A generalization of the notion of helix

TL;DR

This paper extends the classical concept of helices in 3D space by introducing a generalized definition involving an $F$-constant vector field, and derives their natural equations and geometric properties.

Contribution

It introduces a broader definition of helices using an $F$-constant vector field and provides their natural equations and geometric integration.

Findings

Derived the natural equation of generalized helices.

Established the geometric integration for these helices.

Extended classical helix theory to a more general setting.

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.