Multidimensional scaling of simplex shapes

摘要

Bookstein (Statist. Sci. 1 (1986) 181–242; Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991) has proposed a method for the representation of triangle shape as points in the Poincaré half plane – a space of constant negative curvature. Small (The Statistical Theory of Shape, Springer, New York, 1996) provided an extension of the Bookstein representation by representing the shapes of n-simplexes on manifolds. These manifolds are quite distinct from those proposed by D.G. Kendall based upon Procrustes arguments. In this paper, we examine the geometrical properties of these simplex shape spaces in greater detail. In particular, explicit formulas are given for the geodesic distance between any two points in these spaces. Such formulas permit the implementation of multidimensional scaling methods for the statistical shape analysis of two- and three-dimensional objects. In addition, the curvatures of the simplex shape spaces are examined. It is shown that the spaces of n-simplex shapes are not of constant curvature unless n=2.