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

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

Figures (4)

• Figure 1: Profile curves of ${\bf z}$ with $\xi_{\bf z}(t) = 0$, for some $t$. The dashed line is the axis of ${\bf z}$.
• Figure 2: An example illustrating a case where the profile curve could be a front, but the corresponding helicoidal surface is not. The dashed line represents the axis of the helicoidal surface.
• Figure 3: Deformation $\gamma_{\lambda,c}$, for $\gamma=(t+2,\frac{t^2}{2})$ and $\lambda=-2\sqrt{2}$.
• Figure 4: Figures of the Example \ref{['ex:focalcirc']}.

Theorems & Definitions (21)

• Proposition 2.1
• Proposition 3.1
• Remark 3.2
• Example 3.3
• Proposition 4.1
• Example 5.1
• Theorem 5.2
• Lemma 5.3
• Remark 5.4
• Proposition 5.5