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Discrete & Computational Geometry

, Volume 36, Issue 3, pp 393–418 | Cite as

Unfolding of Surfaces

  • Jean-Marie Morvan
  • Boris Thibert
Article

Abstract

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of \({\Bbb E}^2\)) with a triangulation. We prove the following result: let Tn be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of Tn tend to the normals of S, then the shape of the unfolding of Tn tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges.

Keywords

Smooth Surface Interior Point Gauss Curvature Discrete Comput Geom Hausdorff Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Institut Girard Desargues, Universite Claude Bernard Lyon1, bat. 21, 43 Bd du 11 novembre 1918, 69622Villeurbanne CedexFrance
  2. 2.Laboratoire de Modelisation et Calcul, Universite Joseph Fourier, 51 rue des Mathematiques, BP 53, 38041Grenoble Cedex 9France

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