Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces

Abstract

The shape-space ∑ m k whose points σ represent the shapes of not totally degenerate k-ads in Rm is introduced as a quotient space carrying the quotient metric. When m = 1, we find that ∑ 1 k = S K − 2 when m ⩾ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space ∑ 2 k can be identified with a version of CPk−2. Of special importance are the shape-measures induced on CPk−2 by any assigned diffuse law of distribution for the k vertices. We determine several such shape-measures, we resolve some of the technical problems associated with the graphic presentation and statistical analysis of empirical shape distributions, and among applications we discuss the relevance of these ideas to testing for the presence of non-accidental multiple alignments in collections of (i) neolithic stone monuments and (ii) quasars. Finally the recently introduced Ambartzumian density is examined from the present point of view, its norming constant is found, and its connexion with random Crofton polygons is established.