Statistical Shape Analysis, 2nd Edition

2 Size measures and shape coordinates

In this chapter we provide some brief historical background to traditional and geometrical methods of shape analysis. We then define what is meant by a configuration of landmarks and the configuration space, discuss measures of the size, and then consider some simple coordinate systems for representing shapes.

2.1 History

Shape analysis has a long history, especially in biology. In order to analyse size or shape, a biologist traditionally selects ratios of distances between landmarks or angles, and then carries out multivariate analysis (e.g. Rao 1948). This approach has been called ‘multivariate morphometrics’ in biology and a review is given by Reyment et al. (1984, p. 120). Early researchers using such methods include Pearson (1926), who studied a measure of similarity between skulls based on many lengths between landmarks. In fact, various studies have been cited in the journal Biometrika, starting from studies in volume 1 itself (Fawcett and Lee 1902). The data were usually distances between landmarks (such as lengths and widths) or angles between landmarks, rather than landmarks themselves. The definitions of landmarks on skulls follow the Frankfurt Concordat (Frankfurter Verständigung) of the 13th Congress of the German Anthropological Society which met in Frankfurt-am-Main, 14–17 August 1882 (e.g. see Trevor 1950).

Another large area of work in biology has been in the study of allometry, that is differences in shape associated ...