2026-07-25T03:04:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13813 2024-03-06T10:52:47Z ddc:004 open_access

Orientation Preserving Maps of the Square Grid Bárány, Imre Pór, Attila Valtr, Pavel square grid plane order type 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}). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Imre Bárány and Attila Pór and Pavel Valtr 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.14 urn:nbn:de:0030-drops-138130 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.14 eng https://creativecommons.org/licenses/by/4.0/legalcode