Abstract
AbstractFejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.
Highlights
In [3] Fejes Toth introduced inscribed triangulations approximating convex surfaces in R3 optimally and the approximation parameter A2 (Approximierbarkeit)
By a triangulation we shall mean a geometric realization of a simplicial complex in Euclidean space homeomorphic to the surface, that is piecewise linear in ambient space
Fejes Toth claimed that the approximation of ruled surfaces embedded in three dimensional Euclidean space would be entirely different from the approximation of convex surfaces
Summary
In [3] Fejes Toth introduced inscribed triangulations approximating convex surfaces in R3 optimally and the approximation parameter A2 (Approximierbarkeit). [3] Fejes Toth introduced inscribed triangulations approximating convex surfaces in R3 optimally and the approximation parameter A2 (Approximierbarkeit). K dA, where K is the Gaussian curvature, for the approximation parameter for convex surfaces in three dimensional Euclidean space.
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