[Paper Review] Special Smarandache Curves According To Darboux Frame In E3

Özcan Bektaş, Salîm Yüce|arXiv (Cornell University)|Mar 20, 2012

Axon Guidance and Neuronal Signaling4 citations

TL;DR

This paper introduces and analyzes special Smarandache curves—Tg, Tn, gn, and Tgn—defined via the Darboux frame on a surface in Euclidean 3-space. Using the Darboux frame's geodesic curvature $k_g$, normal curvature $k_n$, and geodesic torsion $ au_g$, the authors derive explicit formulas for the curvature and torsion of these curves, establishing their differential geometric properties and characterizing their behavior relative to the surface.

ABSTRACT

In this study, we determine some special Smarandache curves according to Darboux frame in E3. We give some characterizations and consequences of Smarandache curves.

Motivation & Objective

• To define and investigate special Smarandache curves—Tg, Tn, gn, and Tgn—based on the Darboux frame in $E^3$.
• To characterize the differential geometric properties of these curves, particularly their curvature and torsion.
• To compute the normal curvature, geodesic curvature, and geodesic torsion of the Smarandache curves in terms of the original curve's Darboux invariants.
• To establish relationships between the Darboux frame invariants of the base curve and the induced invariants of the Smarandache curves.

Proposed method

• The study uses the Darboux frame $\{\mathbf{T}, \mathbf{g}, \mathbf{n}\}$ on a unit-speed curve $\alpha(s)$ lying on a surface $M$ in $E^3$, where $\mathbf{T}$ is the tangent, $\mathbf{g}$ is the surface-normal orthogonal to $\mathbf{T}$, and $\mathbf{n}$ is the surface normal.
• Smarandache curves are defined as normalized linear combinations of Darboux frame vectors: $\beta(s^*) = \frac{1}{\sqrt{2}}(\mathbf{T} + \mathbf{g})$ for Tg, and similar for others.
• The Frenet-Serret equations are adapted to the Darboux frame, with the Darboux derivative formula $\dot{\mathbf{T}} = k_g \mathbf{g} + k_n \mathbf{n}$, $\dot{\mathbf{g}} = -k_g \mathbf{T} + \tau_g \mathbf{n}$, $\dot{\mathbf{n}} = -k_n \mathbf{T} - \tau_g \mathbf{g}$.
• The curvature and torsion of the Smarandache curves are derived using the Darboux frame invariants $k_g$, $k_n$, and $\tau_g$, along with their derivatives.
• The geodesic curvature $k_g^*$, normal curvature $k_n^*$, and geodesic torsion $\tau_g^*$ of the Smarandache curves are computed via vector projections and trigonometric relations involving the angle $\varphi^*$ between $\mathbf{g}^*$ and $\mathbf{n}^*$.
• Explicit formulas for $k_g^*$, $k_n^*$, and $\tau_g^*$ are derived in terms of the original curve’s Darboux invariants and their derivatives, using normalization and vector decomposition techniques.

Experimental results

Research questions

• RQ1 How are the Smarandache curves Tg, Tn, gn, and Tgn defined in the context of the Darboux frame in $E^3$?
• RQ2 What are the expressions for the geodesic curvature, normal curvature, and geodesic torsion of these Smarandache curves in terms of the base curve’s Darboux invariants?
• RQ3 Under what conditions do these Smarandache curves become geodesic, asymptotic, or principal curves on the surface?
• RQ4 How do the differential geometric properties of the Smarandache curves relate to the original curve’s curvature and torsion?

Key findings

• The geodesic curvature $k_g^*$ of the Tg-Smarandache curve is given by $k_g^* = \frac{\sqrt{2(\delta_1^2 + \delta_2^2 + \delta_3^2)}}{\left((k_n + k_g)^2 + (\tau_g - k_g)^2 + (\tau_g + k_n)^2\right)^2} \cos\varphi^*$, where $\delta_i$ are components of the derivative of the Darboux frame.
• The normal curvature $k_n^*$ of the Tg-Smarandache curve is $k_n^* = \frac{\sqrt{2(\delta_1^2 + \delta_2^2 + \delta_3^2)}}{\left((k_n + k_g)^2 + (\tau_g - k_g)^2 + (\tau_g + k_n)^2\right)^2} \sin\varphi^*$, with $\varphi^*$ being the angle between $\mathbf{g}^*$ and $\mathbf{n}^*$.
• The geodesic torsion $\tau_g^*$ of the Tg-Smarandache curve is derived as a complex expression involving $k_g$, $k_n$, $\tau_g$, their derivatives, and trigonometric functions of $\varphi^*$, plus the derivative $d\varphi^*/ds^*$.
• The unit normal vector $\mathbf{n}^*$ and the unit vector $\mathbf{g}^*$ of the Smarandache curve are expressed as linear combinations of $\mathbf{T}$, $\mathbf{g}$, and $\mathbf{n}$, with coefficients depending on $k_g$, $k_n$, $\tau_g$, and $\varphi^*$.
• The expressions for the Darboux invariants of the Smarandache curves are explicitly computed and shown to depend on the original curve’s $k_g$, $k_n$, $\tau_g$, and their derivatives.
• The study establishes a complete framework for computing the differential geometry of Smarandache curves via the Darboux frame, enabling analysis of their geometric nature (e.g., whether they are geodesic or asymptotic) based on the derived invariants.