Advances in Applied Probability

Full- and half-Gilbert tessellations with rectangular cells

James Burridge, Richard Cowan, and Isaac Ma

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In the full rectangular version, lines extend either horizontally (east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east- and south-growing rays do not interact with west and north rays. For the half rectangular tessellation, we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst, for the full rectangular version, we develop an accurate simulation technique, based in part on the stopping-set theory for Poisson processes (see Zuyev (1999)), to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. In this paper we explore this coincidence, mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moments of the ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model—as much better approximations than those provided by Mackisack and Miles (1996).

Article information

Adv. in Appl. Probab., Volume 45, Number 1 (2013), 1-19.

First available in Project Euclid: 15 March 2013

Permanent link to this document

Digital Object Identifier

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: 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

Random tessellation point process crack formation fragmentation division of space


Burridge, James; Cowan, Richard; Ma, Isaac. Full- and half-Gilbert tessellations with rectangular cells. Adv. in Appl. Probab. 45 (2013), no. 1, 1--19. doi:10.1239/aap/1363354100.

Export citation


  • Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. Dover Publications, New York.
  • Burridge, J. (2010). Simulation of the rectangular Gilbert tessellation. Unpublished Tech. Rep.
  • Cowan, R. and Ma, I. S. W. (2002). Solving a simplified version of the Gilbert tessellation. Available at
  • Cowan, R., Quine, M. and Zuyev, S. (2003). Decomposition of gamma–distributed domains constructed from Poisson point processes. Adv. Appl. Prob. 35, 56–69.
  • Gilbert, E. N. (1967). Surface films of needle-shaped crystals. In Applications of Undergraduate Mathematics in Engineering, ed. B. Noble, Macmillan, pp. 329–346.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Academic Press, Amsterdam.
  • Mackisack, M. S. and Miles, R. E. (1996). Homogeneous rectangular tessellations. Adv. Appl. Prob. 28, 993–1013.
  • Mathews, J. and Walker, R. L. (1973). Mathematical Methods of Physics. Addison-Wesley.
  • Schreiber, T. and Soja, N. (2011). Limit theory for planar Gilbert tessellations. Prob. Math. Statist. 31, 149–160.
  • Zuyev, S. (1999). Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.