Shape manifolds, Procrustean metrics, and complex projective spaces. (English) Zbl 0579.62100

Kendall, David G.

Bull. Lond. Math. Soc. 16, 81-121 (1984).

Summary: The shape-space \(\Sigma^ k_ m\) whose points \(\sigma\) represent the shapes of not totally degenerate k-ads in \({\mathbb{R}}^ m\) is introduced as a quotient space carrying the quotient metric. When \(m=1\), we find that \(\Sigma^ k_ 1=S^{k-2}:\) when \(m\geq 3\), the shape-space contains singularities. This paper deals mainly with the case \(m=2\), when the shape-space \(\Sigma^ k_ 2\) can be identified with a version of \({\mathbb{C}}P^{k-2}.\)
Of special importance are the shape-measures induced on \({\mathbb{C}}P^{k- 2}\) 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.

MSC:

62P99	Applications of statistics
51M99	Real and complex geometry
60D05	Geometric probability and stochastic geometry
51M10	Hyperbolic and elliptic geometries (general) and generalizations
57N25	Shapes (aspects of topological manifolds)

Keywords:

Procrustean metrics; complex projective spaces; shape-manifolds; shape- densities; non-isotropic Gaussian generator; archaeology; shape-space; quotient space; quotient metric; shape-measures; empirical shape distributions; neolithic stone monuments; quasars; Ambartzumian density; random Crofton polygons