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

Abstract

This work investigates helicoidal surfaces in three-dimensional Euclidean space whose profile curves are frontals. Within the framework of Legendre curves and framed surfaces, conditions are established under which helicoidal surfaces generated by frontals are themselves frontals or fronts. Curvature formulas are derived in terms of the invariants of the generating Legendre curve, extending classical results on parallel and focal surfaces of surfaces of revolution to the helicoidal case. It is shown that both the parallel and focal surfaces of a helicoidal surface are again helicoidal, with generating curves obtained from one-parameter deformations of the corresponding parallel and evolute curves. We prove that the singularities of these curves persist under such deformations, revealing geometric rigidity and stability. Finally, the behavior of Gaussian and mean curvatures near singular points of helicoidal surfaces is analyzed.