Title: Accurate approximation of the distributions of the 3D Poisson-Voronoi typical cell geometrical features,
Abstract: Although Poisson-Voronoi diagrams have interesting mathematical properties,
there is still much to discover about the geometrical properties of its grains.
Through simulations, many authors were able to obtain numerical approximations
of the moments of the distributions of more or less all geometrical
characteristics of the grain. Furthermore, many proposals on how to get close
parametric approximations to the real distributions were put forward by several
authors. In this paper we show that exploiting the scaling property of the
underlying Poisson process, we are able to derive the distribution of the main
geometrical features of the grain for every value of the intensity parameter.
Moreover, we use a sophisticated simulation program to construct a close Monte
Carlo based approximation for the distributions of interest. Using this, we
also determine the closest approximating distributions within the mentioned
frequently used parametric classes of distributions and conclude that these
approximations can be quite accurate.