On extending symmetry sets for 2D shapes

Abstract

Many attempts have been made to represent families of 2D shapes in a simpler way. These approaches lead to so-called structures as the Symmetry Set (SS) and a subset of it, the Medial Axes (MA). While the latter is commonly used, the former is still in the mathematical research stage. One reason for this is that in contrast to the SS, the MA can be computed efficiently and fast, and yields one connected component for a closed shape. In this paper a natural complement of the symmetry set, called the AntiSymmetry Set (ASS), is used to connect components bearing the full richness of the symmetry set. Secondly, new ways are presented to visualize these sets. One uses the radius of the describing circle as extra dimension, the other, the so-called pro-Symmetry Set (pre-55), uses the parameter space. Example shapes show the extra information carried in the ASS and the pro-SS in determining the special points on the SS as well as revealing the structure of the SS in more detail. They are also capable of distinguishing between different shapes where the SS and the MA in some cases fail. © Springer-Verlag 2004.