Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

Gert Vegter, Mathijs Wintraecken

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Abstract

Fejes 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.

Original languageEnglish
Pages (from-to)193-199
Number of pages7
Journal Studia Scientiarum Mathematicarum Hungarica
Volume57
Issue number2
DOIs
Publication statusPublished - 2020

Keywords

  • Approximation
  • Fejes Tóth
  • Hausdorff distance
  • Surface meshes

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