Solving distance problems with concave bodies using simulated annealing

Determining the minimum distance between two convex objects is a problem that has been solved using many di¤erent approaches. Some methods rely on computational geometry techniques, while others rely on optimization techniques to ...nd the solution. Some fast algorithms sacri...ce precision for speed while others are limited in the types of objects that they can handle (e.g. linearly bound objects, quadratically bound objects, etc.). 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 subobject 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. In this paper 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 (i.e., has more than one local minimum point), 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 (one on each body) that minimizes the distance will be the solution. Some examples using this novel method are presented.