JOURNAL OF ALGORITHMS 9,151-161 (1988) Improved Algorithms for Discs and Balls Using Power Diagrams F. AURENHAMMER* Institutes for Information Processing, Technical University of Graz and Austrian Computer Socie& Schiesstattgasse 4a, A-801 0 Graz, Awtria Received July 15,1985 The properties of a particular generalization of Voronoi diagrams called power diagrams are exploited to obtain new and improved algorithms for union, intersec- tion, and measure problems for discs and balls. 0 1988 AC&XICC PEW, IW. 1. POWER DIAGRAMS In the last few years, a particular generalization of the Voronoi diagram called power diagram (or also Laguerre diagram, Dirichlet cell complex) has received attention in computational geometry. For a set S of n spheres in Euclidean d-space Ed, the power diagram PD( S) of S is a partition of Ed into convex polyhedra (i.e., a cell complex in Ed) defined as follows. Each s E S is associated with its power cell cell(s) = {x E Ed(pow(x, S) I pOw(X, t),Vl E S}, where pow(x, s) = a2(x, z) - r2, for z and r the center and the radius of s and 6 the Euclidean distance; see Fig. 1 for an illustration. Note that PD(S) coincides with the (standard) Voronoi diagram of the centers if the spheres in S are pairwise congruent. At this early point, let us take a look at the combinatorial structure of PD(S). The power cells (also called d-faces) are closed and convex, but possibly unbounded, and cover E d. The boundary of cell(s) consists of various j-dimensional polyhedra called j-faces (0 I j s d - l), and cell(s) n cell(t) (s, t E S) is either empty or some face. O-faces and l-faces are also called vertices and edges, respectively. *Research was supported by the Austrian Fonds zur Foerderung der wlssenschaftlichen Forschung. 151 0196-6774/88 $3.00 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in my form reserved.