Caging Polygons with Two and Three Fingers

Abstract

We study two- and three-finger caging grasps of a given polygonal object with n edges. A grasp is said to cage an object when it is impossible to take the object to a distant location without penetrating a finger. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging grasps of two disk fingers in O (n 2 log n) time. Our result extends and improves a recent solution for point fingers (Pipattanasomporn and Sudsang 2006). In addition, we construct a data structure in O (n 2 log n) time requiring O (n 2 ) space that can be queried in O (log n) time whether a given two-finger grasp cages the polygon. We also establish a relation between two-finger caging grasps and two-finger immobilizing grasps of polygons without parallel edges. We also study caging grasps with three point fingers. Given the placements of two so-called base fingers, the caging region is the set of all placements of the third finger that jointly with the base fingers forms a caging grasp of a polygonal object. Using the relation between equilibrium grasps and the boundary of the caging region, we present an algorithm that reports the entire caging region in O (n 6 log2 n) time. Our result extends a previous solution that only applies to convex polygons (Erickson et al. 2007).