Advances in Applied Probability

Planar tessellations that have the half-Gilbert structure

James Burridge and Richard Cowan

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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.

Article information

Adv. in Appl. Probab., Volume 48, Number 2 (2016), 574-584.

First available in Project Euclid: 9 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05B45: Tessellation and tiling problems [See also 52C20, 52C22]
Secondary: 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

Random tessellation point process crack formation division of space


Burridge, James; Cowan, Richard. Planar tessellations that have the half-Gilbert structure. Adv. in Appl. Probab. 48 (2016), no. 2, 574--584.

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