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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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References

Gilbert, E. N. (1967) Random plane networks and needle-shaped crystals. In Applications of Undergraduate Mathematics in Engineering. ed. Noble, B. Macmillan, New York.Google Scholar
Mceldowney, P. H. (1996) Subsistence intensification in the recent prehistory of Manus (Papua-New Guinea). PhD thesis. Australian National University, Canberra.Google Scholar
Mecke, J. (1981) Formulas for stationary planar fibre processes III - Intersections with fibre systems. Math. Operat. Statist. 12, 201–210.Google Scholar
Mecke, J. (1984) Parametric representation of mean values for stationary random mosaics. Math. Operat. Statist. 15, 437–442.Google Scholar
Miles, R. E. (1970) On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.CrossRefGoogle Scholar
Miles, R. E. (1974) A synopsis of ‘Poisson flats in Euclidean spaces’. In Stochastic Geometry. ed. Harding, E. F. and Kendall, D. G. Wiley, Chichester. pp 202–227.Google Scholar
Miles, R. E. (1986) Random tessellations. In Encyclopaedia of Statistical Sciences. Vol. 7. ed. Kotz, S. and Johnson, N. L. Wiley, New York. pp. 567–572.Google Scholar
Serra, J. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Watson, D. F. (1982) ACORD: Automatic contouring of raw data. Comput. Geosci. 8, 97–101.Google Scholar