Symmetry sets

Abstract

SynopsisFor a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).