Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces

The shape-space whose points represent the shapes of not totally degenerate k-ads in R^mis introduced as a quotient space carrying the quotient metric.When m = 1, we find that when m 3, the shape-space contains singularities. This paper dealsmainly with the case m = 2, when the shape-space can be identified with a version of CP^k–2.Of special importance are the shape-measures induced on CP^k–2by any assigned diffuse law of distribution for the k vertices.We determine several such shape-measures, we resolve some ofthe technical problems associated with the graphic presentationand statistical analysis of empirical shape distributions, andamong applications we discuss the relevance of these ideas totesting for the presence of non-accidental multiple alignmentsin collections of (i) neolithic stone monuments and (ii) quasars.Finally the recently introduced Ambartzumian density is examinedfrom the present point of view, its norming constant is found,and its connexion with random Crofton polygons is established.