eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-06-02
14:1
14:12
10.4230/LIPIcs.SoCG.2021.14
article
Orientation Preserving Maps of the Square Grid
Bárány, Imre
1
2
Pór, Attila
3
Valtr, Pavel
4
https://orcid.org/0000-0002-3102-4166
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Department of Mathematics, University College London, UK
Department of Mathematics, Western Kentucky University, Bowling Green, KY, USA
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Praha 1, Czech Republic
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}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol189-socg2021/LIPIcs.SoCG.2021.14/LIPIcs.SoCG.2021.14.pdf
square grid
plane
order type