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Aligning two convex figures to minimize area or perimeter (2009)
| Citations: | 4 - 2 self |
BibTeX
@MISC{Ahn09aligningtwo,
author = {Hee-kap Ahn and Otfried Cheong},
title = {Aligning two convex figures to minimize area or perimeter },
year = {2009}
}
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Abstract
Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact nearlinear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1 + ε)-approximation in time O(ε −1/2 log n + ε −3/2 log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a ∈ {0, 1, 2} depending on the version of the problem.
