Approximation of the Normal Vector Field and the Area of a Smooth Surface
Approximation of the Normal Vector Field and the Area of a Smooth Surface
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DOI：
10.1007/s00454-004-1096-4
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发表时间：
2004-06
影响因子：
0.8
通讯作者：
J. Morvan;Boris Thibert
中科院分区：
文献类型：
--
作者：
J. Morvan;Boris Thibert
本文讨论光滑曲面S的法向量场与几乎处处可微曲面S的法向量场的比较。主要结果给出了S的法线与靠近S的三角剖分T的法线之间夹角的上界。这个上界表示在T的几何，S的曲率和两个表面之间的Hausdorff距离。这类结果是非常有用的：特别是，光滑曲面S的法向量场的近似结果可以导出面积的近似结果;实际上，在非常普遍情况下，（T仅假设是Lipschitz函数的局部图），如果我们知道两个曲面的法线之间的角度，则我们可以用T的几何不变量、S的曲率和两个曲面之间的Hausdorff距离来明确地表示S的面积。 我们还将所得结果应用于曲面重构中：当T是S的ε-样本的限制Delaunay三角剖分时，我们得到了收敛结果;利用Chew算法，我们还构造了内接于S的曲率测度趋于S的曲率测度的三角剖分序列.
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.