Stacking and Bundling Two Convex Polygons
Given two compact convex sets C 1 and C 2 in the plane, we consider the problem of finding a placement ϕC 1 of C 1 that minimizes the area of the convex hull of ϕC 1 ∪ C 2. We first consider the case where ϕC 1 and C 2 are allowed to intersect (as in “stacking” two flat objects in a convex box), and then add the restriction that their interior has to remain disjoint (as when “bundling” two convex objects together into a tight bundle). In both cases, we consider both the case where we are allowed to reorient C 1, and where the orientation is fixed. In the case without reorientations, we achieve exact near-linear time algorithms, in the case with reorientations we compute a (1 + ε)-approximation in time O(ε − 1/2 log n + ε − 3/2 log ε − 1/2), if two sets are convex polygons with n vertices in total.
KeywordsConvex Hull Convex Polygon Rigid Motion Supporting Line Convex Object
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