Singularities of helicoidal surfaces of frontals in the Euclidean space

TL;DR

This paper studies helicoidal surfaces generated by both regular and singular curves using frontals, providing invariants, curvatures, and criteria for singularities, thus extending the understanding of such surfaces in Euclidean space.

Contribution

It introduces a framework for analyzing helicoidal surfaces generated by singular curves using frontals, including invariants, curvatures, and singularity criteria.

Findings

Helicoidal surfaces of frontals can be considered as framed base surfaces.

Basic invariants and curvatures are derived using Legendre curves.

Criteria for singularities of helicoidal surfaces are established.

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.