Unfolding of Surfaces |
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Authors: | Jean-Marie Morvan Boris Thibert |
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Institution: | (1) Institut Girard Desargues, Universite Claude Bernard Lyon1, bat. 21, 43 Bd du 11 novembre 1918, 69622, Villeurbanne Cedex, France;(2) Laboratoire de Modelisation et Calcul, Universite Joseph Fourier, 51 rue des Mathematiques, BP 53, 38041, Grenoble Cedex 9, France |
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Abstract: | This paper deals with the approximation of the unfolding of a smooth globally
developable surface (i.e. "isometric" to a domain of
) 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. |
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Keywords: | |
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