Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-11T12:11:09.209Z Has data issue: false hasContentIssue false
  • English
  • Français

A Markov chain of triangle shapes

Published online by Cambridge University Press:  01 July 2016

David Mannion
David Mannion*
Affiliation:
Royal Holloway and Bedford New College, London
*
∗Postal address: Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 OEX, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The process of choosing a random triangle inside a compact convex region, K, may be iterated when K itself is a triangle. In this way successive generations of random triangles are created. Properties of scale, location and orientation are filtered out, leaving only the shapes of the triangles as the objects of study. Various simulation investigations indicate quite clearly that, as n increases, the nth-generation triangle shape converges to collinearity. In this paper we attempt to establish such convergence; our results fall slightly short of a complete proof.

Keywords

RANDOM TRIANGLESCOLLINEAR TRIANGLESRANDOM SHAPESSHAPE SPACESHAPE DISTRIBUTIONS: PRODUCTS OF RANDOM MATRICESLYAPUNOV EXPONENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 348 - 370
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baum, L. E. and Katz, M. (1965) Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123.Google Scholar
[2] Kendall, D. G. (1984) Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[3] Kendall, D. G. (1985) Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Prob. 17, 308–329.Google Scholar
[4] Kendall, D. G. and Le, H.-L. (1986) Exact shape-densities for random triangles in convex polygons. Adv. Appl. Prob. 18, 59–72.Google Scholar
[5] Kendall, D. G. and Le, H.-L. (1987) The structure and explicit determination of convex-polygonally generated shape-densities. Adv. Appl. Prob. 19, 896–916.Google Scholar
[6] Le, H.-L. (1987) Explicit formulae for polygonally generated shape-densities in the basic tile. Math. Proc. Camb. Phil. Soc. 101, 313–321.Google Scholar
[7] Le, H.-L. (1987) Singularities of convex-polygonally generated shape-densities. Math. Proc. Camb. Phil. Soc. To appear.Google Scholar
[8] Le, H.-L. (1987) Convex-polygonally generated shape-densities. Essay submitted for J. T. Knight prize.Google Scholar
[9] Watson, G. S. (1983) Random triangles. Proc. 2nd Internat. Workshop on Stochastic Geometry, ed. Jensen, E. B. and Gundersen, H. J. G.. Memoirs No. 6, Dept of Theoretical Statistics, University of Aarhus.Google Scholar
[10] Watson, G. S. (1986) The shapes of a random sequence of triangles. Adv. Appl. Prob. 18, 156–169.Google Scholar