Spinor Equations of Smarandache Curves in E³

Abstract

This study examines the spinor representations of $\mathbf{TN}$ (tangent and normal), $\mathbf{NB}$ (normal and binormal), $\mathbf{TB}$ (tangent and binormal) and $\mathbf{TNB}$ (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space ${\mathbb{E}}^3$. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the ${\mathbb{C}}^3$ complex vector space form a two-dimensional surface in the ${\mathbb{C}}^2$ complex space. Additionally, each isotropic vector in ${\mathbb{C}}^3$ space corresponds to two vectors in ${\mathbb{C}}^2$ space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its ($\mathbf{TN}$, $\mathbf{NB}$, $\mathbf{TB}$ and $\mathbf{TNB}$)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression.

1. Introduction

Smarandache curves are defined as regular curves with position vectors formed by the Frenet vectors of regular curves. Smarandache curves have been the subject of study for many researchers to date. Smarandache curves in semi-real Euclidean space ${\mathbb{E}}_1^4$ were defined by Turgut and Yılmaz [1]. Ali introduced some special Smarandache curves and studied their Frenet–Serret vectors in three-dimensional Euclidean space [2]. Smarandache curves for quaternionic curves were defined in [3,4]. Later, in ${\mathbb{E}}^3$, Bektaş and Yüce obtained Smarandache curves with a Darboux frame [5]. In [6], Taşköprü and Tosun defined Smarandache curves with the Sabban frame. Smarandache curves in three-dimensional Euclidean space ${\mathbb{E}}^3$ according to the Bishop frame were defined, and the centres of the osculator spheres and the curvature spheres of these curves were found in [7]. Moreover, in [8], special Smarandache curves in ${\mathbb{R}}_1^3$ were obtained, and in [9], isotropic Smarandache curves were expressed in complex four-dimensional space. Then, in [10], Smarandache curves were given for three-dimensional Lie groups. In [11], Smarandache curves of the Salkowski (spacelike) curve with a spacelike principal normal were defined. Some characterizations of Smarandache curves with modified orthogonal frames were obtained in [12]. Type-$\pi$ Smarandache surfaces were defined, the basis curves of which are Smarandache curves derived by Darboux frame vectors that minimize the rotation of the curve at ${\mathbb{E}}^3$ in [13].

Spinors are used as vectorlike structures in physics. The spin matrices were introduced by Pauli in 1927 [14]. Later, the relationships between Lorentz groups and spinors were expressed by Dirac [15]. The French mathematician Cartan, one of the pioneers in studying Lie groups, was the first person to geometrically examine spinors [16]. Cartan gave the spinor corresponding to the basic definitions in geometry and expressed that isotropic vectors in ${\mathbb{C}}^3$ create a surface in ${\mathbb{C}}^2$. Moreover, Cartan showed that every isotropic vector in ${\mathbb{C}}^3$ matches two vectors in ${\mathbb{C}}^2$. Cartan also stated that these vectors in ${\mathbb{C}}^2$ are spinors [16]. The theory of curves in three-dimensional Euclidean space using spinors was given by Torres del Castillo and Barrales in another study [17]. The spinor equations of relationships between Bishop and Frenet frames were expressed in [18]. Additionally, the spinor representation of the Darboux frame of a directed surface in ${\mathbb{E}}^3$ was obtained in [19]. Then, the spinor formulas of some curve pairs in Minkowski space were found [20,21,22]. The spinor equations for involute–evolute curves in ${\mathbb{E}}^3$ were given in [23]. Other fascinating studies about spinors and curves have also been conducted [24,25,26,27,28,29,30].

2. Preliminaries

This section briefly expresses the Euclidean space, spinors and Smarandache curves in ${\mathbb{E}}^3$. Therefore, we provide several important definitions and theorems below.

Euclidean space is an affine space over the reals such that the associated vector space is a real vector space. The dimension of an Euclidean space is the dimension of its associated (real) vector space. Let ${\mathbb{E}}^3$ be three-dimensional Euclidean space and $\alpha$ be a differentiable function in ${\mathbb{E}}^3$ such that $\alpha :I\to {\mathbb{E}}^3$, $s\to \alpha \left(s\right)=\left({\alpha }_1,{\alpha }_2,{\alpha }_3\right)$, where $I\subseteq \mathbb{R}$ is an open subinterval. Therefore, $\alpha \left(I\right)\subset {\mathbb{E}}^3$ is called a differentiable curve in Euclidean space ${\mathbb{E}}^3$. $\left(I,\alpha \right)$ is called the coordinate neighborhood, I is called the parameter range, and $s\in I$ is called the parameter of the curve $\left(\alpha \right)$ [31].

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve or the geometric properties of the curve itself irrespective of any motion in three-dimensional Euclidean space. Specifically, the formulas describe the derivatives of the so-called tangent, normal and binormal unit vectors in terms of each other. The formulas were discovered by French mathematicians J.F. Frenet and J.A. Serret. The tangent, normal and binormal unit vectors are often called $\mathbf{T}$, $\mathbf{N}$ and $\mathbf{B}$ or collectively the Frenet–Serret frame ($\mathbf{TNB}$ frame), together forming an orthonormal basis that spans real vector space ${\mathbb{R}}^3$ and is defined as follows:

$$\begin{array}{c}{\displaystyle \frac{d\mathbf{T}}{ds}}=\kappa \mathbf{N}\hfill \\ {\displaystyle \frac{d\mathbf{N}}{ds}}=-\kappa \mathbf{T}+\tau \mathbf{B}\hfill \\ {\displaystyle \frac{d\mathbf{B}}{ds}}=-\tau \mathbf{N}\hfill \end{array}$$

where ${\displaystyle \frac{d}{ds}}$ is the derivative with respect to arc-length parameter s, $\kappa$ is the curvature and $\tau$ is the torsion of the curve $\left(\alpha \right)$ [31].

Spinors generally form a vector space over complex numbers with the help of linear group representations of the spin group. Let an isotropic vector in ${\mathbb{C}}^3$ be $\mathbf{x}=(x_1,x_2,x_3)$. Specifically, the vector is a non-zero vector that is orthogonal to itself. Moreover, assume that a vector $\phi$ with two complex components in ${\mathbb{C}}^2$ corresponds to the isotropic vector $\mathbf{x}$ [16]. Therefore, the isotropic vectors in ${\mathbb{C}}^3$ create a surface in ${\mathbb{C}}^2$. If we consider that the components of the vector $\phi$ with two complex components are ${\phi }_1$ and ${\phi }_2$, then we obtain that the relationships between the components of the vector $\phi$ and the isotropic vector $\mathbf{x}$ are $x_1={\phi }_1^2-{\phi }_2^2,x_2=i\left({\phi }_1^2+{\phi }_2^2\right),x_3=-2{\phi }_1{\phi }_2$ and ${\phi }_1=\pm \sqrt{{\displaystyle \frac{x_1-i{x}_2}{2}}},{\phi }_2=\pm \sqrt{{\displaystyle \frac{-x_1-i{x}_2}{2}}}$ [16]. Cartan expressed that these complex vectors $\phi$ are called spinors such that [16].

$$\phi ={\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\end{array}\right]}_{2\times 1}$$

Now, we obtain the $2x2$ complex symmetric matrices $\sigma$ via Pauli matrices $\left(P_1,P_2,P_3\right)$ such that

$${\sigma }_1=C{P}_1=\left[\begin{array}{c}10\\ 0-1\end{array}\right],{\sigma }_2=C{P}_2=\left[\begin{array}{c}i0\\ 0i\end{array}\right],{\sigma }_3=C{P}_3=\left[\begin{array}{c}0-1\\ -10\end{array}\right]$$

where $C=\left[\begin{array}{c}01\\ -10\end{array}\right]$ [14,17]. Moreover, in [17], for $\mathbf{a},\mathbf{b},\mathbf{c}\in {\mathbb{R}}^3$, the spinor formulations are

$$\begin{array}{c}\mathbf{a}+i\mathbf{b}={\phi }^t\sigma \phi ,\hfill \\ \mathbf{c}=-{\widehat{\phi }}^t\sigma \phi \hfill \end{array}$$

where $\mathbf{a},\mathbf{b},\mathbf{c}\in {\mathbb{R}}^3,\mathbf{a}+i\mathbf{b}\in {\mathbb{C}}^3$ are isotropic vectors. Then, we can easily obtain the first, second and third components of the vectors $\mathbf{a}+i\mathbf{b}\in {\mathbb{C}}^3$ and $\mathbf{c}\in {\mathbb{R}}^3$, respectively, such that

$$\begin{array}{c}\mathrm{for}{\phi }_1,{\phi }_1^2-{\phi }_2^2,{\phi }_1{\overline{\phi }}_2+{\overline{\phi }}_1{\phi }_2\hfill \\ \mathrm{for}{\phi }_2,i\left({\phi }_1^2+{\phi }_2^2\right),i\left({\phi }_1{\overline{\phi }}_2-{\overline{\phi }}_1{\phi }_2\right)\hfill \\ \mathrm{for}{\phi }_3,-2{\phi }_1{\phi }_2,{\left|{\phi }_1\right|}^2-{\left|{\phi }_2\right|}^2.\hfill \end{array}$$

Specifically,

$$\begin{array}{c}\mathbf{a}+i\mathbf{b}=\left({\phi }_1^2-{\phi }_2^2,i\left({\phi }_1^2+{\phi }_2^2\right),-2{\phi }_1{\phi }_2\right),\hfill \\ \mathbf{c}=\left({\phi }_1{\overline{\phi }}_2+{\overline{\phi }}_1{\phi }_2,i\left({\phi }_1{\overline{\phi }}_2-{\overline{\phi }}_1{\phi }_2\right),{\left|{\phi }_1\right|}^2-{\left|{\phi }_2\right|}^2\right).\hfill \end{array}$$

Moreover, the spinor mate $\widehat{\phi }$ of $\phi$ is defined as

$$\widehat{\phi }=\left[\begin{array}{c}01\\ -10\end{array}\right]\overline{\phi }=-\left[\begin{array}{c}01\\ -10\end{array}\right]\left[\begin{array}{c}{\overline{\phi }}_1\\ {\overline{\phi }}_2\end{array}\right]=\left[\begin{array}{c}-{\overline{\phi }}_2\\ {\overline{\phi }}_1\end{array}\right].$$

One can see that the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ have the same length, $∥\mathbf{a}∥=∥\mathbf{b}∥=∥\mathbf{c}∥={\overline{\phi }}^t\phi$, and these vectors are mutually orthogonal. This relationship between the spinor $\phi$ and mutually orthogonal triad $\left\{\mathbf{a},\mathbf{b},\mathbf{c}\right\}$ is two to one such that the spinors $\phi$ and $-\phi$ correspond to the same ordered orthonormal triad $\left\{\mathbf{a},\mathbf{b},\mathbf{c}\right\}$, with $〈\mathbf{a}\times \mathbf{b},\mathbf{c}〉>0.$ A very important detail here is that the ordered triads $\left\{\mathbf{a},\mathbf{b},\mathbf{c}\right\}$, $\left\{\mathbf{b},\mathbf{c},\mathbf{a}\right\}$ and $\left\{\mathbf{c},\mathbf{a},\mathbf{b}\right\}$ correspond to the different spinors [17].

Proposition 1.

Suppose that two arbitrary spinors are φ and ψ. Therefore, the equations for the spinors can be written as

$$\begin{array}{c}\left(i\right)\overline{{\psi }^t\sigma \phi }=-{\widehat{\psi }}^t\sigma \widehat{\phi }\hfill \\ \left(ii\right)\widehat{\left(\alpha \psi +\lambda \phi \right)}=\overline{\alpha }{\widehat{\psi }}^t+\overline{\lambda }\widehat{\phi }\hfill \\ \left(iii\right){}^{\widehat{^}}\phi =-\phi \hfill \\ \left(iv\right){\psi }^t\sigma \phi ={\phi }^t\sigma \psi \hfill \end{array}$$

where $\alpha ,\lambda \in \mathbb{C}$ [17].

Assume that a regular curve $\beta$ has arc-length parameter s in three-dimensional Euclidean space ${\mathbb{E}}^3$ such that $\parallel {\beta }^{\prime }\left(s\right)\parallel =1$ and the spinor corresponding to the Frenet–Serret frame $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$ of $\beta$ is $\psi$. According to this, we can write spinor equations for the Frenet–Serret vectors of the curve $\beta$ as

$$\begin{array}{c}\mathbf{N}+i\mathbf{B}={\psi }^t\sigma \psi =\left({\psi }_1^2-{\psi }_2^2,i\left({\psi }_1^2+{\psi }_2^2\right),-2{\psi }_1{\psi }_2\right),\hfill \\ \mathbf{T}=-{\widehat{\psi }}^t\sigma \psi =\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2,i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right),{\left|{\psi }_1\right|}^2-{\left|{\psi }_2\right|}^2\right)\hfill \end{array}$$

where $\mathbf{N}+i\mathbf{B}\in {\mathbb{C}}^3$ is an isotropic vector [17].

Theorem 1.

Let $\beta :I\to {\mathbb{E}}^3$ be a regular curve and ψ be a spinor corresponding to Frenet–Serret frame $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ of β. Therefore, the spinor formulation of Frenet–Serret frame $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$ is [17,23].

$$\begin{array}{c}\mathbf{T}=-{\widehat{\psi }}^t\sigma \psi ,\hfill \\ \mathbf{N}={\displaystyle \frac{1}{2}}\left({\psi }^t\sigma \psi -{\widehat{\psi }}^t\sigma \widehat{\psi }\right),\hfill \\ \mathbf{B}=-{\displaystyle \frac{i}{2}}\left({\psi }^t\sigma \psi +{\widehat{\psi }}^t\sigma \widehat{\psi }\right)\hfill \end{array}$$

After some calculations in [23], the spinor equations of $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$ in terms of spinor components are obtained via

$$\begin{array}{c}\mathbf{T}=\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2,i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right),{\left|{\psi }_1\right|}^2-{\left|{\psi }_2\right|}^2\right),\hfill \\ \mathbf{N}={\displaystyle \frac{1}{2}}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2,i\left({\psi }_1^2+{\psi }_2^2-{\overline{{\psi }_1}}^2-{\overline{{\psi }_2}}^2\right),-2{\psi }_1{\psi }_2-2\overline{{\psi }_1{\psi }_2}\right),\hfill \\ \mathbf{B}=-{\displaystyle \frac{i}{2}}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2,i\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_1}}^2+{\overline{{\psi }_2}}^2\right),-2{\psi }_1{\psi }_2+2\overline{{\psi }_1{\psi }_2}\right).\hfill \end{array}$$

(1)

Now, we give definitions and theorems of Smarandache curves in ${\mathbb{E}}^3$.

Definition 1.

Consider the regular curve in ${\mathbb{E}}^3$. If the position vector of this regular curve can be defined with the help of Frenet-Serret vectors on another regular main curve in ${\mathbb{E}}^3$, this curve is called the Smarandache curve of the main curve [2].

Now, we specifically classify these Smarandache curves according to how they are written with the help of Frenet-Serret vectors.

Definition 2.

Let β be a regular curve with the arc-length parameter s in ${\mathbb{E}}^3$. The $\mathbf{TN}$–Smarandache curve generated by the tangent and normal vectors of $\left(\beta \right)$ is denoted by ${\beta }_{TN}$, and it is defined by

$${\beta }_{TN}\left(s\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\mathbf{T}\left(s\right)+\mathbf{N}\left(s\right)\right)$$

where $s_{{\beta }_{TN}}$ is the arc-length parameter of $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ and the Frenet–Serret vectors of $\beta \left(s\right)$ are $\left\{\mathbf{T}\left(s\right),\mathbf{N}\left(s\right),\mathbf{B}\left(s\right)\right\}$ [2].

Theorem 2.

Assume that the $\mathbf{TN}$–Smarandache curve of the curve β parameterized with arc-length parameter s is ${\beta }_{TN}$ in ${\mathbb{E}}^3$. Moreover, the Frenet–Serret vectors of β and ${\beta }_{TN}$ are $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ and $\left\{{\mathbf{T}}_{{\beta }_{TN}},{\mathbf{N}}_{{\beta }_{TN}},{\mathbf{B}}_{{\beta }_{TN}}\right\}$, respectively. Therefore, the Frenet–Serret vectors of the $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ can be written as Frenet–Serret vectors of the curve β as

$${\mathbf{T}}_{{\beta }_{TN}}={\displaystyle \frac{-\kappa \mathbf{T}+\kappa \mathbf{N}+\tau \mathbf{B}}{\sqrt{2{\kappa }^2+{\tau }^2}}},$$

(2)

$${\mathbf{N}}_{{\beta }_{TN}}={\displaystyle \frac{{\delta }_1\mathbf{T}+{\delta }_2\mathbf{N}+{\delta }_3\mathbf{B}}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}},$$

(3)

$${\mathbf{B}}_{{\beta }_{TN}}={\displaystyle \frac{\left(\kappa {\delta }_3-\tau {\delta }_2\right)\mathbf{T}+\left(\tau {\delta }_1+\kappa {\delta }_3\right)\mathbf{N}+\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\mathbf{B}}{\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}$$

(4)

where

$$\begin{array}{c}{\delta }_1=-\left[{\kappa }^2\left(2{\kappa }^2+{\tau }^2\right)+\tau \left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right],\hfill \\ {\delta }_2=-\left[{\kappa }^2\left(2{\kappa }^2+3{\tau }^2\right)-\tau \left({\tau }^3+\kappa {\tau }^{\prime }-\tau {\kappa }^{\prime }\right)\right],\hfill \\ {\delta }_3=\kappa \left[\tau \left(2{\kappa }^2+{\tau }^2\right)-2\left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right]\hfill \end{array}$$

and κ, τ are the curvature and torsion of the curve β, respectively. Moreover, if the arc-length parameters of β and its $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ are s and $s_{{\beta }_{TN}}$, respectively, then the relationship between the differentials of these variables can be given as [2].

$${\displaystyle \frac{d{s}_{{\beta }_{TN}}}{ds}}=\sqrt{{\displaystyle \frac{2{\kappa }^2+{\tau }^2}{2}}}$$

The curve $\beta$ will hereafter be taken as parameterized with arc-length parameter s.

Definition 3.

Assume that the $\mathbf{TB}$–Smarandache curve generated by the tangent and binormal vectors of $\left(\beta \right)$ is ${\beta }_{TB}$ in ${\mathbb{E}}^3$. Then, $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ is defined as

$${\beta }_{TB}\left(s\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\mathbf{T}\left(s\right)+\mathbf{B}\left(s\right)\right)$$

where the Frenet–Serret frame of $\beta \left(s\right)$ is $\left\{\mathbf{T}\left(s\right),\mathbf{N}\left(s\right),\mathbf{B}\left(s\right)\right\}$ [32].

Theorem 3.

Consider that the $\mathbf{TB}$–Smarandache curve of the curve β is ${\beta }_{TB}$ with arc-length parameter $s_{{\beta }_{TB}}$ in ${\mathbb{E}}^3$. Then, Frenet–Serret vectors $\left\{{\mathbf{T}}_{{\beta }_{TB}},{\mathbf{N}}_{{\beta }_{TB}},{\mathbf{B}}_{{\beta }_{TB}}\right\}$ of $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ can be written as Frenet–Serret vectors $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ of β as

$${\mathbf{T}}_{{\beta }_{TB}}=\mathbf{N},$$

(5)

$${\mathbf{N}}_{{\beta }_{TB}}={\displaystyle \frac{-\kappa \mathbf{T}+\tau \mathbf{B}}{\sqrt{{\kappa }^2+{\tau }^2}}}$$

(6)

and

$${\mathbf{B}}_{{\beta }_{TB}}={\displaystyle \frac{\tau \mathbf{T}+\kappa \mathbf{B}}{\sqrt{{\kappa }^2+{\tau }^2}}}$$

(7)

where the relationship between the differentials of variables s and $s_{{\beta }_{TB}}$ is [32].

$${\displaystyle \frac{d{s}_{{\beta }_{TB}}}{ds}}={\displaystyle \frac{\left|\kappa -\tau \right|}{\sqrt{2}}}$$

Definition 4.

Let the $\mathbf{NB}$–Smarandache curve generated by the normal and binormal vectors of the $\left(\beta \right)$ be ${\beta }_{NB}$ in ${\mathbb{E}}^3$. Then, the $\mathbf{NB}$–Smarandache curve is defined as

$${\beta }_{NB}\left(s\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\mathbf{N}\left(s\right)+\mathbf{B}\left(s\right)\right)$$

with the aid of the Frenet–Serret vectors of β, $\left\{\mathbf{T}\left(s\right),\mathbf{N}\left(s\right),\mathbf{B}\left(s\right)\right\}$ [2].

Therefore, we give the following theorem for the $\mathbf{NB}$–Smarandache curve.

Theorem 4.

Assume that $s_{{\beta }_{NB}}$ is the arc-length parameter of the $\mathbf{NB}$–Smarandache curve of the curve β in ${\mathbb{E}}^3$. Therefore, there are relationships between the Frenet–Serret vectors $\left\{{\mathbf{T}}_{{\beta }_{NB}},{\mathbf{N}}_{{\beta }_{NB}},{\mathbf{B}}_{{\beta }_{NB}}\right\}$ and $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ of $\mathbf{TB}$–Smarandache curve ${\beta }_{NB}$ and the curve β, respectively,

$${\mathbf{T}}_{{\beta }_{NB}}={\displaystyle \frac{-\kappa \mathbf{T}-\tau \mathbf{N}+\tau \mathbf{B}}{\sqrt{{\kappa }^2+2{\tau }^2}}},$$

(8)

$${\mathbf{N}}_{{\beta }_{NB}}={\displaystyle \frac{{\mu }_1\mathbf{T}+{\mu }_2\mathbf{N}+{\mu }_3\mathbf{B}}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}$$

(9)

and

$${\mathbf{B}}_{{\beta }_{NB}}={\displaystyle \frac{\left(-\tau {\mu }_2-\tau {\mu }_3\right)\mathbf{T}+\left(\tau {\mu }_1+\kappa {\mu }_3\right)\mathbf{N}+\left(\tau {\mu }_1-\kappa {\mu }_2\right)\mathbf{B}}{\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}$$

(10)

where

$$\begin{array}{c}{\mu }_1=\kappa \tau \left({\kappa }^2+2{\tau }^2\right)+2\tau \left(\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau \right),\hfill \\ {\mu }_2=-\left({\kappa }^2+2{\tau }^2\right)\left({\kappa }^2+{\tau }^2\right)+\kappa \left({\kappa }^{\prime }\tau -{\tau }^{\prime }\kappa \right),\hfill \\ {\mu }_3=\left(-{\tau }^2\right)\left({\kappa }^2+2{\tau }^2\right)+\kappa \left(\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau \right)\hfill \end{array}$$

and the relationship between the differentials of variables s and $s_{{\beta }_{NB}}$ is [2].

$${\displaystyle \frac{d{s}_{{\beta }_{NB}}}{ds}}=\sqrt{{\displaystyle \frac{{\kappa }^2+2{\tau }^2}{2}}}$$

Finally, we give the definition and the theorem of the Frenet–Serret vectors of the $\mathbf{TNB}$–Smarandache curve.

Definition 5.

Assume that the Frenet–Serret frame of $\beta \left(s\right)$ is $\left\{\mathbf{T}\left(s\right),\mathbf{N}\left(s\right),\mathbf{B}\left(s\right)\right\}$ in ${\mathbb{E}}^3$. Thus, the curve ${\beta }_{TNB}$ given with the equation

$${\beta }_{TNB}\left(s\right)={\displaystyle \frac{1}{\sqrt{3}}}\left(\mathbf{T}\left(s\right)+\mathbf{N}\left(s\right)+\mathbf{B}\left(s\right)\right)$$

is defined as the $\mathbf{TNB}$–Smarandache curve of the curve β [2].

Theorem 5.

Suppose that the $\mathbf{TNB}$–Smarandache curve of β is ${\beta }_{TNB}$ in ${\mathbb{E}}^3$. Moreover, the Frenet–Serret vectors of β and ${\beta }_{TNB}$ are $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ and $\left\{{\mathbf{T}}_{{\beta }_{TNB}},{\mathbf{N}}_{{\beta }_{TNB}},{\mathbf{B}}_{{\beta }_{TNB}}\right\}$, respectively. Therefore, the Frenet–Serret vectors of $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$ can be written as Frenet–Serret vectors of β as

$${\mathbf{T}}_{{\beta }_{TNB}}={\displaystyle \frac{-\kappa \mathbf{T}+\left(\kappa -\tau \right)\mathbf{N}+\tau \mathbf{B}}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }}},$$

(11)

$${\mathbf{N}}_{{\beta }_{TNB}}={\displaystyle \frac{{\nu }_1\mathbf{T}+{\nu }_2\mathbf{N}+{\nu }_3\mathbf{B}}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}},$$

(12)

$${\mathbf{B}}_{{\beta }_{TNB}}={\displaystyle \frac{\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)\mathbf{T}+\left(\tau {\nu }_1+{\nu }_3\right)\mathbf{N}+\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\mathbf{B}}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau \sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}}$$

(13)

where

$$\begin{array}{c}{\nu }_1=\kappa \tau \left(4\kappa \left(\kappa -\tau \right)+2\left({\tau }^{\prime }+{\tau }^2\right)+{\kappa }^{\prime }\right)-{\kappa }^2\left(2{\kappa }^2+{\tau }^{\prime }\right)-2{\kappa }^{\prime }{\tau }^2,\hfill \\ {\nu }_2=2\kappa \tau \left({\left(\kappa -\tau \right)}^2+2\tau -2{\tau }^{\prime }\right)-2\left({\kappa }^4+{\tau }^4\right)+{\kappa }^{\prime }{\tau }^2-{\kappa }^2{\tau }^{\prime },\hfill \\ {\nu }_3=\tau \left(2\kappa \left({\kappa }^2+4{\tau }^2-{\kappa }^{\prime }-{\tau }^{\prime }-2\kappa \tau \right)+\left(\tau {\kappa }^{\prime }+{\tau }^{\prime }-2{\tau }^3\right)\right).\hfill \end{array}$$

Moreover, if the arc-length parameter of $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$ is $s_{{\beta }_{TNB}}$, then for the variables s and $s_{{\beta }_{TNB}}$, the equation

$${\displaystyle \frac{d{s}_{{\beta }_{TNB}}}{ds}}={\displaystyle \frac{\sqrt{6}\sqrt{{\kappa }^2+{\tau }^2-\kappa \tau }}{3}}$$

can be written [2].

3. Main Theorems and Results

In this section, the spinor formulations, mostly mentioned in physics terms, of Smarandache curves in Euclidean geometry ${\mathbb{E}}^3$ are obtained. Firstly, two spinors corresponding to the Frenet–Serret frames of any regular curve and its ($\mathbf{TN}$, $\mathbf{TB}$, $\mathbf{NB}$, $\mathbf{TNB}$)–Smarandache curve are considered. Then, the relationships between two spinors corresponding to the Frenet–Serret frames of the regular curve and its Smarandache curve are obtained. Consequently, a geometric interpretation of a physical expression (spinor) is given.

Now, let $\beta$ be a regular curve parametrized by the arc-length parameter s and Frenet frame of the curve $\beta$ be $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$ in ${\mathbb{E}}^3$. Moreover, assume that the spinor corresponding to the Frenet–Serret frame of this curve is $\psi$. The curve $\beta$ is considered as such unless otherwise stated. Therefore, this section can be divided into four subsections for spinor representations of $\mathbf{TN}$–, $\mathbf{TB}$–, $\mathbf{NB}$– and $\mathbf{TNB}$–Smarandache curves.

3.1. Spinor Formulation of $\mathbf{TN}$–Smarandache Curve

Definition 6.

Assume that the $\mathbf{TN}$–Smarandache curve of the regular curve β is ${\beta }_{TN}$ and $\left\{{\mathbf{T}}_{{\beta }_{TN}},{\mathbf{N}}_{{\beta }_{TN}},{\mathbf{B}}_{{\beta }_{TN}}\right\}$ is the Frenet–Serret frame of $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ of β. φ is given as the spinor corresponding to the Frenet–Serret frame of ${\beta }_{TN}$ in ${\mathbb{E}}^3$. In this case, the spinor formulations of the Frenet–Serret frame $\left\{{\mathbf{N}}_{{\beta }_{TN}},{\mathbf{B}}_{{\beta }_{TN}},{\mathbf{T}}_{{\beta }_{TN}}\right\}$ are

$$\begin{array}{c}{\mathbf{T}}_{{\beta }_{TN}}=\left({\phi }_1\overline{{\phi }_2}+\overline{{\phi }_1}{\phi }_2,i\left({\phi }_1\overline{{\phi }_2}-\overline{{\phi }_1}{\phi }_2\right),{\left|{\phi }_1\right|}^2-{\left|{\phi }_2\right|}^2\right),\hfill \\ {\mathbf{N}}_{{\beta }_{TN}}={\displaystyle \frac{1}{2}}\left({\phi }_1^2-{\phi }_2^2-{\overline{{\phi }_2}}^2+{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2-{\overline{{\phi }_1}}^2-{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2-2\overline{{\phi }_1{\phi }_2}\right),\hfill \\ {\mathbf{B}}_{{\beta }_{TN}}=-{\displaystyle \frac{i}{2}}\left({\phi }_1^2-{\phi }_2^2+{\overline{{\phi }_2}}^2-{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2+{\overline{{\phi }_1}}^2+{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2+2\overline{{\phi }_1{\phi }_2}\right).\hfill \end{array}$$

(14)

Theorem 6.

Suppose that the $\mathbf{TN}$–Smarandache curve of β is ${\beta }_{TN}$ and $\left\{\mathbf{T},\mathbf{N},\mathbf{B}\right\}$, $\left\{{\mathbf{T}}_{{\beta }_{TN}},{\mathbf{N}}_{{\beta }_{TN}},{\mathbf{B}}_{{\beta }_{TN}}\right\}$ are Frenet–Serret frames of the curve β and its $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ in ${\mathbb{E}}^3$, respectively. Moreover, φ and ψ are considered the spinors corresponding to Frenet–Serret frames of ${\beta }_{TN}$ and β. Therefore, there are the following relationships between the spinors $\phi ={\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\end{array}\right]}_{2\times 1}$ and $\psi ={\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right]}_{2\times 1}$.

$${\phi }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}{\delta }_1\left(2\overline{{\psi }_2}\sqrt{2{\kappa }^2+{\tau }^2}+\kappa {\psi }_1^2+\kappa {\overline{{\psi }_1}}^2+i\tau {\psi }_1^2\right)\hfill \\ +{\delta }_2\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\left(\sqrt{2{\kappa }^2+{\tau }^2}+\kappa {\psi }_1^2+\kappa {\overline{{\psi }_1}}^2-2i\tau {\psi }_1{\psi }_2\right)\hfill \\ +i{\delta }_3\left(-\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\sqrt{2{\kappa }^2+{\tau }^2}+2\kappa {\psi }_1{\psi }_2-\kappa {\psi }_2^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}}$$

and

$${\phi }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}{\delta }_1\left(-2\overline{{\psi }_1}{\psi }_2+\kappa {\psi }_2^2+\kappa {\overline{{\psi }_2}}^2+\tau i{\psi }_1^2\right)\hfill \\ +{\delta }_2\left({\psi }_2^2-{\overline{{\psi }_1}}^2+\kappa {\psi }_2^2+\kappa {\overline{{\psi }_2}}^2-i\tau \left(2{\psi }_1\overline{{\psi }_2}\right)\right)\hfill \\ -i{\delta }_3\left({\psi }_2^2+{\overline{{\psi }_1}}^2-2\kappa {\psi }_1\overline{{\psi }_2}+\kappa \tau {\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}}$$

where

$$\begin{array}{c}{\delta }_1=-\left[{\kappa }^2\left(2{\kappa }^2+{\tau }^2\right)+\tau \left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right],\hfill \\ {\delta }_2=-\left[{\kappa }^2\left(2{\kappa }^2+3{\tau }^2\right)-\tau \left({\tau }^3+\kappa {\tau }^{\prime }-\tau {\kappa }^{\prime }\right)\right],\hfill \\ {\delta }_3=\kappa \left[\tau \left(2{\kappa }^2+{\tau }^2\right)-2\left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right]\hfill \end{array}$$

and κ, τ are the curvature and torsion of the curve β, respectively.

Proof. 

Assume that $\left\{{\mathbf{T}}_{{\beta }_{TN}},{\mathbf{N}}_{{\beta }_{TN}},{\mathbf{B}}_{{\beta }_{TN}}\right\}$ is a Frenet–Serret frame of the $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ in ${\mathbb{E}}^3$ and $\psi$, $\phi$ are the spinors corresponding to Frenet–Serret frames of $\beta$, ${\beta }_{TN}$, respectively. Then, with the aid of Equations (1) and (3), we can write

$${\mathbf{N}}_{{\beta }_{TN}}={\displaystyle \frac{\left({\delta }_{1}A+\frac{{\delta }_2}{2}B-\frac{i{\delta }_3}{2}C\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}$$

(15)

where

$$\begin{array}{c}A=\left({\phi }_1\overline{{\phi }_2}+\overline{{\phi }_1}{\phi }_2,i\left({\phi }_1\overline{{\phi }_2}-\overline{{\phi }_1}{\phi }_2\right),{\left|{\phi }_1\right|}^2-{\left|{\phi }_2\right|}^2\right),\hfill \\ B=\left({\phi }_1^2-{\phi }_2^2-{\overline{{\phi }_2}}^2+{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2-{\overline{{\phi }_1}}^2-{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2-2\overline{{\phi }_1{\phi }_2}\right),\hfill \\ C=\left({\phi }_1^2-{\phi }_2^2+{\overline{{\phi }_2}}^2-{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2+{\overline{{\phi }_1}}^2+{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2+2\overline{{\phi }_1{\phi }_2}\right).\hfill \end{array}$$

If we use Equations (14) and (15), then we obtain the following equalities for spinors $\psi$ and $\phi$:

$${\displaystyle \frac{1}{2}}\left({\phi }_1^2-{\phi }_2^2-{\overline{{\phi }_2}}^2+{\overline{{\phi }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\frac{i{\delta }_1}{2}\left({\psi }_1^2-{\psi }_2^2-{\psi }_1{\overline{{\psi }_2}}^2-\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +{\delta }_2\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\frac{{\delta }_3}{2}\left({\psi }_1^2-{\psi }_2^2+{\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \end{array}\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}},$$

(16)

$${\displaystyle \frac{i}{2}}\left({\phi }_1^2+{\phi }_2^2-{\overline{{\phi }_2}}^2-{\overline{{\phi }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}{\delta }_1\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)+\frac{{\delta }_2}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ -i\frac{{\delta }_3}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}},$$

(17)

$${\phi }_1{\phi }_2-{\overline{{\phi }_1}}^2{\overline{{\phi }_2}}^2={\displaystyle \frac{{\delta }_1\left({\left|{\psi }_1\right|}^2-{\left|{\psi }_2\right|}^2\right)+{\delta }_2\left(-{\psi }_1{\psi }_2-\overline{{\psi }_1}\overline{{\psi }_2}\right)+{\delta }_{3}i\left({\psi }_1{\psi }_2-\overline{{\psi }_1}\overline{{\psi }_2}\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}.$$

Similarly, we consider Equations (1) and (4). We can write the normal vector ${\mathbf{B}}_{{\beta }_{TN}}$ of the $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ as

$${\mathbf{B}}_{{\beta }_{TN}}={\displaystyle \frac{\left(\begin{array}{c}\left(\kappa {\delta }_3-\tau {\delta }_2\right)\left({\phi }_1\overline{{\phi }_2}+\overline{{\phi }_1}{\phi }_2,i\left({\phi }_1\overline{{\phi }_2}-\overline{{\phi }_1}{\phi }_2\right),{\left|{\phi }_1\right|}^2-{\left|{\phi }_2\right|}^2\right)\hfill \\ +\frac{\left(\tau {\delta }_1+\kappa {\delta }_3\right)}{2}\left({\phi }_1^2-{\phi }_2^2-{\overline{{\phi }_2}}^2+{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2-{\overline{{\phi }_1}}^2-{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2-2\overline{{\phi }_1{\phi }_2}\right)\hfill \\ -\frac{i\left(\kappa {\delta }_2+\kappa {\delta }_1\right)}{2}\left({\phi }_1^2-{\phi }_2^2+{\overline{{\phi }_2}}^2-{\overline{{\phi }_1}}^2,i\left({\phi }_1^2+{\phi }_2^2+{\overline{{\phi }_1}}^2+{\overline{{\phi }_2}}^2\right),-2{\phi }_1{\phi }_2+2\overline{{\phi }_1{\phi }_2}\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}.$$

(18)

Therefore, considering Equations (14) and (18), we can easily obtain some equalities as

$$-{\displaystyle \frac{i}{2}}\left({\phi }_1^2-{\phi }_2^2+{\overline{{\phi }_2}}^2-{\overline{{\phi }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\left(\kappa {\delta }_3-\tau {\delta }_2\right)\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)+\left(\tau {\delta }_1+\kappa {\delta }_3\right)\frac{1}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\left(-\frac{i}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\right)\hfill \end{array}\right)}{\left(\sqrt{2{\kappa }^2+{\tau }^2}\right)\left(\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}\right)}},$$

(19)

$${\displaystyle \frac{1}{2}}\left({\phi }_1^2+{\phi }_2^2+{\overline{{\phi }_1}}^2+{\overline{{\phi }_2}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\left(\kappa {\delta }_3-\tau {\delta }_2\right)i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2+\right)+\left(\tau {\delta }_1+\kappa {\delta }_3\right)\frac{i}{2}\left({\psi }_1^2+{\psi }_2^2-{\overline{{\psi }_1}}^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_1}}^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{\left(\sqrt{2{\kappa }^2+{\tau }^2}\right)\left(\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}\right)}},$$

(20)

$$i\left({\phi }_1{\phi }_2-\overline{{\phi }_1}\overline{{\phi }_2}\right)={\displaystyle \frac{\left(\begin{array}{c}\left(\kappa {\delta }_3-\tau {\delta }_2\right)\left({\left|{\psi }_1\right|}^2-{\left|{\psi }_2\right|}^2\right)+\left(\tau {\delta }_1+\kappa {\delta }_3\right)\left(-{\psi }_1{\psi }_2-\overline{{\psi }_1}\overline{{\psi }_2}\right)\hfill \\ +i\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\left({\psi }_1{\psi }_2-\overline{{\psi }_1}\overline{{\psi }_2}\right)\hfill \end{array}\right)}{\left(\sqrt{2{\kappa }^2+{\tau }^2}\right)\left(\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}\right)}}.$$

Now, we use Equations (16) and (17). Therefore, the spinor equation $({\phi }_1^2-{\overline{{\phi }_2}}^2)$ can be obtained as

$${\phi }_1^2-{\overline{{\phi }_2}}^2={\displaystyle \frac{{\delta }_1\left(2{\psi }_1\overline{{\psi }_2}\right)+{\delta }_2(\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+(-i{\delta }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}.$$

(21)

Similarly, from Equations (19) and (20), we can easily obtain the equality $({\phi }_1^2+{\overline{{\phi }_2}}^2)$ as

$${\phi }_1^2+{\overline{{\phi }_2}}^2={\displaystyle \frac{i\left(\kappa {\delta }_3-\tau {\delta }_2\right)\left(2{\psi }_1\overline{{\psi }_2}\right)+i\left(\tau {\delta }_1+\kappa {\delta }_3\right)\left({\psi }_1^2-{\psi }_2^2\right)+\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}.$$

(22)

Consequently, if Equations (21) and (22) are added together, the first component ${\phi }_1$ of spinor $\phi$ corresponding to Frenet–Serret frame $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ is obtained as

$${\phi }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}{\delta }_1\left(2\overline{{\psi }_2}\sqrt{2{\kappa }^2+{\tau }^2}+\kappa {\psi }_1^2+\kappa {\overline{{\psi }_1}}^2+i\tau {\psi }_1^2\right)\hfill \\ +{\delta }_2\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\left(\sqrt{2{\kappa }^2+{\tau }^2}+\kappa {\psi }_1^2+\kappa {\overline{{\psi }_1}}^2-2i\tau {\psi }_1{\psi }_2\right)\hfill \\ +i{\delta }_3\left(-\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\sqrt{2{\kappa }^2+{\tau }^2}+2\kappa {\psi }_1{\psi }_2-\kappa {\psi }_2^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}}.$$

Now, we obtain the second component ${\phi }_2$ of spinor $\phi$ corresponding to Frenet–Serret frame $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$. First of all, if we take Equations (16), (17), (19) and (20) then we obtain easily the following equations as

$$-{\phi }_2^2+{\overline{{\phi }_1}}^2={\displaystyle \frac{{\delta }_1\left(2\overline{{\psi }_1}{\psi }_2\right)+{\delta }_2\left(-{\psi }_2^2+{\overline{{\psi }_1}}^2\right)+{\delta }_{3}i\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)}{\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}$$

(23)

and

$${\phi }_2^2+{\overline{{\phi }_1}}^2={\displaystyle \frac{\left(\begin{array}{c}i\left(\kappa {\delta }_3-\tau {\delta }_2\right)\left(2{\psi }_1\overline{{\psi }_2}\right)+\tau i\left({\delta }_1+\kappa {\delta }_3\right)\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\kappa {\delta }_2+\kappa {\delta }_1\right)\left({\psi }_2^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}.$$

(24)

Consequently, if Equations (23) and (24) are considered, the second component ${\phi }_2$ of spinor $\phi$ corresponding to Frenet–Serret frame $\mathbf{TN}$–Smarandache curve ${\beta }_{TN}$ is obtained as

$${\phi }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}{\delta }_1\left(-2\overline{{\psi }_1}{\psi }_2+\kappa {\psi }_2^2+\kappa {\overline{{\psi }_2}}^2+\tau i{\psi }_1^2\right)\hfill \\ +{\delta }_2\left({\psi }_2^2-{\overline{{\psi }_1}}^2+\kappa {\psi }_2^2+\kappa {\overline{{\psi }_2}}^2-i\tau \left(2{\psi }_1\overline{{\psi }_2}\right)\right)\hfill \\ -i{\delta }_3\left({\psi }_2^2+{\overline{{\psi }_1}}^2-2\kappa {\psi }_1\overline{{\psi }_2}+\kappa \tau {\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+{\tau }^2}\sqrt{{\delta }_1^2+{\delta }_2^2+{\delta }_3^2}}}}$$

where

$$\begin{array}{c}{\delta }_1=-\left[{\kappa }^2\left(2{\kappa }^2+{\tau }^2\right)+\tau \left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right],\hfill \\ {\delta }_2=-\left[{\kappa }^2\left(2{\kappa }^2+3{\tau }^2\right)-\tau \left({\tau }^3+\kappa {\tau }^{\prime }-\tau {\kappa }^{\prime }\right)\right],\hfill \\ {\delta }_3=\kappa \left[\tau \left(2{\kappa }^2+{\tau }^2\right)-2\left(\tau {\kappa }^{\prime }-\kappa {\tau }^{\prime }\right)\right].\hfill \end{array}$$

□

3.2. Spinor Formulation of $\mathbf{TB}$–Smarandache Curve

Definition 7.

Let $\mathbf{TB}$–Smarandache curve of β be ${\beta }_{TB}$ and $\left\{{\mathbf{T}}_{{\beta }_{TB}},{\mathbf{N}}_{{\beta }_{TB}},{\mathbf{B}}_{{\beta }_{TB}}\right\}$ be Frenet–Serret frame of $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ of β. Then, ξ is given as the spinor corresponding to Frenet–Serret frame $\left\{{\mathbf{N}}_{{\beta }_{TB}},{\mathbf{B}}_{{\beta }_{TB}},{\mathbf{T}}_{{\beta }_{TB}}\right\}$ of ${\beta }_{TB}$. Therefore, the spinor formulations of the Frenet–Serret frame are

$$\begin{array}{c}{\mathbf{T}}_{{\beta }_{TB}}=\left({\xi }_1\overline{{\xi }_2}+\overline{{\xi }_1}{\xi }_2,i\left({\xi }_1\overline{{\xi }_2}-\overline{{\xi }_1}{\xi }_2\right),{\left|{\xi }_1\right|}^2-{\left|{\xi }_2\right|}^2\right),\hfill \\ {\mathbf{N}}_{{\beta }_{TB}}={\displaystyle \frac{1}{2}}\left({\xi }_1^2-{\xi }_2^2-{\overline{{\xi }_2}}^2+{\overline{{\xi }_1}}^2,i\left({\xi }_1^2+{\xi }_2^2-{\overline{{\xi }_1}}^2-{\overline{{\xi }_2}}^2\right),-2{\xi }_1{\xi }_2-2\overline{{\xi }_1{\xi }_2}\right),\hfill \\ {\mathbf{B}}_{{\beta }_{TB}}=-{\displaystyle \frac{i}{2}}\left({\xi }_1^2-{\xi }_2^2+{\overline{{\xi }_2}}^2-{\overline{{\xi }_1}}^2,i\left({\xi }_1^2+{\xi }_2^2+{\overline{{\xi }_1}}^2+{\overline{{\xi }_2}}^2\right),-2{\xi }_1{\xi }_2+2\overline{{\xi }_1{\xi }_2}\right).\hfill \end{array}$$

(25)

Theorem 7.

Suppose that $\mathbf{TB}$–Smarandache curve of β is ${\beta }_{TB}$ and $\left\{{\mathbf{T}}_{{\beta }_{TB}},{\mathbf{N}}_{{\beta }_{TB}},{\mathbf{B}}_{{\beta }_{TB}}\right\}$ is Frenet–Serret frame of $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ in ${\mathbb{E}}^3$, respectively. Moreover, ξ, ψ are considered the spinors corresponding to Frenet–Serret frames of ${\beta }_{TB}$, β. Therefore, there are the following relationships between the spinors $\xi ={\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]}_{2\times 1}$ and $\psi ={\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right]}_{2\times 1}$

$${\xi }_1=\pm \sqrt{{\displaystyle \frac{\left(-2\kappa +2\tau i\right)\left({\psi }_1\overline{{\psi }_2}\right)+\left(-\tau i+\kappa i\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}}$$

and

$${\xi }_2=\pm \sqrt{{\displaystyle \frac{\left(\kappa -\tau i\right)\left(2\overline{{\psi }_1}{\psi }_2+{\psi }_1^2+{\psi }_2^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}}.$$

Proof. 

Assume that $\xi$ and $\psi$ are the spinors corresponding to Frenet–Serret frames $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$ and $\left\{{\mathbf{N}}_{{\beta }_{TB}},{\mathbf{B}}_{{\beta }_{TB}},{\mathbf{T}}_{{\beta }_{TB}}\right\}$ of ${\beta }_{TB}$ and $\beta$ in ${\mathbb{E}}^3$, respectively. Then, with the aid of Equations (1) and (6), we can write

$${\mathbf{N}}_{{\beta }_{TB}}={\displaystyle \frac{\left(\begin{array}{c}-\kappa \left({\xi }_1\overline{{\xi }_2}+\overline{{\xi }_1}{\xi }_2,i\left({\xi }_1\overline{{\xi }_2}-\overline{{\xi }_1}{\xi }_2\right),{\left|{\xi }_1\right|}^2-{\left|{\xi }_2\right|}^2\right)\hfill \\ -\frac{i\tau }{2}\left({\xi }_1^2-{\xi }_2^2+{\overline{{\xi }_2}}^2-{\overline{{\xi }_1}}^2,i\left({\xi }_1^2+{\xi }_2^2+{\overline{{\xi }_1}}^2+{\overline{{\xi }_2}}^2\right),-2{\xi }_1{\xi }_2+2\overline{{\xi }_1{\xi }_2}\right)\hfill \end{array}\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}.$$

(26)

If we use Equations (25) and (26), then we obtain the following equalities for spinors $\xi$ and $\psi$:

$${\displaystyle \frac{1}{2}}\left({\xi }_1^2-{\xi }_2^2-{\overline{{\xi }_2}}^2+{\overline{{\xi }_1}}^2\right)={\displaystyle \frac{1}{\sqrt{{\kappa }^2+{\tau }^2}}}\left(\begin{array}{c}-\kappa i\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\tau \left(-{\displaystyle \frac{i}{2}}\right)\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right),$$

(27)

$${\displaystyle \frac{i}{2}}\left({\xi }_1^2+{\xi }_2^2-{\overline{{\xi }_2}}^2-{\overline{{\xi }_1}}^2\right)={\displaystyle \frac{1}{\sqrt{{\kappa }^2+{\tau }^2}}}\left(\begin{array}{c}-\kappa i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\tau \left(-{\displaystyle \frac{1}{2}}\right)\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)$$

(28)

Similarly, we consider Equations (7) and (25). We can write the normal vector ${\mathbf{B}}_{{\beta }_{TB}}$ of $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ as

$${\mathbf{B}}_{{\beta }_{TB}}={\displaystyle \frac{\left(\begin{array}{c}\tau \left({\xi }_1\overline{{\xi }_2}+\overline{{\xi }_1}{\xi }_2,i\left({\xi }_1\overline{{\xi }_2}-\overline{{\xi }_1}{\xi }_2\right),{\left|{\xi }_1\right|}^2-{\left|{\xi }_2\right|}^2\right)\hfill \\ +\kappa -\frac{i}{2}\left({\xi }_1^2-{\xi }_2^2+{\overline{{\xi }_2}}^2-{\overline{{\xi }_1}}^2,i\left({\xi }_1^2+{\xi }_2^2+{\overline{{\xi }_1}}^2+{\overline{{\xi }_2}}^2\right),-2{\xi }_1{\xi }_2+2\overline{{\xi }_1{\xi }_2}\right)\hfill \end{array}\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}$$

(29)

and considering Equations (25) and (29),

$$-{\displaystyle \frac{i}{2}}\left({\xi }_1^2-{\xi }_2^2+{\overline{{\xi }_2}}^2-{\overline{{\xi }_2}}^2\right)={\displaystyle \frac{1}{\sqrt{{\kappa }^2+{\tau }^2}}}\left(\begin{array}{c}\tau \left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\kappa \left(-{\displaystyle \frac{i}{2}}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\right)\hfill \end{array}\right),$$

(30)

$$-{\displaystyle \frac{1}{2}}\left({\xi }_1^2+{\xi }_2^2+{\overline{{\xi }_2}}^2+{\overline{{\xi }_2}}^2\right)={\displaystyle \frac{1}{\sqrt{{\kappa }^2+{\tau }^2}}}\left(\begin{array}{c}\tau i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +{\displaystyle \frac{1}{2}}\kappa \left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right).$$

(31)

Now, if we use Equations (27), (28), (30) and (31), then the spinor equality $({\xi }_1^2-{\overline{{\xi }_2}}^2)$ and $({\xi }_1^2+{\overline{{\xi }_2}}^2)$ can be obtained as

$${\xi }_1^2-{\overline{{\xi }_2}}^2={\displaystyle \frac{-2\kappa {\psi }_1\overline{{\psi }_2}+\tau \left(-i\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}$$

(32)

and

$${\xi }_1^2+{\overline{{\xi }_2}}^2={\displaystyle \frac{2\tau i{\psi }_1\overline{{\psi }_2}+\kappa i\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}.$$

(33)

If Equations (32) and (33) are considered, then the first component ${\xi }_1$ of spinor $\xi$ corresponding to Frenet–Serret frame $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ is obtained as

$${\xi }_1=\pm \sqrt{{\displaystyle \frac{\left(-2\kappa +2\tau i\right)\left({\psi }_1\overline{{\psi }_2}\right)+\left(-\tau i+\kappa i\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}}.$$

Now, we obtain the second component ${\xi }_2$ of spinor $\xi$ corresponding to the Frenet–Serret frame $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$. First of all, if we take Equations (27), (28), (30) and (31), then we easily obtain the following equations as

$${\xi }_2^2-{\overline{{\xi }_1}}^2={\displaystyle \frac{2\kappa \overline{{\psi }_1}{\psi }_2-\tau i\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}$$

(34)

and

$${\xi }_2^2+{\overline{{\xi }_1}}^2={\displaystyle \frac{\tau i\left(-2\overline{{\psi }_1}{\psi }_2\right)+\kappa \left({\psi }_1^2+{\psi }_2^2\right)}{\sqrt{{\kappa }^2+{\tau }^2}}}.$$

(35)

Consequently, if Equations (34) and (35) are considered, the second component ${\xi }_2$ of spinor $\xi$ corresponding to the Frenet–Serret frame $\mathbf{TB}$–Smarandache curve ${\beta }_{TB}$ is obtained as

$${\xi }_2=\pm \sqrt{{\displaystyle \frac{\left(\kappa -\tau i\right)\left(2\overline{{\psi }_1}{\psi }_2+{\psi }_1^2+{\psi }_2^2\right)}{2\sqrt{{\kappa }^2+{\tau }^2}}}}.$$

□

3.3. Spinor Formulation of $\mathbf{NB}$–Smarandache Curve

Definition 8.

Assume that the $\mathbf{NB}$–Smarandache curve of β is ${\beta }_{NB}$ and $\left\{{\mathbf{T}}_{{\beta }_{TB}},{\mathbf{N}}_{{\beta }_{NB}},{\mathbf{B}}_{{\beta }_{NB}}\right\}$ is the Frenet–Serret frame of $\mathbf{NB}$–Smarandache curve ${\beta }_{NB}$ of the curve β in ${\mathbb{E}}^3$. Then, ϑ is given as the spinor corresponding to the Frenet–Serret frame $\left\{{\mathbf{N}}_{{\beta }_{NB}},{\mathbf{B}}_{{\beta }_{NB}},{\mathbf{T}}_{{\beta }_{NB}}\right\}$ of ${\beta }_{NB}$. Therefore, the spinor formulations of these Frenet–Serret vectors are

$$\begin{array}{c}{\mathbf{T}}_{{\beta }_{NB}}=\left({\vartheta }_1\overline{{\vartheta }_2}+\overline{{\vartheta }_1}{\vartheta }_2,i\left({\vartheta }_1\overline{{\vartheta }_2}-\overline{{\vartheta }_1}{\vartheta }_2\right),{\left|{\vartheta }_1\right|}^2-{\left|{\vartheta }_2\right|}^2\right),\hfill \\ {\mathbf{N}}_{{\beta }_{NB}}={\displaystyle \frac{1}{2}}\left({\vartheta }_1^2-{\vartheta }_2^2-{\overline{{\vartheta }_2}}^2+{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2-{\overline{{\vartheta }_1}}^2-{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2-2\overline{{\vartheta }_1{\vartheta }_2}\right),\hfill \\ {\mathbf{B}}_{{\beta }_{NB}}=-{\displaystyle \frac{i}{2}}\left({\vartheta }_1^2-{\vartheta }_2^2+{\overline{{\vartheta }_2}}^2-{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2+{\overline{{\vartheta }_1}}^2+{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2+2\overline{{\vartheta }_1{\vartheta }_2}\right).\hfill \end{array}$$

(36)

Theorem 8.

Suppose that the $\mathbf{NB}$–Smarandache curve of β is ${\beta }_{NB}$ and ϑ, ψ are the spinors corresponding to Frenet–Serret frames $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$, $\left\{{\mathbf{N}}_{{\beta }_{NB}},{\mathbf{B}}_{{\beta }_{NB}},{\mathbf{T}}_{{\beta }_{NB}}\right\}$ of ${\beta }_{NB}$, β in ${\mathbb{E}}^3$. Therefore, there are the following relationships between the spinors $\vartheta ={\left[\begin{array}{c}{\vartheta }_1\\ {\vartheta }_2\end{array}\right]}_{2\times 1}$ and $\psi ={\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right]}_{2\times 1}$ such that

$${\vartheta }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\left({\mu }_1\left({\psi }_1\overline{{\psi }_2}\right)+{\mu }_2\left({\psi }_1^2-2{\overline{{\psi }_2}}^2\right)+{\mu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\right)\left(\sqrt{{\kappa }^2+2{\tau }^2}\right)+\hfill \\ \left(-\tau {\mu }_2-\tau {\mu }_3\right)i\left(2{\psi }_1{\psi }_2\right)+\left(\tau {\mu }_1+\kappa {\mu }_3\right)i\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{2\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}}$$

and

$${\vartheta }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\tau \left({\mu }_2-\tau {\mu }_3\right)i\left(-2\overline{{\psi }_1}{\psi }_2\right)+\left(\tau {\mu }_1+\kappa {\mu }_3\right)i\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)-{\mu }_1\left(2\overline{{\psi }_1}{\psi }_2\right)+{\mu }_2\left(-{\psi }_2^2+{\overline{{\psi }_1}}^2\right)+{\mu }_{3}i\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{2\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}}$$

where

$$\begin{array}{c}{\mu }_1=\kappa \tau \left(4\kappa \left(\kappa -\tau \right)+2\left({\tau }^{\prime }+{\tau }^2\right)+{\kappa }^{\prime }\right)-{\kappa }^2\left(2{\kappa }^2+{\tau }^{\prime }\right)-2{\kappa }^{\prime }{\tau }^2,\hfill \\ {\mu }_2=2\kappa \tau \left({\left(\kappa -\tau \right)}^2+2\tau -2{\tau }^{\prime }\right)-2\left({\kappa }^4+{\tau }^4\right)+{\kappa }^{\prime }{\tau }^2-{\kappa }^2{\tau }^{\prime },\hfill \\ {\mu }_3=\tau \left(2\kappa \left({\kappa }^2+4{\tau }^2-{\kappa }^{\prime }-{\tau }^{\prime }-2\kappa \tau \right)+\left(\tau {\kappa }^{\prime }+{\tau }^{\prime }-2{\tau }^3\right)\right).\hfill \end{array}$$

Proof. 

Let $\vartheta$, $\psi$ be the spinors corresponding to Frenet–Serret frames $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$, $\left\{{\mathbf{N}}_{{\beta }_{NB}},{\mathbf{B}}_{{\beta }_{NB}},{\mathbf{T}}_{{\beta }_{NB}}\right\}$ of ${\beta }_{NB}$, $\beta$ in ${\mathbb{E}}^3$. Then, with the aid of Equations (9) and (36), we can write

$${\mathbf{N}}_{{\beta }_{NB}}={\displaystyle \frac{\left(\begin{array}{c}{\mu }_1\left({\vartheta }_1\overline{{\vartheta }_2}+\overline{{\vartheta }_1}{\vartheta }_2,i\left({\vartheta }_1\overline{{\vartheta }_2}-\overline{{\vartheta }_1}{\vartheta }_2\right),{\left|{\vartheta }_1\right|}^2-{\left|{\vartheta }_2\right|}^2\right)\hfill \\ +\frac{{\mu }_2}{2}\left({\vartheta }_1^2-{\vartheta }_2^2-{\overline{{\vartheta }_2}}^2+{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2-{\overline{{\vartheta }_1}}^2-{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2-2\overline{{\vartheta }_1{\vartheta }_2}\right)\hfill \\ -\frac{i{\mu }_3}{2}\left({\vartheta }_1^2-{\vartheta }_2^2+{\overline{{\vartheta }_2}}^2-{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2+{\overline{{\vartheta }_1}}^2+{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2+2\overline{{\vartheta }_1{\vartheta }_2}\right)\hfill \end{array}\right)}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}.$$

(37)

If we use Equations (36) and (37), then we obtain the equations for spinors $\psi$ and $\vartheta$:

$${\displaystyle \frac{1}{2}}\left({\vartheta }_1^2-{\vartheta }_2^2-{\overline{{\vartheta }_1}}^2+{\overline{{\vartheta }_2}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}{\mu }_1\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)+\frac{{\mu }_2}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ -\frac{i{\mu }_3}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}},$$

(38)

$${\displaystyle \frac{i}{2}}\left({\vartheta }_1^2-{\vartheta }_2^2-{\overline{{\vartheta }_1}}^2+{\overline{{\vartheta }_2}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}{\mu }_1\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)+\frac{{\mu }_2}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ -\frac{i{\mu }_3}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}$$

(39)

and considering Equations (10) and (36), the normal vector ${\mathbf{B}}_{{\beta }_{NB}}$ of $\mathbf{NB}$–Smarandache curve ${\beta }_{NB}$ can be obtained as

$${\mathbf{B}}_{{\beta }_{NB}}={\displaystyle \frac{\left(\begin{array}{c}\left(-\tau {\mu }_2-\tau {\mu }_3\right)\left({\vartheta }_1\overline{{\vartheta }_2}+\overline{{\vartheta }_1}{\vartheta }_2,i\left({\vartheta }_1\overline{{\vartheta }_2}-\overline{{\vartheta }_1}{\vartheta }_2\right),{\left|{\vartheta }_1\right|}^2-{\left|{\vartheta }_2\right|}^2\right)\hfill \\ +\left(\frac{\tau {\mu }_1+\kappa {\mu }_3}{2}\right)\left({\vartheta }_1^2-{\vartheta }_2^2-{\overline{{\vartheta }_2}}^2+{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2-{\overline{{\vartheta }_1}}^2-{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2-2\overline{{\vartheta }_1{\vartheta }_2}\right)\hfill \\ -\frac{i\left(\tau {\mu }_1-\kappa {\mu }_2\right)}{2}\left({\vartheta }_1^2-{\vartheta }_2^2+{\overline{{\vartheta }_2}}^2-{\overline{{\vartheta }_1}}^2,i\left({\vartheta }_1^2+{\vartheta }_2^2+{\overline{{\vartheta }_1}}^2+{\overline{{\vartheta }_2}}^2\right),-2{\vartheta }_1{\vartheta }_2+2\overline{{\vartheta }_1{\vartheta }_2}\right)\hfill \end{array}\right)}{\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}.$$

(40)

Then, we obtain

$$-{\displaystyle \frac{i}{2}}\left({\vartheta }_1^2-{\vartheta }_2^2+{\overline{{\vartheta }_2}}^2-{\overline{{\vartheta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}\left(-\tau {\mu }_2-\tau {\mu }_3\right)\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\frac{\left(\tau {\mu }_1+\kappa {\mu }_3\right)}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\hfill \\ -\frac{i\left(\tau {\mu }_1-\kappa {\mu }_2\right)}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\left(\sqrt{{\kappa }^2+2{\tau }^2}\right)\left(\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}\right)}},$$

(41)

$${\displaystyle \frac{1}{2}}\left({\vartheta }_1^2+{\vartheta }_2^2+{\overline{{\vartheta }_2}}^2+{\overline{{\vartheta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}\left(-\tau {\mu }_2-\tau {\mu }_3\right)i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\frac{i\left(\tau {\mu }_1+\kappa {\mu }_3\right)}{2}\left({\psi }_1^2+{\psi }_2^2-{\overline{{\psi }_1}}^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\hfill \\ +\frac{\left(\tau {\mu }_1-\kappa {\mu }_2\right)}{2}\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\left(\sqrt{{\kappa }^2+2{\tau }^2}\right)\left(\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}\right)}}.$$

(42)

Now, we use Equations (38) and (39). The spinor equality $({\vartheta }_1^2-{\overline{{\vartheta }_2}}^2)$ can be obtained as

$${\vartheta }_1^2-{\overline{{\vartheta }_2}}^2={\displaystyle \frac{{\mu }_1\left({\psi }_1\overline{{\psi }_2}\right)+{\mu }_2\left({\psi }_1^2-2{\overline{{\psi }_2}}^2\right)+{\mu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}.$$

(43)

Similarly, from Equations (41) and (42), we can easily obtain the equality $({\vartheta }_1^2+{\overline{{\vartheta }_2}}^2)$ as

$${\vartheta }_1^2+{\overline{{\vartheta }_2}}^2={\displaystyle \frac{\left(\begin{array}{c}\left(-\tau {\mu }_2-\tau {\mu }_3\right)i\left(2{\psi }_1{\psi }_2\right)\hfill \\ +\left(\tau {\mu }_1+\kappa {\mu }_3\right)i\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}.$$

(44)

Consequently, if Equations (43) and (44) are added together, the first component ${\vartheta }_1$ of spinor $\vartheta$ corresponding to the Frenet–Serret frame $\mathbf{NB}$–Smarandache curve ${\beta }_{NB}$ is obtained as

$${\vartheta }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}\left({\mu }_1\left({\psi }_1\overline{{\psi }_2}\right)+{\mu }_2\left({\psi }_1^2-2{\overline{{\psi }_2}}^2\right)+{\mu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\right)\left(\sqrt{{\kappa }^2+2{\tau }^2}\right)\hfill \\ +2i\left(-\tau {\mu }_2-\tau {\mu }_3\right){\psi }_1{\psi }_2+i\left(\tau {\mu }_1+\kappa {\mu }_3\right)\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\hfill \end{array}\hfill \\ +\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{2\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}}.$$

Now, we obtain the second component ${\vartheta }_2$ of spinor $\vartheta$ corresponding to Frenet–Serret frame $\mathbf{NB}$–Smarandache curve ${\beta }_{NB}$. First of all, if we take Equations (38), (39), (41) and (42), then we obtain easily the following equations as

$${\vartheta }_2^2-{\overline{{\vartheta }_1}}^2={\displaystyle \frac{-{\mu }_1\left(2\overline{{\psi }_1}{\psi }_2\right)-{\mu }_2\left(-{\psi }_2^2+{\overline{{\psi }_1}}^2\right)-i{\mu }_3\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)}{\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}$$

(45)

and

$${\vartheta }_2^2+{\overline{{\vartheta }_1}}^2={\displaystyle \frac{\left(\begin{array}{c}-2i\left(\tau {\delta }_2-\tau {\mu }_3\right)\left(\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +i\left(\tau {\mu }_1+\kappa {\mu }_3\right)\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}.$$

(46)

Consequently, if Equations (45) and (46) are considered, the second component ${\vartheta }_2$ of spinor $\vartheta$ corresponding to Frenet–Serret frame $\mathbf{NB}$–Smarandache curve ${\beta }_{NB}$ is obtained as

$${\vartheta }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}-2i\tau \left({\mu }_2-\tau {\mu }_3\right)\overline{{\psi }_1}{\psi }_2+i\left(\tau {\mu }_1+\kappa {\mu }_3\right)\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\tau {\mu }_1-\kappa {\mu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)-{\mu }_1\left(2\overline{{\psi }_1}{\psi }_2\right)\hfill \end{array}\hfill \\ +{\mu }_2\left(-{\psi }_2^2+{\overline{{\psi }_1}}^2\right)+i{\mu }_3\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{2\sqrt{{\kappa }^2+2{\tau }^2}\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2}}}}$$

where

$$\begin{array}{c}{\mu }_1=\kappa \tau \left(4\kappa \left(\kappa -\tau \right)+2\left({\tau }^{\prime }+{\tau }^2\right)+{\kappa }^{\prime }\right)-{\kappa }^2\left(2{\kappa }^2+{\tau }^{\prime }\right)-2{\kappa }^{\prime }{\tau }^2,\hfill \\ {\mu }_2=2\kappa \tau \left({\left(\kappa -\tau \right)}^2+2\tau -2{\tau }^{\prime }\right)-2\left({\kappa }^4+{\tau }^4\right)+{\kappa }^{\prime }{\tau }^2-{\kappa }^2{\tau }^{\prime },\hfill \\ {\mu }_3=\tau \left(2\kappa \left({\kappa }^2+4{\tau }^2-{\kappa }^{\prime }-{\tau }^{\prime }-2\kappa \tau \right)+\left(\tau {\kappa }^{\prime }+{\tau }^{\prime }-2{\tau }^3\right)\right).\hfill \end{array}$$

□

3.4. Spinor Formulation of $\mathbf{TNB}$–Smarandache Curve

Definition 9.

Assume that the $\mathbf{TNB}$–Smarandache curve of β is ${\beta }_{TNB}$ and Frenet–Serret frame of $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$ of β is $\left\{{\mathbf{T}}_{{\beta }_{TB}},{\mathbf{N}}_{{\beta }_{TNB}},{\mathbf{B}}_{{\beta }_{TNB}}\right\}$ in ${\mathbb{E}}^3$. Then, ζ is given as the spinor corresponding to Frenet–Serret frame $\left\{{\mathbf{N}}_{{\beta }_{TNB}},{\mathbf{B}}_{{\beta }_{TNB}},{\mathbf{T}}_{{\beta }_{TNB}}\right\}$ of ${\beta }_{TNB}$. Therefore, the spinor formulations of Frenet–Serret vectors are

$$\begin{array}{c}{\mathbf{T}}_{{\beta }_{TBN}}=\left({\zeta }_1\overline{{\zeta }_2}+\overline{{\zeta }_1}{\zeta }_2,i\left({\zeta }_1\overline{{\zeta }_2}-\overline{{\zeta }_1}{\zeta }_2\right),{\left|{\zeta }_1\right|}^2-{\left|{\zeta }_2\right|}^2\right),\hfill \\ {\mathbf{N}}_{{\beta }_{TBN}}={\displaystyle \frac{1}{2}}\left({\zeta }_1^2-{\zeta }_2^2-{\overline{{\zeta }_2}}^2+{\overline{{\zeta }_1}}^2,i\left({\zeta }_1^2+{\zeta }_2^2-{\overline{{\zeta }_1}}^2-{\overline{{\zeta }_2}}^2\right),-2{\zeta }_1{\zeta }_2-2\overline{{\zeta }_1{\zeta }_2}\right),\hfill \\ {\mathbf{B}}_{{\beta }_{TBN}}=-{\displaystyle \frac{i}{2}}\left({\zeta }_1^2-{\zeta }_2^2+{\overline{{\zeta }_2}}^2-{\overline{{\zeta }_1}}^2,i\left({\zeta }_1^2+{\zeta }_2^2+{\overline{{\zeta }_1}}^2+{\overline{{\zeta }_2}}^2\right),-2{\zeta }_1{\zeta }_2+2\overline{{\zeta }_1{\zeta }_2}\right).\hfill \end{array}$$

(47)

Theorem 9.

Suppose that the $\mathbf{TNB}$–Smarandache curve of β is ${\beta }_{TNB}$ and $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$, $\left\{{\mathbf{N}}_{{\beta }_{TNB}},{\mathbf{B}}_{{\beta }_{TNB}},{\mathbf{T}}_{{\beta }_{TNB}}\right\}$ are Frenet–Serret frames of the curve β and its $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$ in ${\mathbb{E}}^3$, respectively. Moreover, ζ, ψ are considered as the spinors corresponding to Frenet–Serret frames of ${\beta }_{TNB}$, β. Therefore, there are the following relationships between the spinors $\zeta ={\left[\begin{array}{c}{\zeta }_1\\ {\zeta }_2\end{array}\right]}_{2\times 1}$ and $\psi ={\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right]}_{2\times 1}$

$${\zeta }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}\left({\nu }_1{\psi }_1\overline{{\psi }_2}+{\nu }_2\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+{\nu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\right)\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\hfill \\ +2i\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right){\psi }_1\overline{{\psi }_2}\hfill \end{array}\hfill \\ +i\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}}$$

and

$${\zeta }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\begin{array}{c}\left(-2{\nu }_1\overline{{\psi }_1}{\psi }_2+{\nu }_2\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)+{\nu }_3\left(-{\psi }_1^2-{\overline{{\psi }_2}}^2\right)\right)\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\hfill \\ +\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)\left(-2\overline{{\psi }_1}{\psi }_2\right)\hfill \end{array}\hfill \\ +\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)+\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}}$$

where

$$\begin{array}{c}{\nu }_1=\kappa \tau \left(4\kappa \left(\kappa -\tau \right)+2\left({\tau }^{\prime }+{\tau }^2\right)+{\kappa }^{\prime }\right)-{\kappa }^2\left(2{\kappa }^2+{\tau }^{\prime }\right)-2{\kappa }^{\prime }{\tau }^2,\hfill \\ {\nu }_2=2\kappa \tau \left({\left(\kappa -\tau \right)}^2+2\tau -2{\tau }^{\prime }\right)-2\left({\kappa }^4+{\tau }^4\right)+{\kappa }^{\prime }{\tau }^2-{\kappa }^2{\tau }^{\prime },\hfill \\ {\nu }_3=\tau \left(2\kappa \left({\kappa }^2+4{\tau }^2-{\kappa }^{\prime }-{\tau }^{\prime }-2\kappa \tau \right)+\left(\tau {\kappa }^{\prime }+{\tau }^{\prime }-2{\tau }^3\right)\right).\hfill \end{array}$$

Proof. 

Let $\zeta$, $\psi$ be the spinors corresponding to Frenet–Serret frames $\left\{\mathbf{N},\mathbf{B},\mathbf{T}\right\}$, $\left\{{\mathbf{N}}_{{\beta }_{TNB}},\right.$$\left.{\mathbf{B}}_{{\beta }_{TNB}},{\mathbf{T}}_{{\beta }_{TNB}}\right\}$ of ${\beta }_{TNB}$, $\beta$. Then, with the aid of Equations (12) and (47), we can write

$${\mathbf{N}}_{{\beta }_{TNB}}={\displaystyle \frac{\left(\begin{array}{c}{\nu }_1\left({\zeta }_1\overline{{\zeta }_2}+\overline{{\zeta }_1}{\zeta }_2,i\left({\zeta }_1\overline{{\zeta }_2}-\overline{{\zeta }_1}{\zeta }_2\right),{\left|{\zeta }_1\right|}^2-{\left|{\zeta }_2\right|}^2\right)\hfill \\ +\frac{{\nu }_2}{2}\left({\zeta }_1^2-{\zeta }_2^2-{\overline{{\zeta }_2}}^2+{\overline{{\zeta }_1}}^2,i\left({\zeta }_1^2+{\zeta }_2^2-{\overline{{\zeta }_1}}^2-{\overline{{\zeta }_2}}^2\right),-2{\zeta }_1{\zeta }_2-2\overline{{\zeta }_1{\zeta }_2}\right)\hfill \\ -\frac{i{\nu }_3}{2}\left({\zeta }_1^2-{\zeta }_2^2+{\overline{{\zeta }_2}}^2-{\overline{{\zeta }_1}}^2,i\left({\zeta }_1^2+{\zeta }_2^2+{\overline{{\zeta }_1}}^2+{\overline{{\zeta }_2}}^2\right),-2{\zeta }_1{\zeta }_2+2\overline{{\zeta }_1{\zeta }_2}\right)\hfill \end{array}\right)}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}.$$

(48)

If we use Equations (47) and (48), then we obtain the following equalities for spinors $\zeta$ and $\psi$

$${\displaystyle \frac{1}{2}}\left({\zeta }_1^2-{\zeta }_2^2-{\overline{{\zeta }_2}}^2+{\overline{{\zeta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}{v}_1\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)+\frac{{\nu }_2}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ -\frac{i{\nu }_3}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}},$$

(49)

$${\displaystyle \frac{i}{2}}\left({\zeta }_1^2+{\zeta }_2^2-{\overline{{\zeta }_2}}^2-{\overline{{\zeta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}{\nu }_1\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right)+\frac{i{\nu }_2}{2}\left({\psi }_1^2+{\psi }_2^2-{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\frac{{\nu }_3}{2}\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}$$

(50)

and from Equations (10) and (47)

$$-{\displaystyle \frac{i}{2}}\left({\zeta }_1^2-{\zeta }_2^2+{\overline{{\zeta }_2}}^2-{\overline{{\zeta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)\left({\psi }_1\overline{{\psi }_2}+\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\frac{\tau {\nu }_1+\kappa {\nu }_3}{2}\left({\psi }_1^2-{\psi }_2^2-{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \\ -\frac{i\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)}{2}\left({\psi }_1^2-{\psi }_2^2+{\overline{{\psi }_2}}^2-{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}$$

(51)

and

$${\displaystyle \frac{1}{2}}\left({\zeta }_1^2+{\zeta }_2^2+{\overline{{\zeta }_2}}^2+{\overline{{\zeta }_1}}^2\right)={\displaystyle \frac{\left(\begin{array}{c}\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)i\left({\psi }_1\overline{{\psi }_2}-\overline{{\psi }_1}{\psi }_2\right)\hfill \\ +\frac{i\left(\tau {\nu }_1+\kappa {\nu }_3\right)}{2}\left({\psi }_1^2+{\psi }_2^2-{\overline{{\psi }_1}}^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\frac{\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2}{2}\right)\left({\psi }_1^2+{\psi }_2^2+{\overline{{\psi }_2}}^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}.$$

(52)

Then, we can easily obtain

$${\zeta }_1^2-{\overline{{\zeta }_2}}^2={\displaystyle \frac{{\delta }_1{\psi }_1\overline{{\psi }_2}+{\nu }_2\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+{\nu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}$$

(53)

and

$${\zeta }_1^2+{\overline{{\zeta }_2}}^2={\displaystyle \frac{\left(\begin{array}{c}2i\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right){\psi }_1\overline{{\psi }_2}+i\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}.$$

(54)

Therefore, the first component ${\zeta }_1$ of spinor $\zeta$ corresponding to Frenet–Serret frame $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$ is obtained as

$${\zeta }_1=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\left(v_1{\psi }_1\overline{{\psi }_2}+{\nu }_2\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)+{\nu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\right)\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\hfill \\ \begin{array}{c}+2i\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right){\psi }_1\overline{{\psi }_2}+i\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_1^2-{\overline{{\psi }_2}}^2\right)\hfill \\ +\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)\hfill \end{array}\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}}.$$

Now, we obtain the second component ${\zeta }_2$ of spinor $\zeta$ corresponding to Frenet–Serret frame $\mathbf{TNB}$–Smarandache curve ${\beta }_{TNB}$. Therefore, we obtain

$${\zeta }_2^2-{\overline{{\zeta }_1}}^2={\displaystyle \frac{-2{\nu }_1\overline{{\psi }_1}{\psi }_2-{\nu }_2\left(-{\psi }_2^2+{\overline{{\psi }_1}}^2\right)-{\nu }_3\left({\psi }_1^2+{\overline{{\psi }_2}}^2\right)}{\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}$$

(55)

and

$${\zeta }_2^2+{\overline{{\zeta }_1}}^2={\displaystyle \frac{\left(\begin{array}{c}-2\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)\overline{{\psi }_1}{\psi }_2+\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\right)}{\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}$$

(56)

and finally

$${\zeta }_2=\pm \sqrt{{\displaystyle \frac{\left(\begin{array}{c}\left(-2{\nu }_1\overline{{\psi }_1}{\psi }_2+{\nu }_2\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)+{\nu }_3\left(-{\psi }_1^2-{\overline{{\psi }_2}}^2\right)\right)\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\hfill \\ \begin{array}{c}-2\left(\left(\kappa -\tau \right){\nu }_3-\tau {\nu }_2\right)\overline{{\psi }_1}{\psi }_2+\left(\tau {\nu }_1+\kappa {\nu }_3\right)\left({\psi }_2^2-{\overline{{\psi }_1}}^2\right)\hfill \\ +\left(\left(\tau -\kappa \right){\nu }_1-\kappa {\nu }_2\right)\left({\psi }_2^2+{\overline{{\psi }_1}}^2\right)\hfill \end{array}\hfill \end{array}\right)}{2\sqrt{2{\kappa }^2+2{\tau }^2-2\kappa \tau }\sqrt{{\nu }_1^2+{\nu }_2^2+{\nu }_3^2}}}}$$

where

$$\begin{array}{c}{\nu }_1=\kappa \tau \left(4\kappa \left(\kappa -\tau \right)+2\left({\tau }^{\prime }+{\tau }^2\right)+{\kappa }^{\prime }\right)-{\kappa }^2\left(2{\kappa }^2+{\tau }^{\prime }\right)-2{\kappa }^{\prime }{\tau }^2,\hfill \\ {\nu }_2=2\kappa \tau \left({\left(\kappa -\tau \right)}^2+2\tau -2{\tau }^{\prime }\right)-2\left({\kappa }^4+{\tau }^4\right)+{\kappa }^{\prime }{\tau }^2-{\kappa }^2{\tau }^{\prime },\hfill \\ {\nu }_3=\tau \left(2\kappa \left({\kappa }^2+4{\tau }^2-{\kappa }^{\prime }-{\tau }^{\prime }-2\kappa \tau \right)+\left(\tau {\kappa }^{\prime }+{\tau }^{\prime }-2{\tau }^3\right)\right).\hfill \end{array}$$

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4. Conclusions

It was once thought that some systems, such as scalars, vectors and tensors, form a mathematical system sufficient to describe physics, independent of the frame of reference. However, mathematical physics is not that simple. For example, in elementary particles, spin-0 particles can be described by scalars, spin-1 particles by vectors and spin-2 particles by tensors. Spin-$1/2$ particles describe electrons, protons and neutrons, which are not included in this list.

Spinors are two-dimensional vectors with complex components in mathematics. They were first used in physics by Wolfgang Pauli, and the term spinor was introduced by Paul Ehrenfest. Spinors have a mathematical structure similar to tensors and can also be defined as entities that allow the concept of invariance under rotation and Lorentz boosts to be considered more generally. In this case, each tensor of rank k corresponds to a spinor of rank $2k$. Some types of tensors can even be associated with a spinor of the same rank. For example, a general 4-vector corresponds to a Hermitian spinor of rank 2, represented by a $2\ast 2$ Hermitian matrix of complex numbers. Spinors can be used without reference to relativity, but they naturally arise in discussions of the Lorentz group. We can say that a spinor is the most fundamental type of mathematical object that can be Lorentz-transformed. This study successfully establishes a new mathematical frame that bridges the gap between differential geometry and spinor space, providing a unique perspective on curve representations in three-dimensional Euclidean space. By mapping Frenet–Serret frame vectors to spinors and investigating the relationships between spinors of various Smarandache curves ($\mathbf{TN}$, $\mathbf{NB}$, $\mathbf{TB}$ and $\mathbf{TNB}$), we demonstrate a convincing geometric interpretation that transcends traditional mathematical boundaries. This research contributes significantly to our understanding of the complex connections between geometric curves and complex vector spaces. By relating isotropic vectors in ${\mathbb{C}}^3$ space to spinors in ${\mathbb{C}}^2$ space, we uncover a complex mathematical relationship that provides insights into both geometric and physical representations.

• The main contributions of this work are as follows:

• A systematic approach to mapping curve vectors to spinor representations is developed;
• Mathematical relationships between spinors of different Smarandache curves are established;
• A conceptual bridge between geometric and physical-mathematical domains is created.

The methodological approach presented here provides a path for future research. Mathematicians and physicists can extend these findings to explore more complex curve representations, investigate possible applications in quantum mechanics and develop more detailed models of geometric transformations. This study not only advances our theoretical understanding but also demonstrates the deep interconnectedness of mathematical disciplines. By uncovering unexpected connections between differential geometry and spinor theory, we invite further interdisciplinary research and innovative mathematical thinking.