Caging Polygons with Two and Three Fingers

We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(n log n) time. Our result extends and improves a recent solution for point fingers. In addition, we construct a data structure requiring O(n) storage that can answer in O(log n) whether two fingers in a query placement cage the polygon. We also study caging with three point fingers. Given the placements of two so-called base fingers, we report all placements of the third finger so that the three fingers jointly cage the polygon. Using the fact that the boundary of the set of placements for the third finger consists of equilibrium grasps, we present an algorithm that reports all placements of the third finger in O(n log n) deterministic time and O(n log n(log log n)) expected time. Our results extend previous solutions that only apply to convex polygons.