A generalization of the notion of helix

TL;DR

This work generalizes the classical helix by replacing the tangent field with an $F$-constant vector field $W$ along a curve $\alpha$ in $\mathbb{R}^3$ and requiring $W$ to form a constant angle $\theta$ with a fixed axis $V$. The authors derive natural equations for three helix families—normal, osculating, and rectifying—expressed in terms of the Frenet data $(\kappa_\alpha,\tau_\alpha)$ and, for normal helices, the Lancret curvature $\rho=\tau_\alpha/\kappa_\alpha$, obtaining explicit characterizations such as $\frac{\kappa_\alpha}{\cos^2\theta\,\kappa_\alpha^2+\tau_\alpha^2}\left(\frac{\tau_\alpha}{\kappa_\alpha}\right)'=-\tan\theta$ for normal helices and $\frac{\tau_\alpha}{\kappa_\alpha^2+\sin^2\theta\,\tau_\alpha^2}\left(\frac{\kappa_\alpha}{\tau_\alpha}\right)'=\cot\theta$ for osculating helices. A duality between normal and osculating helices is established, and rectifying helices are shown to coincide with cylindrical helices under the orthogonality condition. The general case with $W=aT_\alpha+bN_\alpha+cB_\alpha$ yields a unifying differential equation and cylinder-based interpretation, unifying several classical helices (plane, cylindrical, normal, osculating, rectifying) within a single framework.

Abstract

In this paper we generalize the notion of helix in the three-dimensional Euclidean space, which we define as that curve $C$ for which there is an $F$-constant vector field $W$ along $C$ that forms a constant angle with a fixed direction $V$ (called an axis of the helix). We find the natural equation and the geometric integration of helices $C$ where the $F$-constant vector field $W$ is orthogonal to its axis.

Figures (1)

• Figure 1: A normal helix with $\theta=\pi/36$ in a circular cylinder

Theorems & Definitions (12)

• Definition 1.1
• Proposition 1.2
• Definition 1.3
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2