Singularities of helicoidal surfaces of frontals in the Euclidean space

Abstract

We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a frontal can naturally be considered as a generalised framed base surface. Moreover, we show that it is also a framed base surface under a mild condition. We give basic invariants and curvatures for helicoidal surfaces of frontals by using the curvatures of Legendre curves. Moreover, we also give criteria for singularities of helicoidal surfaces.