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DOI: 10.4230/LIPIcs.SoCG.2021.14

Orientation Preserving Maps of the Square Grid

Authors: Imre Bárány, Attila Pór, and Pavel Valtr

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

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}).

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Imre Bárány, Attila Pór, and Pavel Valtr. Orientation Preserving Maps of the Square Grid. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@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/entities/document/10.4230/LIPIcs.SoCG.2021.14},
  URN =		{urn:nbn:de:0030-drops-138130},
  doi =		{10.4230/LIPIcs.SoCG.2021.14},
  annote =	{Keywords: square grid, plane, order type}
}

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Orientation Preserving Maps of the Square Grid

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