License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.75
URN: urn:nbn:de:0030-drops-122337
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12233/
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Fekete, Sándor P. ; Keldenich, Phillip ; Scheffer, Christian

Covering Rectangles by Disks: The Video (Media Exposition)

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LIPIcs-SoCG-2020-75.pdf (0.8 MB)

Abstract

In this video, we motivate and visualize a fundamental result for covering a rectangle by a set of non-uniform circles: For any λ ≥ 1, the critical covering area A^*(λ) is the minimum value for which any set of disks with total area at least A^*(λ) can cover a rectangle of dimensions λ× 1. We show that there is a threshold value λ₂ = √(√7/2 - 1/4) ≈ 1.035797…, such that for λ < λ₂ the critical covering area A^*(λ) is A^*(λ) = 3π(λ²/16 + 5/32 + 9/256λ²), and for λ ≥ λ₂, the critical area is A^*(λ) = π(λ²+2)/4; these values are tight. For the special case λ=1, i.e., for covering a unit square, the critical covering area is 195π/256 ≈ 2.39301…. We describe the structure of the proof, and show animations of some of the main components.

BibTeX - Entry

@InProceedings{fekete_et_al:LIPIcs:2020:12233,
  author =	{S{\'a}ndor P. Fekete and Phillip Keldenich and Christian Scheffer},
  title =	{{Covering Rectangles by Disks: The Video (Media Exposition)}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{75:1--75:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12233},
  URN =		{urn:nbn:de:0030-drops-122337},
  doi =		{10.4230/LIPIcs.SoCG.2020.75},
  annote =	{Keywords: Disk covering, critical density, covering coefficient, tight worst-case bound, interval arithmetic, approximation}
}

Keywords: Disk covering, critical density, covering coefficient, tight worst-case bound, interval arithmetic, approximation
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020
Supplementary Material: https://github.com/phillip-keldenich/circlecover

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