Journal of Mathematical Chemistry

, Volume 15, Issue 1, pp 339–352

• Michael L. Connolly
Article

Abstract

A simple geometrical identity, called the adjoint join formula, is introduced. It allows one to simplify the computation of the volumes of some unions of simple solid objects such as spheres and polyhedra. It involves 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 cones of each face of the surface over the point. The computation of the volume of each of these cones can sometimes be simplified by applying the adjoint join formula. The adjoint join formula states that if two geometrical objects in space have dimensions that sum to three, then the join of the boundary of the first object with the second object is equal to the join of the first object with the boundary of the second object (up to sign). There are occasions when the volume of the first join is difficult to compute, but the volume of the second join is easy to compute, so applying the adjoint join formula simplifies the volume computation. The method is applied to the union of a group of spheres. This provides a simple way to compute the volume of a molecule analytically, provided that one can compute its van der Waals surface analytically. This is not the first analytical and exact method to compute the volume of a hard-sphere representation of a molecule, but it is conceptually the simplest.

Keywords

Physical Chemistry Exact Method Geometrical Object Volume Computation Solid Object
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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