The Voronoi diagram of half-balls and its application to the prediction of the 3D structure of proteins

The calculation of the overlap volume of half-spheres or ellipsoids is of direct interest in structural bioinformatics, which is concerned with the computational study of biological macromolecules on a genomic scale. We present an algorithm for computing the Delaunay graph and the overlap volume of a set of half-balls using exact predicates that detect the disjointness or non-disjointness of two half-balls and the validity of the generalized Voronoi vertex of four half-balls using geometric invariants and action (multiplication map) matrices. We prove the correctness of the algorithm and the optimality of the degree of the predicates by using geometric invariants and Grobner bases. The main application of these certified computations is to predict the 3D structure of proteins.