A Large Class of Random Tessellations with the Classic Poisson Polygon Distributions

Roger E. Miles¹* and Margaret S. Mackisack²

¹1476 Sutton Road, Sutton, NSW 2620, Australia
²Department of Mathematical Sciences, University of Technology Sydney, P.O. Box 123 Broadway, NSW 2007, Australia
*E-mail address: fgh@netspeed.com.au

(Received March 18, 2002; Accepted June 17, 2002)

Keywords: Random Tessellation of Convex Polygons, Poisson Anisotropic Lines in the Plane, Ergodic Polygon Distributions, Rectangular Tessellation, Superposition and Nesting

Abstract. One special case of Arak, Clifford and Surgailis' 1993 point-based polygon models for random graphs yields an isotropic random tessellation of convex polygons, with all vertices T-vertices. It is shown that its polygon distributions coincide with those of the random tessellation determined by Poisson isotropic random lines in the plane, for which all vertices are X-vertices (cf. Fig. 4). This surprising property extends to general orientation distributions, e.g. to rectangular tessellations stemming from a two atom distribution. Applying this property, it is shown that a wide variety of distinct random tessellations obtained from these two by superposition, nesting, etc. possess those very same polygon distributions.