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

Abstract We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> and torsion $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> . By identifying a fundamental function $$\psi = \sin ^2 \alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ψ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> , 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 $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> . This approach yields a single ODE involving only $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> and $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> 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 $$\kappa _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , obtaining an explicit ODE for the torsion. Remarkably, we prove that if $$\kappa _0 = 1/\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:math> , this equation admits an explicit, exact solution for $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> .