An elegant algorithm of the analytical calculation for the volume of fused spheres with different radii Original Research Article

We propose a new algorithm for calculating the volume of fused spheres. The new algorithm is superior to the old methods in two aspects. One is the separation of the algorithm into topological and analytical parts. This makes the mathematical nature of the problem “what is the minimum topological information required to calculate the volume of fused spheres?” clear. The number of conditional branches is drastically reduced. The remaining conditional branches are those which are topologically essential. For practical use, only the list of the spheres forming isolated spheres, isolated sphere circles, exposed vertices, and buried vertices are required. The other aspect which makes the new algorithm superior to the old ones is the analytical expression which is independent of the choice of the coordinate frame. All equations are written only in terms of radii of spheres and distances between spheres. This enables us to calculate derivatives of the excluded volume with respect to a scaling parameter which is used in the scaled particle theory of liquids.