Extremal polygon containment problems

Abstract

Given a convex polygonal object P and an environment consisting of polygonal obstacles, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo [Me], and the fixed size polygon placement algorithms developed by Leven and Sharir [LS,LS1], to obtain an algorithm that runs in time O(k 2 nλ 4 (kn) log 3 (kn) log log(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon.

Diameter, width, closest line pair, and parametric searching

“…We apply his Although the application of Megiddo's parametric searching to the first three problems is rather natural, we are not aware of any previous attempt to apply the technique to these problems. The number of efficient geometric algorithms that make use of this technique is still rather small (see [11 [3]- [6], [15], and [23]- [28] for results of this kind); we believe that the potential of the technique in computational geometry is still quite far from being exhausted, as this paper well demonstrates.…”

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

“…Chazelle [31] gave an algorithm to decide whether one convex polygon fits into another. Other papers dealing with polygon containment include [16,20,21,58]; applications in manufacturing, cartography and image processing are described in [41,61,62], respectively.…”

Largest similar copies of convex polygons amidst polygonal obstacles

“…Toledo [21], and Sharir and Toledo [19] studied this problem (they called this problem the extremal polygon containment problem) and applied the motion-planning algorithm [14] to solve this problem. They gave an algorithm with running time O(k 2 nλ 4 (kn) log 3 (kn) log log(kn)) that uses the parametric search technique of Megiddo [17].…”