Research on properties of power diagram and its construction algorithm

Power diagram of a weighted point set is the generalization of the Voronoi diagram of a point set. It is especially appropriate for solving the geometric problems concerning spheres (circles). This paper first gives a geometric proof of the properties of the Power diagram. And then, it studies the minus weights' influence on the Power diagram, and points out that the weights of the points can be minus in the theory of Power diagram. In this sense, the application of the Power diagram can be extended to the field containing minus weights. In the end, this paper develops an algorithm for computing the Power diagram, which is based on the principle that the Power diagram of a weighted point set is the line dual graph of the regular triangulation of it, and examples are given to illustrate its efficiency.