Inclusion-exclusion formulas from independent complexes (Q866971)

The volume of a union of spheres has many practical applications. Unfortunately, the general inclusion-exclusion principle becomes exponentially difficult to use as the number of spheres increases. This can be overcome if certain terms in the inclusion-exclusion principle are ignored. Methods of determining which terms to ignore, via Delauney triangulations and dual complexes have been derived in the past. This article generalises some such methods. The result is a method of deriving an abstract simplicial complex from a collection of spheres, and a way to use this complex to determine a formula for the volume of the union of this collection of spheres. Careful proofs are given of the validity of the methods, and some suggestions are given on how the results could be extended to unions of other collections of other shapes (ellipsoids, for example).