Determination of the position vectors of general helices from intrinsic equations in $\e^3$

TL;DR

This paper derives a fourth-order vector differential equation for space curves and provides a method to explicitly determine the position vectors of general helices from their curvature and torsion functions, extending previous results.

Contribution

It introduces a new approach to find the position vectors of general helices directly from intrinsic equations, extending known results in differential geometry.

Findings

Derived a fourth-order vector differential equation for space curves.

Provided a parametric representation of general helices from intrinsic equations.

Presented four examples illustrating the method.

Abstract

In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector $\psi=\Big(\psi_1,\psi_2,\psi_3\Big)$ of general helices from the intrinsic equations $\kappa=\kappa(s)$ and $\tau=\tau(s)$ where $\kappa$ and $\tau$ are the curvature and torsion of the space curve $\psi$, respectively. Our result extends some knwown results. Moreover, we give four examples to illustrate how to find the position vector from the intrinsic equations of general helices.