Dynamic shape

Abstract

Many useful notions of partial order and/or similarity and relatedness of different geometrical features of smooth shapes that occur in psychologically valid descriptions of shape have no equivalents in the usual geometrical shape theories. This is especially true where similarities are noted between objects of different connectivity: in almost all of the present theories the topological type generates the primary categorization. It is argued that such relations find a logical place only in shape theories that involve morphogenesis. Any object can be embedded uniquely in a morphogenetic sequence if one takes resolution as the parameter of the sequence. A theory of measurement is presented that allows one to define surfaces and (boundary-) curves on multiple levels of resolution. The embedding is essentially unique and is generated via a partial differential equation that governs the evolution. A canonical projection connects any high resolution specimen to lower resolution versions. The bifurcation set of the projection generates natural part boundaries. Singularities of the evolution are completely characterized as emergence, accretion and versification processes (involving topological change) and singularities by which inflections (inflection points for curves, parabolic curves for surfaces) are generated. The latter singularities involve a single process for the generation of inflections and three other processes by which the existing inflection structure may be changed. Relations with existing theories in vogue in robotics and AI, as well as in psychophysics are discussed.