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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 approximationsof the moments of the distributions of more or less all geometricalcharacteristics of the grain. Furthermore, many proposals on how to get closeparametric approximations to the real distributions were put forward by severalauthors. In this paper we show that exploiting the scaling property of theunderlying Poisson process, we are able to derive the distribution of the maingeometrical features of the grain for every value of the intensity parameter.Moreover, we use a sophisticated simulation program to construct a close MonteCarlo based approximation for the distributions of interest. Using this, wealso determine the closest approximating distributions within the mentionedfrequently used parametric classes of distributions and conclude that theseapproximations can be quite accurate.

Accurate approximation of the distributions of the 3D Poisson-Voronoi typical cell geometrical features