Balls and spheres are amongst the simplest 3 D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm. Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed. Experimental results are presented on hand-crafted models illustrating various difficulties, as well as on the 58,898 models found in the tenth of July 2009 release of the Protein Data Bank.

Intervor can be run from the web site http://cgal.inria.fr/abs/Intervor or upon downloading the binary file. Plugins are also made available for VMD and Pymol.

Abstract:We propose two ways to compute the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. We use the so-called space of circles as mathematical background for this work. We present a fully robust implementation built upon existing generic algorithms provided by the cgal library. The eciency of the implementation is established by benchmarks. Nous proposons deux façons de calculer la triangulation de Delaunay d'un ensemble de points qui appartiennent soit à la sphère, soit à son voisinage. Ces deux méthodes reposent sur l'algorithme incrémental classique, tel qu'il a été créé à l'origine pour calculer les triangulations de Delaunay planaires. Le cadre mathématique classique justiant cette approche est rappelé, à l'aide de l'espace des cercles. Ces deux approches ont été implantées de façon robuste en s'appuyant sur les algorithmes génériques fournis par la bibliothèque CGAL. Des tests comparatifs montrent l'ecacité de nos implantations sur des jeux de données de taille variée.