Application of Procrustes Distance to Shape Analysis of Delaunay Simplexes

The concept of Procrustes distance is applied to the shape analysis of the Delaunay simplexes. Procrustes distance provides a measure of coincidence of two point sets {x_i} and {y_i}, i=1..N. For this purpose the variance of point deviations is calculated at the optimal superposition of the sets. It allows to characterize the shape proximity of a given simplex to shape of a reference one, e.g. to the shape of the regular tetrahedron. This approach differs from the method used in physics, where the variations of edge lengths are calculated in order to characterize the simplex shape. We compare both methods on an example of structure analysis of dense packings of hard spheres. The method of Procrustes distance reproduces known structural results; however, it allows to distinguish more details because it deals with simplex vertices, which define the simplex uniquely, in contrast to simplex edges.