Skip to main content Accessibility help

Anisotropic Growth of Voronoi Cells

  • Thomas H. Scheike (a1)


This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.


Corresponding author

* Present address: Schlegels Alle 9, 1807 Frederiksberg, Denmark.


Hide All
Cowan, R. (1978) The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 4757.
Cowan, R. (1980) Properties of ergodic random mosaic processes. Math. Nachr. 97, 89102.
Dirichlet, G. L. (1850) über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math. 40, 209227.
Gilbert, E. N. (1962) Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958972.
Green, P. J. and Sibson, R. (1978) Computing Dirichlet tessellations in the plane. Computer J. 21, 168173.
Hinde, A. L. and Miles, R. E. (1980) Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul. 10, 205223.
Mecke, J. (1980) Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V., pp. 124132. Armenian Academy of Sciences Publishing House, Erevan.
Meijering, J. L. (1953) Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270290.
Miles, R. E. (1969) Poisson flats in Euclidean spaces, part I: a finite number of random uniform flats. Adv. Appl. Prob. 1, 211237.
Miles, R. E. (1970) On the homogeneous planar Poisson point process. Math. Biosci. 6, 85127.
Miles, R. E. (1972) The random division of space. Suppl. Adv. Appl. Prob. 4, 243266.
Miles, R. E. (1974) A synopsis of Poisson flats in Euclidean spaces. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., pp. 202227, Wiley, New York.
Møller, J. (1989) Random tessellations in d. Adv. Appl. Prob. 21, 3773.
Okabe, A., Boots, B. and Sugihara, K. (1992) Spatial Tessellations Concepts and Applications of Voronoi Diagrams. Wiley, New York.
Radecke, W. (1980) Some mean value relations on stationary random mosaics in the space. Math. Nachr. 97, 203210.
Rogers, C. A. (1964) Packing and Covering. Cambridge University Press.
Spiegel, M. R. (1968) Mathematical Handbook of Formulas and Tables. McGraw-Hill, New York.
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, New York.
Thiessen, A. H. (1911) Precipitation averages for large areas. Monthly Weather Rev. 39, 10821084.
Voronoi, G. (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques, deuxième memoire, recherches sur les parallelloèdres primitifs. J. Reine Angew. Math. 134, 287–198.


MSC classification

Anisotropic Growth of Voronoi Cells

  • Thomas H. Scheike (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed