A weighted tessellation of Voronoi with Poisson fields of centroids

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Tome 153 (1986), pp. 160-172

Résumé

A weighted tessellation of Voronoi generated by a system of $n$ Poisson fields of centroids is considered. A composition and boundary fields of the structure are investigated. The intensity of the boundary field between grains of types $i$ and $j$ $(1\leqslant i\leqslant j\leqslant n)$ is proved to be $$ q_{ij}=36^{1/3}\pi^{1/3}\Gamma\left(\frac23\right)p_ip_j\frac{(\alpha_i+\alpha_j)^3-|\alpha_i-\alpha_j|^3}{9\alpha_i\alpha_jc} $$ where $p_i$ is a volume part, $\alpha_i>0$ is a weight (for comparison of distances) of the $i$-th component, $c$ is a scale. Formulae for boundary field intensities in flat and line sections are obtained $q_{ij}^{(2)}=\frac\pi4q_{ij}$, $q_{ij}^{(1)}=\frac12q_{ij}$. Estimations for parameters $p_i$ and $\alpha_i$ dependent on line observations are proposed.