results are illustrated by discrete isotropic minimal surfaces and meshes computed by a combination of optimization and subdivision.

Original languageEnglish (US)
Title of host publicationMathematics of Surfaces XII - 12th IMA International Conference, Proceedings
PublisherSpringer Verlag
Pages341-363
Number of pages23
ISBN (Print)9783540738428
StatePublished - Jan 1 2007
Event12th IMA International Conference on the Mathematics of Surfaces - Sheffield, United Kingdom
Duration: Sep 4 2007Sep 6 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4647 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th IMA International Conference on the Mathematics of Surfaces
CountryUnited Kingdom
CitySheffield
Period09/4/0709/6/07

Keywords

  • Conical mesh
  • Discrete differential geometry
  • Edge offset mesh
  • Isotropic geometry
  • Isotropic minimal surface
  • Surfaces in architecture

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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