On the area of overlap of translated polygons
Computer Vision and Image Understanding
Rotational polygon containment and minimum enclosure
Proceedings of the fourteenth annual symposium on Computational geometry
Lectures on Discrete Geometry
Approximation of Convex Polygons
ICALP '90 Proceedings of the 17th International Colloquium on Automata, Languages and Programming
Approximating extent measures of points
Journal of the ACM (JACM)
Maximizing the overlap of two planar convex sets under rigid motions
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets
Computational Geometry: Theory and Applications
Maximum overlap and minimum convex hull of two convex polyhedra under translations
Computational Geometry: Theory and Applications
Bundling three convex polygons to minimize area or perimeter
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Given two compact convex sets C1 and C2 in the plane, we consider the problem of finding a placement ϕC1 of C1 that minimizes the area of the convex hull of ϕC1∪C2. We first consider the case where ϕC1 and C2 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 C1, 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.