@article{53a2da69b54f422c8780443aa7da0b9e,
title = "Refutation of a claim made by Fejes T{\'o}th on the accuracy of surface meshes",
abstract = "Fejes T{\'o}th [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.",
keywords = "Approximation, Fejes T{\'o}th, Hausdorff distance, Surface meshes",
author = "Gert Vegter and Mathijs Wintraecken",
note = "Funding Information: Acknowledgements. The authors are greatly indebted to Dror Atariah, G{\"u}n-ther Rote and John Sullivan for discussion and suggestions. The authors also thank Jean-Daniel Boissonnat, Ramsay Dyer, David de Laat and Rien van de Weijgaert for discussion. This work has been supported in part by the European Union{\textquoteright}s Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL Computational Geometry Learning) and ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions), the European Union{\textquoteright}s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement number 754411, and the Austrian Science Fund (FWF): Z00342 N31. Funding Information: Acknowledgements. The authors are greatly indebted to Dror Atariah, G?nther Rote and John Sullivan for discussion and suggestions. The authors also thank Jean-Daniel Boissonnat, Ramsay Dyer, David de Laat and Rien van de Weijgaert for discussion. This work has been supported in part by the European Union?s Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL Computational Geometry Learning) and ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions), the European Union?s Horizon 2020 research and innovation pro-gramme under the Marie Sklodowska-Curie grant agreement number 754411, and the Austrian Science Fund (FWF): Z00342 N31. Publisher Copyright: {\textcopyright} 2020 The Authors",
year = "2020",
doi = "10.1556/012.2020.57.2.1454",
language = "English",
volume = "57",
pages = "193--199",
journal = " Studia Scientiarum Mathematicarum Hungarica ",
issn = "0081-6906",
publisher = "Akad{\'e}miai Kiad{\'o}",
number = "2",
}