Deformations, isosymmetric manifolds, and higher dimensional form space symmetries for point ensembles (polygonal forms) under O(2) symmetry III. Motifs in geometric figures and constraints in form spaces

Abstract

A polygonal form F specified by a set of N points in E n (here: a plane) can be represented by a point in an n · N -dimensional form space F . A motif M within such a form has been defined as a subset M ⊆ F on which certain constraints are imposed. Different types of constraints are discussed. In any case, every independent constraint reduces the dimension of the form space by one so that the resulting forms correspond to points in a constrained form space F c ⊂ F . The subspaces F c are investigated for some forms containing rigid and—as a novel aspect— deformable motifs, and they are correlated with the full form spaces F described in previous papers [1–3]. Furthermore, constraints which do not lead to motifs are treated. Generally, constraints of distances define circles in certain planes of F so that angular coordinates have to be used. As a special case, cyclic polygons have been found to lie on hyperspheres in F . On the other hand, if the distances between points are constrained to such an extent that only two degrees of freedom (the global rotation and one degree of shape variation) are left, then the resulting forms have been found to lie on hyperellipsoids in F . These constitute special cases of isodiastemic manifolds consisting of forms with fixed point distances.