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On the induced distribution of the shape of the projection of a randomly rotated configuration

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

H. Le  and
D. Barden
H. Le*
Affiliation:
University of Nottingham
D. Barden*
Affiliation:
University of Cambridge
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: huiling.le@nottingham.ac.uk
∗∗ Postal address: DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWB, UK.
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Abstract

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Using the geometry of the Kendall shape space, in this paper we study the shape, as well as the size-and-shape, of the projection of a configuration after it has been rotated and, when the given configuration lies in a Euclidean space of an arbitrary dimension, we obtain expressions for the induced distributions of such shapes when the rotation is uniformly distributed.

Keywords

Kendall shape spaceshapesize-and-shapehorizontal liftRiemannian submersion

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 331 - 346
Copyright
Copyright © Applied Probability Trust 2010 

References

Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.CrossRefGoogle Scholar
Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.CrossRefGoogle Scholar
Kendall, W. S. and Le, H. (2009). Statistical shape theory. In New Perspectives in Stochastic Geometry, eds Kendall, W. S. and Molchanov, I., Oxford University Press, pp. 384–373.CrossRefGoogle Scholar
O'Neill, B. (1983). Semi-Riemannian Geometry. Academic Press, New York.Google Scholar
Panaretos, V. M. (2006). The diffusion of Radon shape. Adv. Appl. Prob. 38, 320–335.CrossRefGoogle Scholar
Panaretos, V. M. (2008). Representation of Radon shape diffusions via hyperspherical Brownian motion. Math. Proc. Camb. Phil. Soc. 145, 457–470.CrossRefGoogle Scholar
Panaretos, V. M. (2009). On random tomography with unobservable projection angles. Ann. Statist. 37, 3272–3306.CrossRefGoogle Scholar