Framed Slant Helices And Singularities Of Developable Surfaces

In this paper,we investigate the geometric properties of singular curves in 3-dimensional Euclidean space,a new Frenet type frame along singular curves and two important invariants are presented.Meanwhile,the notion of framed slant helices and the principal normal rectifying developable surface and the Darboux normal developable surface which generated by a Frenet type framed base curve are given.Furthermore,the local topological structure of two developable surfaces is revealed by means of unfolding theory,and some characterisations of the framed slant helices are described via certain equivalent conditions.It is found that these developable surfaces surfaces have some singularities whose types can be determined by the corresponding geometric invariants.More significantly,there is a close relationship between the framed slant helices and the two developable surfaces,e.g.the principal normal rectifying developable surface will be a cylinder surface if the original curve is a framed slant helix.Finally,several examples are provided to illustrate the main results.