A necessary condition for cylindrical curves in terms of curvature and torsion

TL;DR

This paper derives necessary conditions for a regular curve to lie on a circular cylinder using curvature and torsion, reducing the problem to a specific differential equation involving these quantities.

Contribution

It introduces a new framework linking curvature and torsion to cylindrical inclusion via an explicit ODE, including solutions for constant curvature cases.

Findings

Derived an eighth-degree polynomial compatibility condition.

Reduced the problem to a single ODE involving curvature and torsion.

Provided explicit solutions for curves with constant curvature.

Abstract

We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $\kappa$ and torsion $\tau$. By identifying a fundamental function $\psi = \sin^2 \alpha$, representing the squared sine of the angle between the tangent vector and the axis of the cylinder, we reduce the geometric inclusion problem to a compatibility condition between an explicit eighth-degree polynomial equation and a differential equation for $\psi$. This approach yields a single ODE involving only $\kappa$ and $\tau$ that governs the inclusion of the curve in the cylinder. The robustness of this framework is demonstrated through specific examples of cylindrical curves. Furthermore, we analyze the case of curves with constant curvature $\kappa_0$, obtaining an explicit ODE for the torsion. Remarkably, we prove that if $\kappa_0 = 1/\rho$, this equation admits an explicit, exact solution for $\tau$.