Local Symmetry of Plane Curves

1. How can we perceive, recognize, describe or represent the shape of an object? These questions are of great interest to workers in theoretical biology and in the perception of patterns -whether by people or by computers. Organisms have simple shapes which are not usually equilateral triangles, squares or other basic mathematical shapes, so new concepts are needed to capture their simplicity. Organisms move and grow: what is preserved as their shapes change in these ways? Shapes that we all immediately recognize as the same (such as all the instances of the letter m in this journal) are in fact subtly different: what do they have in common? In order to tackle questions such as these, a number of ways have been suggested for reducing the huge amount of information carried by a shape down to a "skeleton" of crucial, possibly discrete, information which can be more readily assimilated. For example, H. Blum [1] suggested that a planar shape could be studied by fitting circular disks inside it and noting the locus of their centres, which he called the "sym-ax" or symmetric axis transform (Fig. 1, left). A wiggling worm can be described, perhaps, by keeping the radii fixed and flexing the sym-ax; a growing worm by changing the radii but leaving the sym-ax unaltered.