Helicoidal surfaces of frontals in Euclidian space as deformations of surfaces of revolution, with singularities

TL;DR

This paper explores the geometry of helicoidal surfaces generated by frontals in Euclidean space, analyzing their curvature, singularities, and deformations, extending classical surface theory to include singular cases.

Contribution

It introduces conditions for helicoidal frontals to remain frontals, derives curvature formulas, and studies the persistence of singularities under deformations.

Findings

Helicoidal frontals can be characterized using Legendre curves.

Parallel and focal surfaces of helicoidal surfaces are also helicoidal.

Singularities of generating curves persist under deformations.

Abstract

We investigate helicoidal surfaces in three-dimensional Euclidean space whose profile curves are frontals. Using the framework of Legendre curves and framed surfaces, we establish conditions under which helicoidal surfaces generated by frontals are themselves frontals or fronts. We then derive curvature expressions in terms of the invariants of the generating Legendre curve. Our study extends classical results on parallel and focal surfaces of surfaces of revolution to the helicoidal setting. In particular, we show that both parallel and focal surfaces of a helicoidal surface are helicoidal, with their generating curves arising from one-parameter deformations of the corresponding parallel and evolute curves. We prove that singularities of these curves persist under such deformations, revealing geometric rigidity and stability of singularities. Finally, we examine the behavior of the Gaussian and mean curvatures near singular points of helicoidal surfaces