Approximation of the normal vector field and the area of a smooth surface

Jean-Marie Morvan*, Boris Thibert

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)383-400
Number of pages18
JournalDiscrete and Computational Geometry
Volume32
Issue number3
DOIs
StatePublished - Jan 1 2004

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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