---
_id: '8163'
abstract:
- lang: eng
  text: 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.
acknowledgement: "The authors are greatly indebted to Dror Atariah, Günther 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\r\nEuropean 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 programme under the Marie Sk lodowska-Curie
  grant agreement number 754411,and the Austrian Science Fund (FWF): Z00342 N31."
article_processing_charge: No
article_type: original
author:
- first_name: Gert
  full_name: Vegter, Gert
  last_name: Vegter
- first_name: Mathijs
  full_name: Wintraecken, Mathijs
  id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
  last_name: Wintraecken
  orcid: 0000-0002-7472-2220
citation:
  ama: Vegter G, Wintraecken M. Refutation of a claim made by Fejes Tóth on the accuracy
    of surface meshes. <i>Studia Scientiarum Mathematicarum Hungarica</i>. 2020;57(2):193-199.
    doi:<a href="https://doi.org/10.1556/012.2020.57.2.1454">10.1556/012.2020.57.2.1454</a>
  apa: Vegter, G., &#38; Wintraecken, M. (2020). Refutation of a claim made by Fejes
    Tóth on the accuracy of surface meshes. <i>Studia Scientiarum Mathematicarum Hungarica</i>.
    Akadémiai Kiadó. <a href="https://doi.org/10.1556/012.2020.57.2.1454">https://doi.org/10.1556/012.2020.57.2.1454</a>
  chicago: Vegter, Gert, and Mathijs Wintraecken. “Refutation of a Claim Made by Fejes
    Tóth on the Accuracy of Surface Meshes.” <i>Studia Scientiarum Mathematicarum
    Hungarica</i>. Akadémiai Kiadó, 2020. <a href="https://doi.org/10.1556/012.2020.57.2.1454">https://doi.org/10.1556/012.2020.57.2.1454</a>.
  ieee: G. Vegter and M. Wintraecken, “Refutation of a claim made by Fejes Tóth on
    the accuracy of surface meshes,” <i>Studia Scientiarum Mathematicarum Hungarica</i>,
    vol. 57, no. 2. Akadémiai Kiadó, pp. 193–199, 2020.
  ista: Vegter G, Wintraecken M. 2020. Refutation of a claim made by Fejes Tóth on
    the accuracy of surface meshes. Studia Scientiarum Mathematicarum Hungarica. 57(2),
    193–199.
  mla: Vegter, Gert, and Mathijs Wintraecken. “Refutation of a Claim Made by Fejes
    Tóth on the Accuracy of Surface Meshes.” <i>Studia Scientiarum Mathematicarum
    Hungarica</i>, vol. 57, no. 2, Akadémiai Kiadó, 2020, pp. 193–99, doi:<a href="https://doi.org/10.1556/012.2020.57.2.1454">10.1556/012.2020.57.2.1454</a>.
  short: G. Vegter, M. Wintraecken, Studia Scientiarum Mathematicarum Hungarica 57
    (2020) 193–199.
date_created: 2020-07-24T07:09:18Z
date_published: 2020-07-24T00:00:00Z
date_updated: 2023-10-10T13:05:27Z
day: '24'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1556/012.2020.57.2.1454
ec_funded: 1
external_id:
  isi:
  - '000570978400005'
file:
- access_level: open_access
  content_type: application/pdf
  creator: mwintrae
  date_created: 2020-07-24T07:09:06Z
  date_updated: 2020-07-24T07:09:06Z
  file_id: '8164'
  file_name: 57-2-05_4214-1454Vegter-Wintraecken_OpenAccess_CC-BY-NC.pdf
  file_size: 1476072
  relation: main_file
file_date_updated: 2020-07-24T07:09:06Z
has_accepted_license: '1'
intvolume: '        57'
isi: 1
issue: '2'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 193-199
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
publication: Studia Scientiarum Mathematicarum Hungarica
publication_identifier:
  eissn:
  - 1588-2896
  issn:
  - 0081-6906
publication_status: published
publisher: Akadémiai Kiadó
quality_controlled: '1'
scopus_import: '1'
status: public
title: Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes
tmp:
  image: /images/cc_by_nc.png
  legal_code_url: https://creativecommons.org/licenses/by-nc/4.0/legalcode
  name: Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)
  short: CC BY-NC (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 57
year: '2020'
...