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A diffusion model for Bookstein triangle shape

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Warwick
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Extract

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3} is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi; determines a triple {X1 X2, X3} whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 334 - 335
Copyright
Copyright © Applied Probability Trust 1996 

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References

Dynkin, Ε. B., (1961) Non-negative eigenfunctions of the Laplace-Beltrami operator and Brownian motion in certain symmetric spaces. Dokl. Akad. Nauk SSS 141, 288–291.Google Scholar
Dyson, F. J. (1962) A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198.CrossRefGoogle Scholar
Kendall, D. G. (1977) The diffusion of shape. Adv. Appl. Prob. 9, 428–430.CrossRefGoogle Scholar
Norris, J. R., Rogers, L. C. G. and Williams, D. (1986) Brownian motion of ellipsoids. Trans. Amer. Math. Soc. 294, 757–765.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales II: Ito Calculus. Wiley, Chichester.Google Scholar