Smooth surface and triangular mesh: comparison of the area, the normals and the unfolding

J. Morvan, B. Thibert
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引用次数: 20

Abstract

Replacing a smooth surface with a triangular mesh (i.e., a polyedron) "close to it" leads to some errors. The geometric properties of the triangular mesh can be very different from the geometric properties of the smooth surface, even if both surfaces are very close from one another. In this paper, we give examples of "developable" triangular meshes (the discrete Gaussian curvature is equal to 0 at each interior vertex) inscribed in a sphere (whose Gaussian curvature is equal to 1 at every point). However, if we make assumptions on the geometry of the triangular mesh, on the curvature of the smooth surface and on the Hausdorff distance between both surfaces, we get an estimate of several properties of the smooth surface in terms of the properties of the triangular mesh. In particular, we give explicit approximations of the normals and of the area of the smooth surface. Furthermore, if we suppose that the smooth surface is developable (i.e., "isometric" to a surface of the plane), we give an explicit approximation of the "unfolding" of this surface. Just notice that in some of our approximations, we do not suppose that the vertices of the triangular mesh belong to the smooth surface. Oddly, the upper bounds on the errors are better when triangles are right-angled (even if there are small angles): we do not need every angle of the triangular mesh to be quite large. We just need each triangle of the triangular mesh to contain at least one angle whose sine is large enough. Besides, approximations are better if the triangles of the triangular mesh are quite small where the smooth surface has a large curvature. Some proofs will be omitted.
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光滑表面和三角形网格:面积,法线和展开的比较
用三角形网格(即聚体)代替光滑表面“接近它”会导致一些错误。三角形网格的几何性质可能与光滑表面的几何性质非常不同,即使这两个表面彼此非常接近。在本文中,我们给出了“可展开”三角形网格(离散高斯曲率在每个内部顶点都等于0)内切在一个球体(其高斯曲率在每个点都等于1)中的例子。然而,如果我们对三角形网格的几何形状、光滑表面的曲率以及两个表面之间的豪斯多夫距离做出假设,我们就可以根据三角形网格的性质对光滑表面的几种性质进行估计。特别地,我们给出了光滑表面的法线和面积的显式近似。此外,如果我们假设光滑表面是可展开的(即,与平面的一个表面“等距”),我们给出了这个表面的“展开”的显式近似。请注意,在我们的一些近似中,我们没有假设三角形网格的顶点属于光滑表面。奇怪的是,当三角形是直角时(即使有小角度),误差的上界更好:我们不需要三角形网格的每个角度都很大。我们只需要三角形网格的每个三角形包含至少一个正弦值足够大的角。此外,如果三角形网格的三角形相当小,而光滑表面具有较大的曲率,则近似值更好。一些证明将被省略。
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