Computation of certain measures of proximity between convex polytopes: a complexity viewpoint

The quantification of proximity between a pair of objects whose point descriptions are given is considered. Four problems of proximity between two convex polytopes in R³ are considered. The convex polytopes are represented as convex hulls of finite sets of points. The authors discuss the complexity of solving the four problems. They analyze algorithms for the four problems in terms of two complexity types. Let the total number of points in the two finite sets be n . It is shown that three of the proximity problems, checking intersection, checking whether the polytopes are just touching, and finding the distance between them, can be solved in O(n) time for fixed s and in polynomial time for varying s. It is also shown that the fourth proximity problem of finding the intensity of collision for varying s is NP-complete