Approximation of the Normal Vector Field and the Area of a Smooth
Surface |
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Authors: | Email author" target="_blank">Jean-Marie ?MorvanEmail author Email author" target="_blank">Boris?ThibertEmail author |
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Institution: | (1) INRIA, 2004 Route des Lucioles, B.P 93, 06904 Sophia-Antipolis, France;(2) Institut Girard Desargues, Université Claude Bernard Lyon1, bât. 21, 43 Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France |
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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 -sample of S; using Chews
algorithm, we also build sequences of triangulations inscribed in S whose
curvature measures tend to the curvatures measures of S. |
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Keywords: | |
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