Shape statistics: Procrustes superimpositions and tangent spaces

The shape of a set of labeled points corresponds to those attributes of the configuration that are invariant to the effects of translation, rotation, and scale. Procrustes distance may be used to compare different shapes and also serve as a metric that may be used to define multidimensional shape sp aces. This paper demonstrates that the preshape space of planar triangles P rocrustes aligned to a reference triangle corresponds to a unit hemisphere. An overview of methods used as linear approximations of D. G. Kendall's no n-Euclidean shape space is given, and the equivalence of several methods ba sed on orthogonal projections is shown. Some problems with approximations b ased on stereographic projections are also discussed. A simple example usin g artificial data is included.