License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.14
URN: urn:nbn:de:0030-drops-138130
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13813/
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Bárány, Imre ; Pór, Attila ; Valtr, Pavel

Orientation Preserving Maps of the Square Grid

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LIPIcs-SoCG-2021-14.pdf (0.9 MB)

Abstract

For a finite set A ⊂ ℝ², a map φ: A → ℝ² is orientation preserving if for every non-collinear triple u,v,w ∈ A the orientation of the triangle u,v,w is the same as that of the triangle φ(u),φ(v),φ(w). We prove that for every n ∈ ℕ and for every ε > 0 there is N = N(n,ε) ∈ ℕ such that the following holds. Assume that φ:G(N) → ℝ² is an orientation preserving map where G(N) is the grid {(i,j) ∈ ℤ²: -N ≤ i,j ≤ N}. Then there is an affine transformation ψ :ℝ² → ℝ² and a ∈ ℤ² such that a+G(n) ⊂ G(N) and ‖ψ∘φ (z)-z‖ < ε for every z ∈ a+G(n). This result was previously proved in a completely different way by Nešetřil and Valtr, without obtaining any bound on N. Our proof gives N(n,ε) = O(n⁴ε^{-2}).

BibTeX - Entry

@InProceedings{barany_et_al:LIPIcs.SoCG.2021.14,
  author =	{B\'{a}r\'{a}ny, Imre and P\'{o}r, Attila and Valtr, Pavel},
  title =	{{Orientation Preserving Maps of the Square Grid}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13813},
  URN =		{urn:nbn:de:0030-drops-138130},
  doi =		{10.4230/LIPIcs.SoCG.2021.14},
  annote =	{Keywords: square grid, plane, order type}
}

Keywords: square grid, plane, order type
Collection: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue Date: 2021
Date of publication: 02.06.2021

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