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Discrete & Computational Geometry

, Volume 32, Issue 3, pp 383–400 | Cite as

Approximation of the Normal Vector Field and the Area of a Smooth Surface

  • Jean-Marie  Morvan
  • Boris Thibert
Article

Abstract

This paper deals with the comparison of the normal vector field of a smooth surface S with the normal vector field of another surface differentiable almost everywhere. The main result gives an upper bound on angles between the normals of S and the normals of a triangulation T close to S. This upper bound is expressed in terms of the geometry of T, the curvature of S and the Hausdorff distance between both surfaces. This kind of result is really useful: in particular, results of the approximation of the normal vector field of a smooth surface S can induce results of the approximation of the area; indeed, in a very general case (T is only supposed to be locally the graph of a lipschitz function), if we know the angle between the normals of both surfaces, then we can explicitly express the area of S in terms of geometrical invariants of T, the curvature of S and of the Hausdorff distance between both surfaces. We also apply our results in surface reconstruction: we obtain convergence results when T is the restricted Delaunay triangulation of an ε-sample of S; using Chew’s algorithm, we also build sequences of triangulations inscribed in S whose curvature measures tend to the curvatures measures of S.

Keywords

Vector Field Smooth Surface Normal Vector Convergence Result Lipschitz Function 
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-Verlag 2004

Authors and Affiliations

  1. 1.INRIA, 2004 Route des Lucioles, B.P 93, 06904 Sophia-AntipolisFrance
  2. 2.Institut Girard Desargues, Université Claude Bernard Lyon1, bât. 21, 43 Bd du 11 novembre 1918, 69622 Villeurbanne CedexFrance

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