On affine evolutoids

Abstract

The envelope of straight lines affine normal to a plane curve C is its affine evolute; the envelope of the affine lines tangent to C is the original curve, together with the entire affine tangent line at each inflexion of C. In this paper, we consider plane curves without inflexions. We use some techniques of singularity theory, such as unfoldings, discriminants and functions on discriminants, to explain how the first envelope turns into the second, as the (constant) slope between the set of lines forming the envelope and the set of affine tangents to C changes from 0 to 1. In particular, we guarantee the existence of the first slope for which singularities occur. Moreover, we explain how these singularities evolve on discriminant surface.