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Homogeneous rectangular tessellations

Part of: Geometric probability and stochastic geometry Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Margaret S. Mackisack  and
Roger E. Miles
Margaret S. Mackisack*
Affiliation:
University of Queensland
Roger E. Miles*
Affiliation:
The Australian National University
*
∗ Postal address: Department of Mathematics, University of Queensland, Brisbane, Qld 4072, Australia.
∗∗ Visiting Fellow, Centre for Mathematics and its Applications, Australian National University, Canberra. Postal address: RMB 345, Queanbeyan, NSW 2620, Australia.
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Abstract

A rectangular tessellation is a covering of the plane by non-overlapping rectangles. A basic theory for general homogeneous random rectangular tessellations is developed, and it is shown that many first-order mean values may be expressed in terms of just three basic quantities. Corresponding values for independent superpositions of two or more such tessellations are derived. The most interesting homogeneous rectangular tessellations are those with only T-vertices (i.e. no X-vertices). Gilbert's (1967) isotropic model adapted to this two-orthogonal-orientations case, although simply specified, appears theoretically intractable, due to a complex ‘blocking' effect. However, the approximating penetration model, also introduced by Gilbert, is found to be both tractable and informative about the true model. A multi-stage method for simulating the model is developed, and the distributions of important characteristics estimated.

Keywords

PLANAR HOMOGENEITYRANDOM RECTANGULAR TESSELLATIONST- AND X-VERTICES: I-SEGMENTSINDEPENDENT SUPERPOSITIONSCHRONOLOGYGILBERT'S MODEL AND PENETRATION APPROXIMATIONSIMULATION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 65C20: Models, numerical methods
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 993 - 1013
Copyright
Copyright © Applied Probability Trust 1996 

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