Caging convex polygons with three fingers

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 build a data structure in polynomial time for a given convex polygon with n edges that allows us to solve two problems. (1) For a given grasp, we give an algorithm that determines in O(log n) time whether the grasp cages the polygon. (2) For a given placement of two fingers we give an algorithm that outputs in O(n⁴ log 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 description.