How to describe pure form and how to measure differences in shapes using shape numbers

Abstract The shape number of a curve is derived for two-dimensional non-intersecting closed curves that are the boundary of simply connected regions. This description is independent of their size, orientation and position, but it depends on their shape. Each curve carries “within it” its own shape number. The order of the shape number indicates the precision with which that number describes the shape of the curve. For a curve, the order of its shape number is the length of the perimeter of a ‘discrete shape’ (a closed curve formed by vertical and horizontal segments, all of equal length) closely corresponding to the curve. A procedure is given that deduces, without table look-up, string matching or correlations, the shape number of any order for an arbitrary curve. To find out how close in shape two curves are, the degree of similarity between them is introduced; dissimilar regions will have a low degree of similarity, while analogous shapes will have a high degree of similarity. Informally speaking, the degree of similarity between the shapes of two curves tells how deep it is necessary to descend into a list of shapes, before being able to differentiate between the shape of those two curves. Again, a procedure is given to compute it, without need for such list or grammatical parsing or least square curve or area fitting. The degree of similarity maps the universe of curves into a tree or hierarchy of shapes. The distance between the shapes of any two curves, defined as the inverse of their degree of similarity, is found to be an ultradistance over this tree. The shape number is a description that changes with skewing, anisotropic dilation and mirror images, as the intuitive psychological concept of “shape” demands. Nevertheless, at the end of the paper a related Theory “B” of shapes is introduced that allows anisotropic changes of scale, thus permitting for instance a rectangle and a square to have the same B shape. These definitions and procedures may facilitate a quantitative study of shape.