ADJOINT JOIN VOLUMES

A simple geometrical identity, called the adjoint join formula, is int roduced. It allows one to simplify the computation of the volumes of s ome unions of simple solid objects such as spheres and polyhedra. It i nvolves cones and a generalization of a cone, called a join. In order to apply the adjoint join formula it is necessary to first compute the surface of the object. The volume of an object is equal to a cone of the object's surface over some point. This cone is the sum of the cone s of each face of the surface over the point. The computation of the v olume of each of these cones can sometimes be simplified by applying t he adjoint join formula. The adjoint join formula states that if two g eometrical objects in space have dimensions that sum to three, then th e join of the boundary of the first object with the second object is e qual to the join of the first object with the boundary of the second o bject (up to sign). There are occasions when the volume of the first j oin is difficult to compute, but the volume of the second join is easy to compute, so applying the adjoint join formula simplifies the volum e computation. The method is applied to the union of a group of sphere s. This provides a simple way to compute the volume of a molecule anal ytically, provided that one can compute its van der Waals surface anal ytically. This is not the first analytical and exact method to compute the volume of a hard-sphere representation of a molecule, but it is c onceptually the simplest.