2026-07-22T12:28:38Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19979 2024-06-06T06:21:14Z ddc:004 open_access

Convex Polygon Containment: Improving Quadratic to Near Linear Time Chan, Timothy M. Hair, Isaac M. Polygon containment convex polygons translations rotations We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0. Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Timothy M. Chan and Isaac M. Hair 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.34 urn:nbn:de:0030-drops-199795 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.34 eng https://creativecommons.org/licenses/by/4.0/legalcode