Singularities of helix surfaces in Euclidean 3-space

In this paper we study helix surfaces whose unit normals make a constant angle with a fixed direction. As an application of the singularity theory, we classify the generic singularities of helix surfaces, which are cuspidaledges and swallowtails. These singularities are deeply related to the order of contact between the generating curves of helix surfaces and the affine tangent bundles over unit spheres. We also show that the unit normals and rulings of helix surfaces are spherical Legendrian dual to each other. Moreover, we investigate the singularities of helix surfaces from the viewpoint of opening map-germ. (C) 2020 Elsevier Ltd. All rights reserved.