Six new mechanics corresponding to further shape theories

摘要

In this paper, suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: (i) Kendall's (metric) shape space with his shape statistics and Barbour's mechanics thereupon. (ii) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour-Bertotti mechanics. This paper's new theories include, using the invariant and group namings, (iii) Angle alias conformal shape mechanics. (iv) Area ratio alias affine shape mechanics. (v) Area alias affine scale-and-shape mechanics. (iii)-(v) rest respectively on angle space, area-ratio space, and area space configuration spaces. Probability and statistics applications are also pointed to in outline.