Geometric Properties and Computation of Three-Finger Caging Grasps of Convex Polygons

We study three-finger caging grasps of convex polygons. A grasp is said to cage a part when the fingers make it impossible for the part to move to a distant location-and, hence, escape the grasp-without penetrating a finger. We present a collection of geometric properties of three-finger caging grasps of convex polygons, and establish a relation between caging grasps and immobilizing grasps. We use these results to derive an algorithm that computes a data structure in roughly O(n6) time for a given convex polygon with n edges and a specified distance between two so-called base fingers. The data structure allows us to efficiently solve two problems. 1) For a given placement of the base fingers and a placement for the third finger, we give an algorithm that determines in O(log n) time whether the resulting grasp cages the polygon. 2) For a given placement of the base fingers on the polygon boundary we give an algorithm that outputs in O(n+K) time all placements of the third finger such that the three fingers together constitute a caging grasp of the polygon, in which K is proportional to the complexity of the output.