A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling

Abstract

We describe a software package for computing and manipulating the subdivision of a sphere by a collection of (not necessarily great) circles and for computing the boundary surface of the union of spheres. We present problems that arise in the implementation of the software and the solutions that we have found for them. At the core of the paper is a novel perturbation scheme to overcome degeneracies and precision problems in computing spherical arrangements while using floating point arithmetic. The scheme is relatively simple, it balances between the efficiency of computation and the magnitude of the perturbation, and it performs well in practice. In one O(n) time pass through the data, it perturbs the inputs necessary to insure no potential degeneracies and then passes the perturbed inputs on to the geometric algorithm. We report and discuss experimental results. Our package is a major component in a larger package aimed to support geometric queries on molecular models; it is currently employed by chemists working in "rational drug design." The spherical subdivisions are used to construct a geometric model of a molecule where each sphere represents an atom. We also give an overview of the molecular modeling package and detail additional features and implementation issues.