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}).
@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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