Power diagrams: properties, algorithms and applications

Reviewer: Tsang C. Huang

In 1908 the French mathematician G. F. Voronoi published a paper on quadratic forms [1], and since then the geometrical aspects of the subject have attracted the attention of many mathematicians and engineers, including Rogers [2] and Lee and Wong [3]. The power diagrams introduced by Aurenhammer in this paper generalize properties of Voronoi diagrams from points to sets. The power pow( x, s) of a point x with respect to a sphere s in Euclidean n-space E n is defined to be d 2( x, z)? r 2, where d is the Euclidean distance function, and z and r are the center and radius of s. The power diagram of a finite set S of spheres in E n is a cell complex that associates each s ? S with the convex domain cell ( s)={ x ? E n) &vbm0; pow( x, t) for all t ? S}-{s}}. Power diagrams enjoy applications in crystallography, metallurgy, and economics. Analyzing power diagrams, the author explains their relation to Voronoi diagrams and gives algorithms for their construction. The author also generalizes power diagrams to order-k power diagrams, analogous to order- k Voronoi diagrams, by extending the definition of cell( ? to sets of cardinality k, but the efficient construction of order- k power diagrams remains an open problem. Although the paper is carefully organized, it has a few unclear definitions and ambiguous notations. For example, d is used for both the dimension and the distance function. Nevertheless, everyone interested in computational geometry is encouraged to read this paper.