On the complexity of the set of three-finger caging grasps of convex polygons

We study three-finger caging grasps of convex polygons. A part is caged with a number of fingers when it is impossible to rigidly move the part to an arbitrary placement far from its initial placement without penetrating any finger. A convex polygon with n vertices and a placement of two fingers —referred to as the base fingers— are given. The caging region is the set of all placements of the third finger that together with the base fingers cage the polygon. We derive a novel formulation of caging in terms of visibility in three-dimensional space. We use this formulation to prove that the worst-case combinatorial complexity of the caging region is close to O(n), which is a significant improvement of the previously known upper bound of O(n). Moreover we provide an algorithm with a running time close to O(n logn) that considerably improves the current best known algorithm, which runs in O(n) time.