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Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces
Author(s) -
Kendall David G.
Publication year - 1984
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/16.2.81
Subject(s) - mathematics , quotient , space (punctuation) , metric (unit) , point (geometry) , distribution (mathematics) , gravitational singularity , similarity (geometry) , degeneracy (biology) , pure mathematics , combinatorics , geometry , mathematical analysis , image (mathematics) , artificial intelligence , computer science , bioinformatics , operations management , economics , biology , operating system
The shape‐space∑ m kwhose points σ represent the shapes of not totally degenerate k ‐ads in R m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that∑ 1 k = S K − 2when m ⩾ 3, the shape‐space contains singularities. This paper deals mainly with the case m = 2, when the shape‐space∑ 2 kcan be identified with a version of CP k−2 . Of special importance are the shape‐measures induced on CP 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.