Affine invariant medial axis and skew symmetry

Affine invariant medial axes and symmetry sets of planar shapes are introduced and studied in this paper. Two different approaches are presented. The first one is based on affine invariant distances, and defines the symmetry set, a set containing the medial axis; as the closure of the locus of points on (at least) two affine normals an affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) three-point contact with two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set (medial axis) are equivalent, this is not the case for the affine group. We then show how to use the symmetry set to detect affine skew symmetry, proving that the contact based symmetry set is a straight line if and only if the given shape is the affine transformation of a symmetric object.