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Planar tessellations that have the half-Gilbert structure

Part of: Designs and configurations Real and complex geometry Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  10 June 2016

James Burridge  and
Richard Cowan
James Burridge*
Affiliation:
University of Portsmouth
Richard Cowan*
Affiliation:
University of Sydney
*
* Postal address: Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth PO1 3HF, UK. Email address: james.burridge@port.ac.uk
** Postal address: School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia.
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Abstract

In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

Keywords

Random tessellationpoint processcrack formationdivision of space

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05B45: Tessellation and tiling problems
Secondary: 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 574 - 584
Copyright
Copyright © Applied Probability Trust 2016 

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References

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