The Weighted Volume Derivative of a Space Filling Diagram

Figure 1: Computing geometric properties of a molecule. As common in biology, atoms are treated as intersecting balls, whose union forms the space-filling diagram. Richards and others proposed to study this union using the Voronoi diagram (or more exactly the power diagram since the balls have different radii). That diagram divides the space into convex cells, one per atom. In the two-dimensional example shown here, the edges that separate these cells are shown as solid inside and dotted outside the union of disks. Note that the power cells of some surface atoms extend to infinity. The superimposed Delaunay triangulation (thick solid and dashed lines) is the dual of the power diagram obtained by drawing a line segment between two ball centers if their convex cells share a common edge. Despite their different appearance, the Delaunay triangulation and the power diagram contain exactly the same information. The key to connect the power diagram to the molecule is to consider the intersection of the two: this defines a convex decomposition of the space-filling diagram (i.e the light shaded area inside the disks, divided into convex regions by the solid part of the Voronoi edges). The dual of this decomposition is the dual complex (thick solid lines and shaded triangles). The dual complex is a subset of the Delaunay triangulation, and contains all simplices (tetrahedra, triangles and edges) that correspond to overlapping atoms. The dual complex contains all information about a molecule required to compute its surface and volume. In this paper, we show that the same dual complex can be used to compute the derivative of the volume. THE WEIGHTED VOLUME DERIVATIVE