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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

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