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Discrete Surfaces in Isotropic Geometry

  • Helmut Pottmann
  • Yang Liu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4647)

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.

Keywords

discrete differential geometry surfaces in architecture isotropic geometry conical mesh edge offset mesh isotropic minimal surface 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Helmut Pottmann
    • 1
  • Yang Liu
    • 2
  1. 1.Geometric Modeling and Industrial Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 WienAustria
  2. 2.Department of Computer Science, University of Hong KongChina

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