Solving distance problems with concave bodies using simulated annealing

Abstract

Determining the minimum distance between two convex objects is a problem that has been solved using many different approaches. On the other hand, computing the minimum distance between combinations of convex and concave objects is known to be a more complicated problem. Some methods propose to partition the concave object into convex sub-objects and then solve the convex problem between all possible sub-object combinations. While this method has been shown to work reliably, it adds a large computational expense when the concave objects in the scene are complicated, or when a quadratically bound object is to be linearized. An optimization approach is used to solve the concave problem without the need for partitioning the concave object into convex sub-objects. Since the optimization problem is no longer unimodal, a global optimization technique is used. Simulated annealing is used to solve the concave problem. In order to reduce the computational expense, it is proposed to replace the objects' geometry by a set of points on the surface of each body. This reduces the problem to a combinatorial problem where the combination of points that minimizes the distance will be the solution. Some examples using this method are presented.