Two finger caging: squeezing and stretching

Abstract

This paper studies the problem of placing two point fingers to cage a mobile rigid body in a Euclidean space of arbitrary dimension. (To cage an object is to arrange obstacles so that all motions of the mobile body are bounded.) This paper shows that if a compact connected contractible object is caged by two points, then it is either stretching caged or squeezing caged or both, where stretching caged means the body is trapped even if the point fingers are given the freedom of moving apart, and squeezing caged means the the body is trapped even if the fingers are given the freedom of moving closer. This result generalizes a previous result by Vahedi and van der Stappen which applied to two points trapping a polygon in the plane. Our use of a topological approach led to the generalization, and may lead to further generalizations and insights.