Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T10:14:13.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Random spherical triangles I: Geometrical background

Published online by Cambridge University Press:  01 July 2016

Huiling Le
Huiling Le*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we identify the shape space Σ(S2, k) for k labelled points on the sphere S2 that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points on S2 and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on Σ(S2, 3). In a companion paper (pp. 581–594) we investigate the statistical properties of such shapes when the k points are generated by a random mechanism.

Keywords

COLLINEARITYCURVATUREGEODESIC DISTANCEPROCRUSTEAN METRICQUASARSRIEMANNIAN METRIC
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 570 - 580
Copyright
Copyright © Applied Probability Trust 1989 

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] Carne, T. K. (1988) The shape space for points on the sphere. Submitted to Proc. London Math. Soc. Google Scholar
[2] Eisenhart, L. P. (1926) Riemannian Geometry. Princeton University Press.Google Scholar
[3] Gallot, S. Hulin, D. and Lafontaine, J. (1987) Riemannian Geometry. Springer-Verlag, Berlin.Google Scholar
[4] Kendall, D. G. (1984) Shape-manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[5] Kendall, D. G. (1988) A survey of the statistical theory of shape. Statistical Science. To appear.CrossRefGoogle Scholar
[6] Kendall, D. G. (1988) Spherical triangles revisited. In preparation.Google Scholar
[7] Le, H. (1988) Shape Theory in Flat and Curved Spaces, and Shape Densities with Uniform Generators. Ph.D. dissertation, Statistical Laboratory, University of Cambridge.Google Scholar
[8] Schwerdtfeger, H. (1962) Geometry of Complex Numbers. University of Toronto Press.CrossRefGoogle Scholar
[9] Sibson, R. (1978) Studies in the robustness of multidimensional scaling: procrustes statistics. JR. Statist. Soc. B 40, 234–238.Google Scholar
[10] Yano, K. and Kon, M. (1984) Structures on Manifolds. World Scientific Publishing, Calcutta.Google Scholar