Landmark Coordinates Aligned by Procrustes Analysis Do Not Lie in Kendall's Shape Space

Abstract

The field of geometric morphometrics is concerned with methods for studying shapes of objects. These methods are increasingly used to address a broad range of ecological and evolutionary questions. A few recent examples include the study of ontogenetic reaction norms (Arnqvist and Johansson, 1998), fluctuating asymmetry (Klingenberg and McIntyre, 1998), and trophic morphology (Caldecutt and Adams, 1998). There has also been recent debate within these pages as to the appropriate place for geometric morpho-metrics in the context of phylogenetic analysis (Adams and Rosenberg, 1998; Rohlf, 1998a; Swiderski et al., 1998; Zelditch and Fink, 1998; Zeldtich et al., 1998). Much of our theoretical understanding of this powerful and relatively new approach to shape analysis is due to work by Kendall on the geometric and statistical properties of shape spaces defined by the Procrustes metric (Kendall, 1984, 1985). It is important that users of these methods appreciate the relationship between the theoretical aspects of geometric morphometrics and their practical application. This report demonstrates that the geometry of sample variation resulting from the most commonly used geometric method, Procrustes analysis (also known as least squares superimposition), is not the same as the geometry of the shape space described by Kendall. The coordinates of landmarks after the Procrustes superimposition (with unit scaling) of a sample onto a reference lie on the surface of a (hyper)hemisphere of unit radius that, at that reference, approximates the relationships between configurations in Kendall’s shape space.