Solving Geometric Optimization Problems

We show how to use graphics hardware for tackling optimization problems arising in the field of computational geometry. We exemplarily discuss three problems, where combinatorial algorithms are inefficient or hard to implement. Given a set S of n point in the plane, the first two problems are to determine the smallest homothetic scaling of a star shaped polygon P enclosing S and to find the largest empty homothetic scaling of P completely contained inside an arbitrary polygonal region. Pixel-exact solutions for both problems are computed in real-time. The third problem is a facility location problem and more difficult to solve. Given the Voronoi diagram VoD S of the n points, we try to position another point p in the plane, such that the resulting Voronoi region of p has maximal area. As far as we know there exists no traditional solution for this problem for which we present pixel-exact solutions.