Computing the volume of the union of spheres David Avis 1, Binay K. Bhattacharya 2, and Hiroshi Imai 3 1 School of Computer Science, McGill University, 805 Sherbrooke Street West, Montreal, PQ, Canada H3A 2K6 2 School of Computer Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 3 Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812, Japan On O (n 2) exact algorithm is given for com- puting the volume of a set of n spheres in space. The algorithm employs the La- guerre Voronoi (power) diagram and a method for computing the volume of the intersection of a simplex and a sphere ex- actly. We give a new proof of a special case of a conjecture, popularized by Klee, concerning the change in volume as the centres of the spheres become further apart. Key words: Union of Spheres Volumes - Laguerre Voronoi diagram - Power dia- gram 1 Introduction In this paper we give an O(n 2) exact algorithm for computing the volume of the union of a set of n spheres in space. This problem is of interest in nuclear physics [7, 6] and urban planning [4]. The only other exact method known to the authors is by an application of the inclusion-exclusion prin- ciple, giving an exponential running time algo- rithm. Our approach is to partition space into poly- gonal cells, with one cell for each sphere, so that the volume of the union can be computed by sim- ply summing the volume of the intersection of each sphere with its corresponding cell. The cell decom- position used is the Laguerre-Voronoi diagram [3, 1], which was used to solve the two dimensional version of the same problem. A critical procedure required is to compute the volume of the intersec- tion of a sphere and three half spaces. Our ap- proach generalized to d-dimensions whenever the measure of the intersection of a hypersphere and d half spaces is computable. This problem is partic- ularly simple for d = 2, but seems hard for d > 4. In the paper, we consider the three-dimensional case in detail. In principle it is possible to obtain a formula for computing the volume of the intersec- tion of a sphere and three half spaces, but such a formula would be extremely complex. We rather adopt the decomposition approach. We decompose the problem, using elementary geometric proper- ties, and show that the volume of the intersection of a sphere and three half spaces can be computed if a formula for computing the volume of the inter- section of a sphere and two half spaces is available. We also present such a formula, thus giving an exact method of computing the volume of the union of n spheres. As our model of computation, we adopt the real RAM in Preparata and Shamos [8], in which each word is capable of holding a single real number, and, besides the fundamental arithmetic operations and comparisons, the square root and inverse trig- onometric functions are available at unit cost. In computing the volume of the union of spheres, those additional operations seem indispensable. We also mention an old problem on the volume of the union of spheres, and prove a special case of the conjecture by means of the Laguerre-Voron- oi diagram. TheVisual Computer (1988) 3:323-328 323 9Spriuger-Verlag 1988