<p>We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> and torsion <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>. By identifying a fundamental function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi = \sin ^2 \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mo>sin</mo> <mn>2</mn> </msup> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, representing the squared sine of the angle between the tangent vector and the axis of the cylinder, we reduce the geometric inclusion problem to a compatibility condition between an explicit eighth-degree polynomial equation and a differential equation for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>. This approach yields a single ODE involving only <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> that governs the inclusion of the curve in the cylinder. The robustness of this framework is demonstrated through specific examples of cylindrical curves. Furthermore, we analyze the case of curves with constant curvature <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, obtaining an explicit ODE for the torsion. Remarkably, we prove that if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\kappa _0 = 1/\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>ρ</mi> </mrow> </math></EquationSource> </InlineEquation>, this equation admits an explicit, exact solution for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>.</p>

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

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2026年5月19日

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Rafael López

摘要

We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $\kappa $ $κ$ and torsion $\tau $ $τ$ . By identifying a fundamental function $\psi = \sin ^2 \alpha $ $ψ = {sin}^{2} α$ , 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 $ $ψ$ . This approach yields a single ODE involving only $\kappa $ $κ$ and $\tau $ $τ$ 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 $\kappa _0$ $κ_{0}$ , obtaining an explicit ODE for the torsion. Remarkably, we prove that if $\kappa _0 = 1/\rho $ $κ_{0} = 1 / ρ$ , this equation admits an explicit, exact solution for $\tau $ $τ$ .