## 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 language | English (US) |
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Pages (from-to) | 383-400 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2004 |

## ASJC Scopus subject areas

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