Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces

TLDR

The shape‑space ∑_m^k, representing non‑degenerate k‑ads in ℝ^m, is introduced as a quotient space equipped with the quotient metric. This study focuses on the case m=2, identifying the shape‑space ∑_2^k with a version of the complex projective space CP^{k-2}. The authors define shape‑measures on CP^{k-2} induced by diffuse vertex‑distribution laws, resolve technical issues in graphical and statistical analysis of empirical shape distributions, and apply these tools to test for non‑accidental multiple alignments in neolithic stone monuments and quasars. They show that for m=1 the shape‑space equals S^{k-2} and contains singularities when m≥3, determine several shape‑measures, establish the norming constant of the Ambartzumian density and its link to random Crofton polygons, and demonstrate the applicability of these measures to detecting alignment patterns.

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.