2026-07-22T14:05:27Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:12233 2024-03-06T10:49:46Z ddc:004 open_access

Covering Rectangles by Disks: The Video (Media Exposition) Fekete, Sándor P. Keldenich, Phillip Scheffer, Christian Disk covering critical density covering coefficient tight worst-case bound interval arithmetic approximation 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sándor P. Fekete and Phillip Keldenich and Christian Scheffer 2020 Is Part Of LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2020.75 urn:nbn:de:0030-drops-122337 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.75 eng https://creativecommons.org/licenses/by/3.0/legalcode