This paper presents a rigorous generalization of the classical Bäcklund transformation (BT) for space curves in Euclidean 3-space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {E}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> using the versatile quasi-frame (q-frame). The novelty lies in introducing the frame rotation angle, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\theta (s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, as a continuous geometric control parameter for designing BTs that satisfy fundamental geometric constraints, notably the constant separation distance r. We derive a coupled system of first-order ordinary differential equations (ODEs) governing the evolution of both the frame rotation <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\theta (s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the inclination angle <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\gamma (s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the transformation. A key theoretical contribution is the demonstration that this class of BTs universally preserves the second quasi-curvature (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\tilde{\kappa }_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>κ</mi> <mo stretchy="false">~</mo> </mover> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>) and the quasi-torsion (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\tilde{\tau }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>) of the q-frame, implying that the geometric action is concentrated solely on the first quasi-curvature (<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\kappa _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>). This invariance offers deep, intrinsic insight into the geometric nature of the transformation. Our formulation naturally encompasses the classical Frenet and Bishop frame BTs as special static cases. Numerical simulations starting from a circular helix illustrate the framework’s ability to generate complex, non-helical curves under specific dynamic geometric constraints, highlighting the utility of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\theta (s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in geometric control theory.

On the Bäcklund transformation according to the $q-$frame in euclidean $3-$space $\mathbb {R}^{3}$

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2026年3月26日

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Ghassan Ali Mahmood Mahmood,
Ufuk Öztürk

摘要

This paper presents a rigorous generalization of the classical Bäcklund transformation (BT) for space curves in Euclidean 3-space $\mathbb {E}^{3}$ $E^{3}$ using the versatile quasi-frame (q-frame). The novelty lies in introducing the frame rotation angle, $\theta (s)$ $θ (s)$ , as a continuous geometric control parameter for designing BTs that satisfy fundamental geometric constraints, notably the constant separation distance r. We derive a coupled system of first-order ordinary differential equations (ODEs) governing the evolution of both the frame rotation $\theta (s)$ $θ (s)$ and the inclination angle $\gamma (s)$ $γ (s)$ of the transformation. A key theoretical contribution is the demonstration that this class of BTs universally preserves the second quasi-curvature ( $\tilde{\kappa }_{2}$ ${\tilde{κ}}_{2}$ ) and the quasi-torsion ( $\tilde{\tau }$ $\tilde{τ}$ ) of the q-frame, implying that the geometric action is concentrated solely on the first quasi-curvature ( $\kappa _1$ $κ_{1}$ ). This invariance offers deep, intrinsic insight into the geometric nature of the transformation. Our formulation naturally encompasses the classical Frenet and Bishop frame BTs as special static cases. Numerical simulations starting from a circular helix illustrate the framework’s ability to generate complex, non-helical curves under specific dynamic geometric constraints, highlighting the utility of $\theta (s)$ $θ (s)$ in geometric control theory.

On the Bäcklund transformation according to the \(q-\)frame in euclidean \(3-\)space \(\mathbb {R}^{3}\)

摘要