Orientation Preserving Maps of the Square Grid

Bárány, Imre; Pór, Attila; Valtr, Pavel

doi:10.4230/LIPIcs.SoCG.2021.14

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}).

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2021.14

Author Details

Imre Bárány

Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Department of Mathematics, University College London, UK

Attila Pór

Department of Mathematics, Western Kentucky University, Bowling Green, KY, USA

Pavel Valtr

Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Praha 1, Czech Republic

Funding

Bárány, Imre: Partially supported by Hungarian National Research Grants No 131529, 132696, and 133819.
Valtr, Pavel: Supported by the grant no. 21-32817S of the Czech Science Foundation (GAČR).

References

Jacob E. Goodman and Richard Pollack. The complexity of point configurations. Discret. Appl. Math., 31(2):167-180, 1991. URL: https://doi.org/10.1016/0166-218X(91)90068-8.
Jacob E. Goodman, Richard Pollack, and Bernd Sturmfels. The intrinsic spread of a configuration in ℝ^d. Journal of the American Mathematical Society, 3(3):639-651, 1990. URL: http://www.jstor.org/stable/1990931.
Jan Kratochvíl and Jiří Matoušek. Intersection graphs of segments. J. Comb. Theory, Ser. B, 62(2):289-315, 1994. URL: https://doi.org/10.1006/jctb.1994.1071.
Alexander Pilz and Emo Welzl. Order on order types. Discret. Comput. Geom., 59(4):886-922, 2018. URL: https://doi.org/10.1007/s00454-017-9912-9.
Jaroslav Nešetřil and Pavel Valtr. A Ramsey-type theorem in the plane. Comb. Probab. Comput., 3:127-135, 1994. URL: https://doi.org/10.1017/S0963548300001024.
Jaroslav Nešetřil and Pavel Valtr. A Ramsey property of order types. J. Comb. Theory, Ser. A, 81(1):88-107, 1998. URL: https://doi.org/10.1006/jcta.1997.2820.

Orientation Preserving Maps of the Square Grid

Authors Imre Bárány, Attila Pór, Pavel Valtr

File

Document Identifiers

Subject Classification

ACM Subject Classification

Keywords

Metrics

Abstract

Cite As Get BibTex

Author Details

References

Thanks for your feedback!

Could not send message