Stably placing piecewise smooth objects

Abstract

Given a piecewise smooth object, its stable poses consist of all the orientations into which other initial orientations of the object will eventually converge under dissipative forces. The capture region for each stable pose is the set of initial orientations converging to the stable pose in question.

We employ duality to solve these two related problems. Our approach produces non-trivial combinatorial bounds on the complexity of these problems as well as asymptotically efficient algorithms. It also allows us to remove the non-singularity constraints in Kriegman's previous work on the latter problem, and to enumerate the degenerate cases in a systematic way. Our analysis leads to a significant reduction in the algebraic complexity for objects consisting of quadratic surface patches cut by planes. The practical value of this approach is demonstrated by the implementation of an efficient approximation algorithm for this subclass of objects.