[1]Zehra İşbilir These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Department of Mathematics, Düzce University, Düzce, 81620, Türkiye

2]Department of Mathematics, Bilecik Şeyh Edebali University, Bilecik, 11100,
Türkiye

3]Department of Mathematics, Sakarya University, Sakarya, 54187, Türkiye

# Hyperbolic Spinor Representations of Non-Null Framed Curves

<EMAIL_ADDRESS>Bahar Doğan Yazıcı<EMAIL_ADDRESS>Mehmet Güner<EMAIL_ADDRESS>*  [  [

###### Abstract

In this paper, we intend to bring together the hyperbolic spinors, which are
useful frameworks from mathematics to physics, and both spacelike and timelike
framed curves in Minkowski 3-space $\mathbb{R}_{1}^{3}$, which are new type
attractive frames and very crucial issue for singularity theory especially.
First, we obtain new adapted frames which are called adapted frames for non-
null (spacelike and timelike) framed curves in $\mathbb{R}_{1}^{3}$. Then, we
investigate the hyperbolic spinor representations of non-null framed curves of
the general and adapted frames. Also, we find some geometric results and
interpretations with respect to them, and we obtain illustrative and numerical
examples with figures in order to support the given theorems and results.

###### keywords:

Hyperbolic spinors, Spacelike framed curves, Timelike framed curves, Adapted
framed frames

###### pacs:

[

MSC Classification]15A66, 53A04, 58K05

##  1 Introduction

Despite its long history, the theory of curves is one of the most popular,
classical, and fundamental subjects of differential geometry, and attracts a
great deal of researchers. At the beginning of the discovery of the moving
frame, which is called as Frenet-Serret frame, which is found by two distinct
researchers Frenet [19] and Serret [36], was a milestone for differential
geometry and researchers interested in them. A lot of researchers studied the
Frenet-Serret frame with respect to several different topics and ongoing.
Frenet-Serret frame, which is a fundamental studying area of differential
geometry and has been a long time since its investigation, has been an
interesting subject, and also has attracted much attention from several
researchers. This frame is convenient to examine, evaluate and interpret the
regular curves for lots of geometric properties. In the existing literature,
several researchers also found and examined different new types moving frames
such as Bishop (type-1, type-2, type-3) [4, 37, 48], Darboux frame [8] and
etc. It is clear that the Frenet-Serret frame is obtained for only regular
curves with the condition non-zero curvature. If curves have singular points,
then we can not construct the Frenet-Serret frame of them. Afterward, the
concept of the framed curve and framed base curve in order to construct the
Frenet-Serret frame for non-regular curves are needed [22]. Framed curves,
which were investigated and defined by Honda and Takahashi, are smooth curves
with a moving frame that may have singular points [22]. This new frame also
has attracted the notice of several researchers because of the fact that it
provides the occasion to obtain a moving frame on curves with singular points.
This new interesting frame has contributed to the literature and attracted
also plenty of researchers. Framed curves substantially contribute to
singularity theory in the existing literature, and the studies interested in
this issue are ongoing.

Moreover, Frenet-type framed curves are a special type of framed curves. In
order to construct the Frenet frame, if tangent vectors of a non-regular curve
vanish at singular points, then a regular spherical curve, which is named as
Frenet-type framed base curve, is considered [23, 47]. Recently, lots of
researchers attached to this topic since this frame have lots of occasion for
establishing a moving frame on curves with singular points. In the literature,
lots of studies have been done and are ongoing despite of the fact that it has
not been a long time since it was discovered. The existence conditions of
framed curves [20], the evolutes of framed curves [24], framed rectifying
curves, and framed helices [46, 23, 9] can be examined. In addition to these,
trajectory ruled surfaces related to the framed base curves are scrutinized
[47]. These studies studied in the Euclidean space. In addition to these,
framed curves also have been studied in the Minkowski $3$-space as spacelike
by Li and Pei who introduced the spacelike framed curves in [29]. Also, Özyurt
has obtained the special singular curves and properties of them in Minkowski
$3$-space in [33].

On the other hand, the concept of the spinor is also basic and attractive, and
many researchers especially mathematicians and physicists. P. Ehrenfest is the
first person who has used the concept of spinor [44]. Cartan said that spinors
satisfy a linear representation of the groups of rotations of a space of any
dimension, and also spinors and geometry are very close topics in addition to
their relationship with physics [21]. É. Cartan who is the first mathematician
and studied spinors in a geometrical sense [7]. Since this study includes
quite important notions and notations related to the geometry of spinor
representation, it is an important study. According to this fundamental study,
in the vector space $\mathbb{C}^{3}$, the set of isotropic vectors establishes
a two-dimensional surface in the space $\mathbb{C}^{2}$. Conversely, these
vectors in $\mathbb{C}^{2}$ correspond to the same isotropic vectors. Cartan
asserted that these vectors are complex as two-dimensional in the space
$\mathbb{C}^{2}$ and he said the spinors including of two complex components
related to the vectors in Euclidean 3-space [15, 14, 7]. Additionally,
Vivarelli is examined the spinors related to the spinors in a geometric sense
[45]. Some relations between the spinors and quaternions are determined by
Vivarelli, and provided the spinor representation of rotations in $E^{3}$ with
the help of the relations between the rotations in $E^{3}$ and quaternions
[45, 14]. To get for more detailed information about the spinors, we can refer
to the studies [12, 21, 5, 7, 11, 30, 31, 44, 45]. Pauli matrices, which have
fundamental importance for spinors, were first studied by W. Pauli [34]. In
the existing literature, lots of studies have been done with respect to
spinors.

The study of del Castillo and Barrales [11] is a milestone for researchers and
literature in order to combine the moving frames and spinors. In [11], authors
study the Frenet-Serret frames and spinors, and they find the spinor
representation of Frenet-Serret frame and vectors of this frame. Inspired by
this study, lots of researchers also studied the spinors, and different frames
and curves. Spinor representations according to the Bishop frame [41], Darboux
frame [26] and Sabban frame [39] are examined. Also, spinor equations are
scrutinized related to the different types curves such as successor curves,
involute-evolute curves, and Bertrand curves [16, 17, 15]. In addition to
these, spinors also are studied with hyperbolic components in Minkowski space
which are called as the hyperbolic spinors [27, 2]. By using the hyperbolic
spinors and frames in the Minkowski space, several studies have been done such
as; hyperbolic spinor representations of alternative frame [18], Frenet frame
[28], Darboux frame [3] and Q-frame [43]. On the other hand, hyperbolic
spinors aree examined related to split quaternions for transformations in
$\mathbb{R}_{1}^{3}$ [40] and Fibonacci spinors are examined [14]. In addition
to these, spinor representation of framed Mannheim curves and framed Bertrand
curves in Euclidean 3-space are introduced by Yazıcı et al. [10] and İşbilir
et al. [25], respectively. We want to also the references with respect to
spinors by Aerts et al. [1]. Moreover, Eren and Erişir [13], and Ünal and
Yenitürk [42] investigate the spinor representations of $Q$-frame.

We started this study with the question “What are the results when non-null
framed curves in Minkowski 3-space and hyperbolic spinors are combined?” and
the results made us wonder. This study contains indeed both hyperbolic spinor
representations of regular and singular non-null curves in Minkowski 3-space.
Since spinors, spinor representations and framed curves are many important
topics, in future studies several studies will be completed. Non-null framed
curves are a very important framework for singularity theory and recently,
lots of studies have been done and are ongoing. We strongly believe that this
study will shed light on future studies with respect to the hyperbolic spinor
representations and non-null framed curves.

This present study has the following structure as follows. In Section 2, we
remind some definitions, notations and notions with respect to both hyperbolic
spinors and non-null framed curves. Then, in Section 3, we construct new
adapted frames for spacelike and timelike framed curves in
$\mathbb{R}_{1}^{3}$. In Section 4, we introduce the hyperbolic spinor
representations of non-null framed curves in $\mathbb{R}_{1}^{3}$ with the
help of both the studies in the existing literature and the newly produced
adapted frames in this study. Also, we give spinor equations of vectors of
non-null framed curves and components of them. Then, we construct two
illustrative examples in order to supply the given theorems and results with
figures in Section 5. Then, we give the conclusions in Section 6.

##  2 Preliminaries

In this section, we briefly remind some basic notions, notations, and required
information with respect to hyperbolic numbers, hyperbolic spinors, notions of
Minkowski 3-space $\mathbb{R}^{3}_{1}$, and spacelike and timelike framed
curves in $\mathbb{R}^{3}_{1}$.

Let the hyperbolic number set is denoted by $\mathbb{H}$ and any hyperbolic
number is written as follows $h=h_{1}+jh_{2}\in\mathbb{H}$ where
$h_{1},h_{2}\in\mathbb{R}$ and $j^{2}=1,j\neq\pm 1$. Then, the Euler formula
for the hyperbolic rotation is given as [38, 18]:

$\displaystyle e^{j\Theta}=\cosh{\Theta}+j\sinh{\Theta}$ (1)

where $\Theta$ is the hyperbolic angle. For more detailed information related
to the hyperbolic number plane can be found in [38].

In ${\mathbb{R}_{1}^{3}}$, the Lorentzian inner product of any two vectors
$\boldsymbol{\omega}=({{\omega}_{1}},{{\omega}_{2}},{{\omega}_{3}})\in{\mathbb{R}_{1}^{3}}$
and
$\boldsymbol{\varsigma}=({{\varsigma}_{1}},{{\varsigma}_{2}},{{\varsigma}_{3}})\in{\mathbb{R}_{1}^{3}}$
is expressed as
$\left\langle\boldsymbol{\omega},\boldsymbol{\varsigma}\right\rangle={-{\omega}_{1}}{{\varsigma}_{1}}+{{\omega}_{2}}{{\varsigma}_{2}}+{\omega_{3}}{{\varsigma}_{3}}$.
If the non-zero vector $\boldsymbol{\omega}$ satisfies
$\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle>0$,
$\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle<0$ and
$\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle=0$, then it is called
as a spacelike, timelike or null (lightlike) vector, respectively. In addition
to these, the vector product of $\boldsymbol{\omega}$ and
$\boldsymbol{\varsigma}$ is given as in $\mathbb{R}_{1}^{3}$:

$\begin{split}\boldsymbol{\omega}\times\boldsymbol{\varsigma}=\begin{vmatrix}-\boldsymbol{e}_{1}&\boldsymbol{e}_{2}&\boldsymbol{e}_{3}\\\
\omega_{1}&\omega_{2}&\omega_{3}\\\
\varsigma_{1}&\varsigma_{2}&\varsigma_{3}\end{vmatrix}\end{split}$

where ${\boldsymbol{e}_{n}}$ for $n=1,2,3$ are canonical basis of
$\mathbb{R}_{1}^{3}$ [29]. Then, the norm in the Minkowski 3-space is defined
as follows related to the inner product in Minkowski 3-space:
$||\boldsymbol{\omega}||=\sqrt{|\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle|}$
[32].

In addition to these, de Sitter 2-space is determined as [29, 33]:

$\mathbb{S}_{1}^{2}=\\{\boldsymbol{\omega}\in\mathbb{R}_{1}^{3}|\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle=1\\}$

and the hyperbolic 2-space are defined as follows [29, 33]:

$\mathbb{H}_{0}^{2}=\\{\boldsymbol{\omega}\in\mathbb{R}_{1}^{3}|\langle\boldsymbol{\omega},\boldsymbol{\omega}\rangle=-1\\}.$

###  2.1 Hyperbolic spinors

Assume that $\flat$ be an $n\times n$ matrix which is defined on the
hyperbolic number system $\mathbb{H}$. $\flat^{\dagger}$ determined as
transposing and conjugating of $\flat$, namely
$\flat^{\dagger}=\overline{\flat^{t}}$, which is an $n\times n$ matrix. If
$\flat$ is a Hermitian matrix related to $\mathbb{H}$, then the equation
$\flat^{t}=\flat$ is satisfied. Additionally, if $\flat$ is an anti-Hermitian
matrix related to $\mathbb{H}$, then $\flat^{t}=-\flat$. Let $\flat$ be a
Hermitian matrix, the statement $UU^{\dagger}=U^{\dagger}U=1$ is valid for
$U=e^{j\flat}$. The set of all $n\times n$ type matrices on $\mathbb{H}$ which
is satisfied the previous statement construct a group, which is named as
hyperbolic unitary group and denoted by $U(n,\mathbb{H})$. Provided that $\det
U=1$, then this type group is represented by $SU(n,\mathbb{H})$ [2, 3, 18].

The group of all Lorentz transformations in the Minkowski space is named as
Lorentz group. The Lorentz group is a subgroup of the Poincaré group.
Additionally, Poincaré group is determined as the group of all isometries in
the Minkowski space. The term “orthochronous” is a Lorentz transformation that
is kept in the direction of time. Moreover, the orthochronous Lorentz group is
determined as that rigid transformation of Minkowski 3-space which kept both
the direction of time and orientation. Provided that they have the determinant
$+1$, then this subgroup is represented as $SO(1,3)$ [6, 3, 18, 28].

In addition to these, there is a homomorphism between the group $SO(1,3)$,
which is the group of the rotation along the origin, and $SU(2,\mathbb{H})$,
which is the group of the unitary $2\times 2$ type matrix. While the elements
of the group $SU(2,\mathbb{H})$ give a fillip to the hyperbolic spinors, the
elements of the group $SO(1,3)$ give a fillip to the vectors with three real
components in Minkowski space [35, 3, 18, 28].

A spinor with two hyperbolic components is represented as follows:

$\eta=\begin{pmatrix}\eta_{1}\\\ \eta_{2}\end{pmatrix}$

by the vectors $a,b,c\in\mathbb{R}_{1}^{3}$ such that

$\begin{array}[]{rl}a+jb=\eta^{t}\sigma\eta,\\\
c=-\widehat{\eta}^{t}\sigma\eta,\end{array}$ (2)

where “t” denotes the transposition, $\overline{\eta}$ is the conjugate of
$\eta$, $\widehat{\eta}$ is the mate of $\eta$. Also, the followings can be
written:

$\widehat{\eta}=-\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\overline{\eta}=-\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}\overline{\eta}_{1}\\\
\overline{\eta}_{2}\end{pmatrix}=\begin{pmatrix}-\overline{\eta}_{2}\\\
\overline{\eta}_{1}\end{pmatrix}.$

Then, $2\times 2$ hyperbolic symmetric matrices which are cartesian components
for the vector $\sigma=(\sigma_{1},\sigma_{2},\sigma_{3})$

$\sigma_{1}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}j&0\\\
0&j\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}$ (3)

can be given [3, 18, 40, 28, 27]. The equation
${\overline{\eta^{\prime}}}^{t}\eta^{\prime}=\overline{\eta}^{t}\eta$ is
satisfied where $U\in SU(2,\mathbb{H})$ which is an arbitrary matrix and
$\eta^{\prime}=U\eta$. Hence, the norms of the non-null (spacelike and
timelike) vectors $a^{\prime},b^{\prime},c^{\prime}$ match up with the
hyperbolic spinor $\eta^{\prime}$ are equal to the norms of the non-null
vectors $a,b,c$ match up with the hyperbolic spinor $\eta$. So, all elements
of the group $SU(2,\mathbb{H})$ establish a transformation which turns into
the orthogonal basis $\\{a,b,c\\}$ of the Minkowski 3-space to the orthogonal
basis $\\{a^{\prime},b^{\prime},c^{\prime}\\}$. The correspondence between the
hyperbolic spinors and orthogonal bases is two-to-one. That is, $U$ and $-U$,
which are any two elements of the group $SU(2,\mathbb{H})$, generate the same
ordered triad in $\mathbb{R}_{1}^{3}$. Also, the ordered triads
$\\{a,b,c\\},\\{b,c,a\\},\\{c,a,b\\}$ correspond to different spinors, and the
hyperbolic spinors $\eta$ and $-\eta$ correspond to the same ordered
orthogonal basis. For the hyperbolic spinors $\eta$ and $\phi$, the following
equations can be given:

$\displaystyle\eta^{t}\sigma\phi$ $\displaystyle=\phi^{t}\sigma\eta,$ (4)
$\displaystyle\overline{\eta^{t}\sigma\phi}$
$\displaystyle=-\widehat{\eta}^{t}\sigma\widehat{\phi},$ (5)
$\displaystyle\widehat{\left(\varrho_{1}\eta+\varrho_{2}\phi\right)}$
$\displaystyle=\overline{\varrho}_{1}\widehat{\eta}+\overline{\varrho}_{2}\widehat{\phi},$
(6)

where $\varrho_{1},\varrho_{2}\in\mathbb{H}$ [3, 28, 18, 43, 27]. Let
$\vartheta=(\vartheta_{1},\vartheta_{2},\vartheta_{3})\in\mathbb{H}^{3}$ be an
isotropic vector (namely, length of this vector is zero:
$\langle\vartheta,\vartheta\rangle=0$, $\vartheta\neq 0$) in
$\mathbb{R}_{1}^{3}$. According to the above notions and notations, the
following equations can be given:

$\displaystyle\vartheta_{1}=\eta_{1}^{2}-\eta^{2}_{2},\quad\vartheta_{2}=j(\eta^{2}_{1}+\eta^{2}_{2}),\quad\vartheta_{3}=-2\eta_{1}\eta_{2}.$

Also, the following equations

$\displaystyle\eta_{1}=\pm\sqrt{\frac{\vartheta_{1}+j\vartheta_{2}}{2}}\quad\text{and}\quad\eta_{2}=\pm\sqrt{\frac{-\vartheta_{1}+j\vartheta_{2}}{2}}$
(7)

can be given. In that case, $||a||=||b||=||c||=\overline{\eta}^{t}\eta$.
According to the (2) and (3), the followings

$\begin{array}[]{rl}\vartheta_{1}=\eta^{t}\sigma_{1}\eta=\eta^{2}_{1}-\eta^{2}_{2},\quad\vartheta_{2}=\eta^{t}\sigma_{2}\eta=j(\eta^{2}_{1}+\eta^{2}_{2}),\quad\vartheta_{3}=\eta^{t}\sigma_{3}\eta=-2\eta_{1}\eta_{2}\end{array}$

and

$\displaystyle a+jb$
$\displaystyle=\left(\eta^{2}_{1}-\eta^{2}_{2},j(\eta^{2}_{1}+\eta^{2}_{2}),-2\eta_{1}\eta_{2}\right),$
(8) $\displaystyle c$
$\displaystyle=\left(\eta_{1}\overline{\eta}_{2}+\overline{\eta}_{1}\eta_{2},j(\eta_{1}\overline{\eta}_{2}-\overline{\eta}_{1}\eta_{2}),\left|\eta_{1}\right|^{2}-\left|\eta_{2}\right|^{2}\right)$
(9)

can be written [3, 28, 18, 43]. For more detailed information with respect to
the hyperbolic spinor (especially related to hyperbolic spinors and moving
frames), we want to refer to the studies [3, 28, 40, 18, 43].

###  2.2 Spacelike framed curves in $\mathbb{R}^{3}_{1}$

######  Definition 1.

Let $\gamma:I\rightarrow\mathbb{R}^{3}_{1}$ be a spacelike curve in
$\mathbb{R}^{3}_{1}$. For all $s\in I$, if the following conditions are
satisfied

$\langle\gamma^{\prime}(s),\nu_{1}(s)\rangle=0\quad\text{and}\quad\langle\gamma^{\prime}(s),\nu_{2}(s)\rangle=0$

then, the map
$(\gamma,\nu_{1},\nu_{2}):I\rightarrow\mathbb{R}^{3}_{1}\times\Delta_{1}$ is
named as a spacelike framed curve. Also, $\gamma$ is named as a base curve of
the spacelike framed curve. Here,

$\Delta_{1}=\\{(\nu_{1},\nu_{2})\in\mathbb{S}_{1}^{2}\times\mathbb{H}_{0}^{2}|\langle\nu_{1}(s),\nu_{2}(s)\rangle=0\\}$

or

$\Delta_{1}=\\{(\nu_{1},\nu_{2})\in\mathbb{H}_{0}^{2}\times\mathbb{S}_{1}^{2}|\langle\nu_{1}(s),\nu_{2}(s)\rangle=0\\}.$

Then, a spacelike vector field is determined as
$\mu(s)=\nu_{1}(s)\times\nu_{2}(s)$, which is a smooth function, and
$\alpha(s)$ satisfies that $\gamma^{\prime}(s)=\alpha(s)\mu(s)$. Moreover, the
base curve $\gamma(s)$ is singular at the point $s_{0}$ if and only if
$\alpha(s_{0})=0$ [29].

Furthermore, the Frenet-type formulas are given for a spacelike framed curve
as follows [29]:

$\begin{pmatrix}\mu^{\prime}(s)\\\ \nu^{\prime}_{1}(s)\\\
\nu^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&-\delta(s)l_{2}(s)&\delta(s)l_{3}(s)\\\
l_{2}(s)&0&l_{1}(s)\\\
l_{3}(s)&l_{1}(s)&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\ \nu_{1}(s)\\\
\nu_{2}(s)\end{pmatrix}$ (10)

where $\delta(s)=\text{sign}(\nu_{1}(s))=\langle\nu_{1}(s),\nu_{1}(s)\rangle$
and the functions $(\alpha(s),l_{1}(s),l_{2}(s),l_{3}(s))$ are the curvature
of the spacelike framed curve. These functions are determined as [29]:

$\begin{array}[]{lcl}\left\\{\begin{array}[]{lcl}l_{1}(s)&=&\langle\nu^{\prime}_{1}(s),\nu_{2}(s)\rangle,\\\
l_{2}(s)&=&\langle\nu^{\prime}_{1}(s),\mu(s)\rangle,\\\
l_{3}(s)&=&\langle\nu^{\prime}_{2}(s),\mu(s)\rangle.\end{array}\right.\end{array}$
(11)

###  2.3 Timelike framed curves in $\mathbb{R}^{3}_{1}$

######  Definition 2.

Let $\gamma:I\rightarrow\mathbb{R}^{3}_{1}$ be a timelike curve in
$\mathbb{R}^{3}_{1}$. For all $s\in I$, if the following conditions are
satisfied

$\langle\gamma^{\prime}(s),\nu_{1}(s)\rangle=0\quad\text{and}\quad\langle\gamma^{\prime}(s),\nu_{2}(s)\rangle=0$

then, the map
$(\gamma,\nu_{1},\nu_{2}):I\rightarrow\mathbb{R}^{3}_{1}\times\Delta_{2}$ is
called as a timelike framed curve. If there exists
$\left(\nu_{1},\nu_{2}\right):I\rightarrow\Delta_{2}$ such that
$(\gamma,\nu_{1},\nu_{2}):I\rightarrow\mathbb{R}^{3}_{1}\times\Delta_{2}$ is a
timelike framed curve, then the curve $\gamma$ is called as timelike framed
type curve. Here,
$\Delta_{2}=\\{(\nu_{1},\nu_{2})|\langle\nu_{1}(s),\nu_{2}(s)\rangle=0\\}\subset\mathbb{S}_{1}^{2}\times\mathbb{S}_{1}^{2}$.
Also, $\left\\{\mu(s),\nu_{1}(s),\nu_{2}(s)\right\\}$ is a moving frame along
the curve $\gamma(s)$ in $\mathbb{R}_{1}^{3}$ such that
$\mu(s)=\nu_{1}(s)\times\nu_{2}(s)\in\mathbb{H}_{0}^{2}$, and $\alpha(s)$
satisfies that $\gamma^{\prime}(s)=\alpha(s)\mu(s)$ [33].

Then, the Frenet-type formulas are presented for a timelike framed curve as
follows [33]:

$\begin{pmatrix}\mu^{\prime}(s)\\\ \nu^{\prime}_{1}(s)\\\
\nu^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&l_{2}(s)&l_{3}(s)\\\
l_{2}(s)&0&l_{1}(s)\\\
l_{3}(s)&-l_{1}(s)&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\ \nu_{1}(s)\\\
\nu_{2}(s)\end{pmatrix}$ (12)

where the functions $(\alpha(s),l_{1}(s),l_{2}(s),l_{3}(s))$ are the curvature
of the timelike framed curve. Then, these functions are written as [33]:

$\begin{array}[]{lcl}\left\\{\begin{array}[]{lcl}l_{1}(s)&=&\langle\nu^{\prime}_{1}(s),\nu_{2}(s)\rangle,\\\
l_{2}(s)&=&-\langle\nu^{\prime}_{1}(s),\mu(s)\rangle,\\\
l_{3}(s)&=&-\langle\nu^{\prime}_{2}(s),\mu(s)\rangle,\\\
\alpha(s)&=&-\langle\gamma^{\prime}(s),\mu(s)\rangle.\end{array}\right.\end{array}$
(13)

##  3 The New Adapted Frames for Non-Null Framed Curves

In this section of this study, we introduce new types adapted frames for both
spacelike framed curves and timelike framed curves in Minkowski $3$-space. We
want to bring to the literature these new types adapted frames, and this
section makes more comprehensive to our study related to the hyperbolic spinor
representations of non-null framed curves. Thanks to the studies [22, 29, 33]
and inspired by the study [46] (by using the method for constructing adapted
frame), we obtain this section.

###  3.1 A new adapted frame for spacelike framed curves

The adapted frame for spacelike framed curves can be constructed and the
followings are satisfied:

$\displaystyle\left(\begin{matrix}\breve{\nu}_{1}(s)\\\
\breve{\nu}_{2}(s)\end{matrix}\right)=\left(\begin{matrix}\cosh{{\Theta}}(s)&-\sinh{\Theta}(s)\\\
-\sinh{\Theta}(s)&\cosh{\Theta}(s)\end{matrix}\right)\left(\begin{matrix}\nu_{1}\\\
\nu_{2}\end{matrix}\right)$ (14)

where $\Theta:I\rightarrow\mathbb{R}$ is a smooth function. Also,
$(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}):I\rightarrow\mathbb{R}^{3}_{1}\times\Delta_{1}$
is also a spacelike framed curve and we get:

$\displaystyle\breve{\mu}(s)$
$\displaystyle=\breve{\nu}_{1}(s)\times\breve{\nu}_{2}(s)$
$\displaystyle=\left(\cosh{\Theta}(s)\nu_{1}(s)-\sinh{\Theta}(s)\nu_{2}(s)\right)\times\left(-\sinh{\Theta}(s)\nu_{1}(s)+\cosh{\Theta}(s)\nu_{2}(s)\right)$
$\displaystyle=\nu_{1}(s)\times\nu_{2}(s)$ $\displaystyle=\mu(s).$

By straightforward calculations, we have:

$\displaystyle\breve{\nu}^{\prime}_{1}(s)$
$\displaystyle=\left(\Theta^{\prime}(s)-l_{1}(s)\right)\sinh{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cosh{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{2}(s)\cosh{\Theta}(s)-l_{3}(s)\sinh{\Theta}(s)\right)\mu(s),$
$\displaystyle\breve{\nu}^{\prime}_{2}(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cosh{\Theta}(s)\nu_{1}(s)+\left(\Theta^{\prime}(s)-l_{1}(s)\right)\sinh{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{3}(s)\cosh{\Theta}(s)-l_{2}(s)\sinh{\Theta}(s)\right)\mu(s).$

Provided that we take a smooth function $\Theta:I\rightarrow\mathbb{R}$ which
holds:

$\Theta^{\prime}(s)=l_{1}(s)$ (15)

then we get this triad $\\{{\mu}(s),\breve{\nu}_{1}(s),\breve{\nu}_{2}(s)\\}$
which is an adapted frame along the spacelike framed base curve $\gamma(s)$.
Then, we have the following Frenet-Serret-type derivative formulas as follows
(Bishop type frame):

$\begin{pmatrix}\mu^{\prime}(s)\\\ \breve{\nu}^{\prime}_{1}(s)\\\
\breve{\nu}^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&-\delta(s)\breve{l}_{2}(s)&\delta(s)\breve{l}_{3}(s)\\\
\breve{l}_{2}(s)&0&0\\\
\breve{l}_{3}(s)&0&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\
\breve{\nu}_{1}(s)\\\ \breve{\nu}_{2}(s)\end{pmatrix}$ (16)

where $\breve{l}_{2}(s)$ and $\breve{l}_{3}(s)$ are written by

$\displaystyle\left(\begin{matrix}\breve{l}_{2}(s)\\\
\breve{l}_{3}(s)\end{matrix}\right)=\left(\begin{matrix}\cosh{{\Theta}}(s)&-\sinh{\Theta}(s)\\\
-\sinh{\Theta}(s)&\cosh{\Theta}(s)\end{matrix}\right)\left(\begin{matrix}l_{2}(s)\\\
l_{3}(s)\end{matrix}\right)$ (17)

On the other hand, let $\Theta:I\rightarrow\mathbb{R}$ be a smooth function
which holds $l_{2}(s)\sinh\Theta(s)=l_{3}(s)\cosh\Theta(s)$. Suppose that

$l_{2}(s)=p(s)\cosh\Theta(s)\quad\text{and}\quad l_{3}(s)=p(s)\sinh\Theta(s)$
(18)

then we get as follows:

$\displaystyle{\mu}^{\prime}(s)$
$\displaystyle=-\delta(s)l_{2}(s)\nu_{1}(s)+\delta(s)l_{3}(s)\nu_{2}(s)$
$\displaystyle=-\delta(s)p(s)\breve{\nu}_{1}(s),$
$\displaystyle\breve{\nu}^{\prime}_{1}(s)$
$\displaystyle=\left(\Theta^{\prime}(s)-l_{1}(s)\right)\sinh{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cosh{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{2}(s)\cosh{\Theta}(s)-l_{3}(s)\sinh{\Theta}(s)\right)\mu(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\breve{\nu}_{2}(s)+p(s)\mu(s),$

and

$\displaystyle\breve{\nu}^{\prime}_{2}(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cosh{\Theta}(s)\nu_{1}(s)+\left(\Theta^{\prime}(s)-l_{1}(s)\right)\sinh{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{3}(s)\cosh{\Theta}(s)-l_{2}(s)\sinh{\Theta}(s)\right)\mu(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\breve{\nu}_{1}(s).$

In that case, the triad $\\{\mu(s),\breve{\nu}_{1}(s),\breve{\nu}_{2}(s)\\}$
become an adapted frame along the spacelike framed curve $\gamma(s)$, and we
get the following Frenet-Serret derivative formula:

$\begin{pmatrix}\mu^{\prime}(s)\\\ \breve{\nu}^{\prime}_{1}(s)\\\
\breve{\nu}^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&-\delta(s)p(s)&0\\\
p(s)&0&q(s)\\\ 0&q(s)&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\
\breve{\nu}_{1}(s)\\\ \breve{\nu}_{2}(s)\end{pmatrix}$ (19)

where the vectors $\mu(s),\nu_{1}(s)$ and $\nu_{2}(s)$ are called as spacelike
generalized tangent vector, spacelike generalized principal normal, and
spacelike generalized binormal vector of the spacelike framed curve,
respectively. Also,

$p(s)=||\mu^{\prime}(s)||\neq 0\quad\text{and}\quad
q(s)=-\Theta^{\prime}(s)+l_{1}(s).$ (20)

Then, the functions $(p(s),q(s),\alpha(s))$ are called as the curvature of the
adapted frame of the spacelike framed curve $\gamma(s)$.

######  Proposition 1.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve. The relation between the first curvature
(curvature) $\kappa(s)$ and the second curvature (torsion) $\tau(s)$, and the
curvature of the spacelike framed curve $(p(s),q(s),\alpha(s))$ of a regular
spacelike curve are written as follows:

$\kappa(s)=\cfrac{\delta(s)p(s)}{\alpha(s)}\quad\text{and}\quad\tau(s)=\cfrac{q(s)}{\alpha(s)}.$

###### Proof.

The followings can be written easily:

$\displaystyle\gamma^{\prime}(s)=$ $\displaystyle\alpha(s)\mu(s),$
$\displaystyle\gamma^{\prime\prime}(s)=$
$\displaystyle\alpha^{\prime}(s)\mu(s)-\delta(s)\alpha(s)p(s)\breve{\nu}_{1}(s),$
$\displaystyle\gamma^{\prime\prime\prime}(s)=$
$\displaystyle\left(\alpha^{\prime\prime}(s)-\delta(s)\alpha(s)p^{2}(s)\right)\mu(s)-\delta(s)\left(2\alpha^{\prime}(s)p(s)+\alpha(s)p^{\prime}(s)\right)\breve{\nu}_{1}(s)$
$\displaystyle-\delta(s)\alpha(s)p(s)q(s)\breve{\nu}_{2}(s).$

Then, we have:

$\displaystyle||\gamma^{\prime}(s)||$ $\displaystyle=|\alpha(s)|,$
$\displaystyle\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)$
$\displaystyle=-\delta(s)\alpha^{2}(s)p(s)\breve{\nu}_{2}(s),$
$\displaystyle\det\left(\gamma^{\prime}(s),\gamma^{\prime\prime}(s),\gamma^{\prime\prime\prime}(s)\right)$
$\displaystyle=\delta^{2}(s)\alpha^{3}(s)p^{2}(s)q(s).$

In that case, we get:

$\displaystyle\kappa(s)$
$\displaystyle=\cfrac{||\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)||}{||\gamma^{\prime}(s)||^{3}}=\cfrac{\delta(s)p(s)}{|\alpha(s)|}$

and

$\displaystyle\tau(s)$
$\displaystyle=\cfrac{\det\left(\gamma^{\prime}(s),\gamma^{\prime\prime}(s),\gamma^{\prime\prime\prime}(s)\right)}{||\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)||^{2}}=\cfrac{q(s)}{\alpha(s)}.$

∎

###  3.2 A new adapted frame for timelike framed curves

We obtain the adapted frame for timelike adapted framed curves and the
followings hold:

$\displaystyle\left(\begin{matrix}\breve{\nu}_{1}(s)\\\
\breve{\nu}_{2}(s)\end{matrix}\right)=\left(\begin{matrix}\cos{{\Theta}}(s)&-\sin{\Theta}(s)\\\
\sin{\Theta}(s)&\cos{\Theta}(s)\end{matrix}\right)\left(\begin{matrix}\nu_{1}(s)\\\
\nu_{2}(s)\end{matrix}\right)$ (21)

where $\Theta(s)$ is a smooth function. Moreover,
$(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}):I\rightarrow\mathbb{R}^{3}_{1}\times\Delta_{2}$
is a timelike framed curve and we have:

$\displaystyle\breve{\mu}(s)$
$\displaystyle=\breve{\nu}_{1}(s)\times\breve{\nu}_{2}(s)=\left(\cos{\Theta}(s)\nu_{1}(s)-\sin{\Theta}(s)\nu_{2}(s)\right)\times\left(\sin{\Theta}(s)\nu_{1}(s)+\cos{\Theta}(s)\nu_{2}(s)\right)$
$\displaystyle=\nu_{1}(s)\times\nu_{2}(s)$ $\displaystyle=\mu(s).$

Then, if the required calculations are done, we get:

$\displaystyle{\breve{\nu}_{1}}^{\prime}(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\sin{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cos{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{2}(s)\cos{\Theta}(s)-l_{3}(s)\sin{\Theta}(s)\right)\mu(s),$
$\displaystyle\breve{\nu}^{\prime}_{2}(s)$
$\displaystyle=\left(\Theta^{\prime}(s)-l_{1}(s)\right)\cos{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\sin{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{2}(s)\sin{\Theta}(s)+l_{3}(s)\cos{\Theta}(s)\right)\mu(s).$

Taking a smooth function $\Theta:I\rightarrow\mathbb{R}$ which is satisfied
$\Theta^{\prime}(s)=l_{1}(s)$, then we have this triad
$\\{{\mu}(s),\breve{\nu}_{1}(s),\breve{\nu}_{2}(s)\\}$ which is an adapted
frame along the timelike framed base curve $\gamma(s)$. Then, we have the
following Frenet-Serret type derivative formulas as follows (Bishop type
frame):

$\begin{pmatrix}\mu^{\prime}(s)\\\ \breve{\nu}^{\prime}_{1}(s)\\\
\breve{\nu}^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&\breve{l}_{2}(s)&\breve{l}_{3}(s)\\\
\breve{l}_{2}(s)&0&0\\\
\breve{l}_{3}(s)&0&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\
\breve{\nu}_{1}(s)\\\ \breve{\nu}_{2}(s)\end{pmatrix}$ (22)

where $\breve{l}_{2}(s)$ and $\breve{l}_{3}(s)$ are written by

$\displaystyle\left(\begin{matrix}\breve{l}_{2}(s)\\\
\breve{l}_{3}(s)\end{matrix}\right)=\left(\begin{matrix}\cos{{\Theta}}(s)&-\sin{\Theta}(s)\\\
\sin{\Theta}(s)&\cos{\Theta}(s)\end{matrix}\right)\left(\begin{matrix}l_{2}(s)\\\
l_{3}(s)\end{matrix}\right)$

Then, let $\Theta:I\rightarrow\mathbb{R}$ be a smooth function which holds
$l_{2}(s)\sin\Theta(s)=-l_{3}(s)\cos\Theta(s)$. Suppose that
$l_{2}(s)=p(s)\cos\Theta(s)$ and $l_{3}(s)=-p(s)\sin\Theta(s)$, then we obtain
as follows:

$\displaystyle{\mu}^{\prime}(s)$
$\displaystyle=l_{2}(s)\nu_{1}(s)+l_{3}(s)\nu_{2}(s)$
$\displaystyle=p(s)\breve{\nu}_{1}(s),$
$\displaystyle\breve{\nu}^{\prime}_{1}(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\sin{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\cos{\Theta}(s)\nu_{2}(s)$
$\displaystyle\,\,\,\,\,\,\,+\left(l_{2}(s)\cos{\Theta}(s)-l_{3}(s)\sin{\Theta}(s)\right)\mu(s)$
$\displaystyle=\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\breve{\nu}_{2}(s)+p(s)\mu(s),$

and

$\displaystyle\breve{\nu}^{\prime}_{2}(s)=$
$\displaystyle\left(\Theta^{\prime}(s)-l_{1}(s)\right)\cos{\Theta}(s)\nu_{1}(s)+\left(-\Theta^{\prime}(s)+l_{1}(s)\right)\sin{\Theta}(s)\nu_{2}(s)$
$\displaystyle+\left(l_{2}(s)\sin{\Theta}(s)+l_{3}(s)\cos{\Theta}(s)\right)\mu(s)$
$\displaystyle=$
$\displaystyle\left(\Theta^{\prime}(s)-l_{1}(s)\right)\breve{\nu}_{1}.$

Then, the triad $\\{\mu(s),\breve{\nu}_{1}(s),\breve{\nu}_{2}(s)\\}$ construct
an adapted frame along the timelike curve $\gamma(s)$, and we have the
following Frenet-Serret derivative formula:

$\begin{pmatrix}\mu^{\prime}(s)\\\ \breve{\nu}^{\prime}_{1}(s)\\\
\breve{\nu}^{\prime}_{2}(s)\end{pmatrix}=\begin{pmatrix}0&p(s)&0\\\
p(s)&0&q(s)\\\ 0&-q(s)&0\end{pmatrix}\begin{pmatrix}\mu(s)\\\
\breve{\nu}_{1}(s)\\\ \breve{\nu}_{2}(s)\end{pmatrix}$ (23)

where the vectors $\mu(s),\nu_{1}(s)$ and $\nu_{2}(s)$ are called as timelike
generalized tangent vector, timelike generalized principal normal, and
timelike generalized binormal vector of the timelike framed curve,
respectively. Also, $p(s)=||\mu^{\prime}(s)||\neq 0$ and
$q(s)=-\Theta^{\prime}(s)+l_{1}(s)$. Then, the functions
$(p(s),q(s),\alpha(s))$ are called as the curvature of the adapted frame of
the timelike framed curve $\gamma(s)$.

######  Proposition 2.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve. The relation between the first curvature
(curvature) $\kappa(s)$ and the second curvature (torsion) $\tau(s)$, and the
curvature of the timelike framed curve $(p(s),q(s),\alpha(s))$ of a regular
timelike curve are expressed as follows:

$\kappa(s)=\cfrac{p(s)}{\alpha(s)}\quad\text{and}\quad\tau(s)=\cfrac{q(s)}{\alpha(s)}.$

###### Proof.

The followings can be obtained:

$\displaystyle\gamma^{\prime}(s)$ $\displaystyle=\alpha(s)\mu(s),$
$\displaystyle\gamma^{\prime\prime}(s)$
$\displaystyle=\alpha^{\prime}(s)\mu(s)+\alpha(s)p(s)\breve{\nu}_{1}(s),$
$\displaystyle\gamma^{\prime\prime\prime}(s)$
$\displaystyle=\left(\alpha^{\prime\prime}(s)+\alpha(s)p^{2}(s)\right)\mu(s)+\left(2\alpha^{\prime}(s)p(s)+\alpha(s)p^{\prime}(s)\right)\breve{\nu}_{1}(s)+\alpha(s)p(s)q(s)\breve{\nu}_{2}(s).$

In that case, we get:

$\displaystyle||\gamma^{\prime}(s)||$ $\displaystyle=|\alpha(s)|,$
$\displaystyle\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)$
$\displaystyle=\alpha^{2}(s)p(s)\breve{\nu}_{2}(s),$
$\displaystyle\det\left(\gamma^{\prime}(s),\gamma^{\prime\prime}(s),\gamma^{\prime\prime\prime}(s)\right)$
$\displaystyle=\alpha^{3}(s)p^{2}(s)q(s).$

Then, we have:

$\displaystyle\kappa(s)$
$\displaystyle=\cfrac{||\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)||}{||\gamma^{\prime}(s)||^{3}}=\cfrac{p(s)}{|\alpha(s)|}$

and

$\displaystyle\tau(s)$
$\displaystyle=\cfrac{\det\left(\gamma^{\prime}(s),\gamma^{\prime\prime}(s),\gamma^{\prime\prime\prime}(s)\right)}{||\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s)||^{2}}=\cfrac{q(s)}{\alpha(s)}.$

∎

##  4 Hyperbolic spinor representations of non-null framed curves

In this section, we introduce the hyperbolic spinor representations of
spacelike and timelike framed curves in Minkowski 3-space
$\mathbb{R}_{1}^{3}$. We organize by seperating this section as two subparts.
First of all, hyperbolic spinor representations of spacelike framed curves are
determined and examined, then hyperbolic spinor representations of timelike
framed curves are given. Also, we obtain some geometric interpretations with
respect to them.

It should be noted that, we do not use the parameter “$s$” all of the
equations in the definitions, theorems and conclusions in this section for the
sake of the brevity.

###  4.1 Hyperbolic spinor representations of spacelike framed curves

In this part of this study, we investigate and scrutinize the hyperbolic
spinor representations of spacelike framed curves in $\mathbb{R}_{1}^{3}$.

######  Definition 3.

Let
$\left(\gamma,{\nu}_{1},{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ represents the
triad $\\{\nu_{1},{\nu_{2}},{\mu}\\}$. Then, the hyperbolic spinor
representations of the spacelike framed curve are defined as follows:

$\displaystyle\nu_{1}+j\nu_{2}$ $\displaystyle=\phi^{t}\sigma\phi,$ (24)
$\displaystyle\mu$ $\displaystyle=-\widehat{\phi}^{t}\sigma\phi,$ (25)

where $\overline{\phi}^{t}\phi=1$.

######  Theorem 3.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ represents the
triad $\\{\nu_{1},\nu_{2},\mu\\}$. The single spinor equation that includes
the curvatures of the spacelike framed curve is written as:

$\cfrac{d\phi}{ds}=\cfrac{1}{2}\left[jl_{1}\phi-\left(l_{2}+jl_{3}\right)\widehat{\phi}\right].$
(26)

###### Proof.

By taking the derivative of the equation (24) with respect to the parameter
$s$, then we have:

$\cfrac{d\nu_{1}}{ds}+j\cfrac{d{\nu_{2}}}{ds}=\left(\cfrac{d\phi}{ds}\right)^{t}\sigma\phi+\phi^{t}\sigma\cfrac{d\phi}{ds}.$
(27)

Since the system $\\{\phi,\widehat{\phi}\\}$ is a basis for the hyperbolic
spinor $\phi$, then $\cfrac{d\phi}{ds}$ can be written as follows:

$\cfrac{d\phi}{ds}=\xi\phi+\chi\widehat{\phi}$ (28)

where $\xi$ and $\chi$ are any two hyperbolic valued functions. By using the
equations (10), (27) and (28), we have:

$\displaystyle
l_{1}{\nu}_{2}+l_{2}{\mu}+j\left(l_{1}{\nu}_{1}+l_{3}{\mu}\right)$
$\displaystyle=\left(\xi\phi+\chi\widehat{\phi}\right)^{t}\sigma\phi+\phi^{t}\sigma\left(\xi\phi+\chi\widehat{\phi}\right)$
$\displaystyle=\xi\phi^{t}\sigma\phi+\chi{\widehat{\phi}}^{t}\sigma\phi+\xi\phi^{t}\sigma\phi+\chi\phi^{t}\sigma\widehat{\phi}.$

Then, with the help of the equations (4), (24) and (25), we get:

$\displaystyle
jl_{1}\left({\nu}_{1}+j{\nu}_{2}\right)+\left({l}_{2}+jl_{3}\right){\mu}$
$\displaystyle=2\xi\left(\nu_{1}+j{\nu}_{2}\right)-2\chi{\mu}.$

In that case, we obtain:

$\xi=\cfrac{jl_{1}}{2}\quad\text{and}\quad\chi=-\cfrac{l_{2}+jl_{3}}{2}.$ (29)

By substituting the equation (29) in the equation (28), then we get the
equation (26). ∎

######  Theorem 4.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ represents the
triad $\\{\nu_{1},\nu_{2},\mu\\}$. The hyperbolic spinor representations of
vectors of the spacelike framed curve are given as:

$\displaystyle{\mu}$ $\displaystyle=-\widehat{\phi}^{t}\sigma\phi,$ (30)
$\displaystyle\nu_{1}$
$\displaystyle=\cfrac{1}{2}\left(\phi^{t}\sigma\phi-{\widehat{\phi}^{t}\sigma\widehat{\phi}}\right),$
(31) $\displaystyle{\nu_{2}}$
$\displaystyle=\cfrac{j}{2}\left(\phi^{t}\sigma\phi+\widehat{\phi}^{t}\sigma\widehat{\phi}\right).$
(32)

###### Proof.

Let the hyperbolic spinor $\phi$ corresponds to the triad
$\\{\nu_{1},\nu_{2},\mu\\}$ of the spacelike framed curve
$\left(\gamma,\nu_{1},\nu_{2}\right)$. According to the equation (24), we get
$\nu_{1}=Re(\phi^{t}\sigma\phi)$ and $\nu_{2}=Im(\phi^{t}\sigma\phi)$. We have
already $\mu=-\widehat{\phi}^{t}\sigma\phi$ from the equation (25). Also, by
using the well-known properties of hyperbolic numbers:
$Re(\rho)=\frac{1}{2}\left(\rho+\overline{\rho}\right)$ and
$Im(\rho)=\frac{j}{2}\left(\rho-\overline{\rho}\right)$ for every
$\rho\in\mathbb{H}$, we can write as follows:

$\nu_{1}=\cfrac{1}{2}\left(\phi^{t}\sigma\phi+\overline{\phi^{t}\sigma\phi}\right),$
$\nu_{2}=\cfrac{j}{2}\left(\phi^{t}\sigma\phi-\overline{\phi^{t}\sigma\phi}\right).$

According to the last two equations and by using the equation(5), we obtain:

$\nu_{1}=\cfrac{1}{2}\left(\phi^{t}\sigma\phi-{\widehat{\phi}^{t}\sigma\widehat{\phi}}\right),$
$\nu_{2}=\cfrac{j}{2}\left(\phi^{t}\sigma\phi+\widehat{\phi}^{t}\sigma\widehat{\phi}\right),$

and complete the proof. ∎

######  Corollary 1.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ represents the
triad $\\{\nu_{1},\nu_{2},\mu\\}$. Then the hyperbolic spinor components for
the spacelike framed vectors are given as:

$\displaystyle\nu_{1}$
$\displaystyle=\cfrac{1}{2}\left(\phi^{2}_{1}-\phi^{2}_{2}+\overline{\phi}^{2}_{1}-\overline{\phi}^{2}_{2},j\left(\phi^{2}_{1}+\phi^{2}_{2}-\overline{\phi}^{2}_{1}-\overline{\phi}^{2}_{2}\right),-2\left(\phi_{1}\phi_{2}+\overline{\phi}_{1}\overline{\phi}_{2}\right)\right),$
$\displaystyle\nu_{2}$
$\displaystyle=\cfrac{j}{2}\left(\phi^{2}_{1}-\phi^{2}_{2}-\overline{\phi}^{2}_{1}+\overline{\phi}^{2}_{2},j\left(\phi^{2}_{1}+\phi^{2}_{2}+\overline{\phi}^{2}_{1}+\overline{\phi}^{2}_{2}\right),2\left(\overline{\phi}_{1}\overline{\phi}_{2}-\phi_{1}\phi_{2}\right)\right),$
$\displaystyle\mu$
$\displaystyle=\left(\phi_{1}\overline{\phi}_{2}+\overline{\phi}_{1}\phi_{2},j\left(\phi_{1}\overline{\phi}_{2}-\overline{\phi}_{1}\phi_{2}\right),|\phi_{1}|^{2}-|\phi_{2}|^{2}\right).$

###### Proof.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ corresponds to
the triad $\\{\nu_{1},\nu_{2},\mu\\}$. Then, for $\phi^{t}\sigma\phi$, we can
write:

$\begin{array}[]{lcl}\left\\{\begin{array}[]{lcl}\phi^{t}\sigma_{1}\phi&=&\begin{pmatrix}\phi_{1}&\phi_{2}\end{pmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=\phi_{1}^{2}-\phi_{2}^{2},\\\
\phi^{t}\sigma_{2}\phi&=&\begin{pmatrix}\phi_{1}&\phi_{2}\end{pmatrix}\begin{pmatrix}j&0\\\
0&j\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=j(\phi_{1}^{2}+\phi_{2}^{2}),\\\
\phi^{t}\sigma_{3}\phi&=&\begin{pmatrix}\phi_{1}&\phi_{2}\end{pmatrix}\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=-2\phi_{1}\phi_{2}.\end{array}\right.\end{array}$ (33)

Also, for $\widehat{\phi}^{t}\sigma\widehat{\phi}$, we have:

$\begin{array}[]{lcl}\left\\{\begin{array}[]{lcl}\widehat{\phi}^{t}\sigma_{1}\widehat{\phi}&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}-\overline{\phi}_{2}\\\
\overline{\phi}_{1}\end{pmatrix}=\overline{\phi}_{2}^{2}-\overline{\phi}_{1}^{2},\\\
\widehat{\phi}^{t}\sigma_{2}\widehat{\phi}&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}j&0\\\
0&j\end{pmatrix}\begin{pmatrix}-\overline{\phi}_{2}\\\
\overline{\phi}_{1}\end{pmatrix}=j\left(\overline{\phi}_{2}^{2}+\overline{\phi}_{1}^{2}\right),\\\
\widehat{\phi}^{t}\sigma_{3}\widehat{\phi}&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix}\begin{pmatrix}-\overline{\phi}_{2}\\\
\overline{\phi}_{1}\end{pmatrix}=2\overline{\phi}_{1}\overline{\phi}_{2}.\end{array}\right.\end{array}$
(34)

Then, for $\widehat{\phi}^{t}\sigma\phi$, we get:

$\begin{array}[]{lcl}\left\\{\begin{array}[]{lcl}\widehat{\phi}^{t}\sigma_{1}\phi&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=-\overline{\phi}_{2}\phi_{1}-\overline{\phi}_{1}\phi_{2},\\\
\widehat{\phi}^{t}\sigma_{2}\phi&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}j&0\\\
0&j\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=i\left(-\overline{\phi}_{2}\phi_{1}+\overline{\phi}_{1}\phi_{2}\right),\\\
\widehat{\phi}^{t}\sigma_{3}\phi&=&\begin{pmatrix}-\overline{\phi}_{2}&\overline{\phi}_{1}\end{pmatrix}\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=-|\overline{\phi}_{1}|^{2}+|\overline{\phi}_{2}|^{2}.\end{array}\right.\end{array}$
(35)

If we substitute the equations (33), (34) and (35) in the equations (30), (31)
and (32), we have desired result. ∎

From the Definition 3 to Corollary 1, all of the notions and notations in the
definition, theorems, and corollaries can be obtained easily for the adapted
frame for spacelike framed curve, but for the sake of the brevity and since
they are very clear, we do not give them. However, we want to present also the
following definition:

######  Definition 4.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\phi$ represents the
triad $\\{\breve{\nu}_{1},{\breve{\nu}_{2}},{\mu}\\}$. Then, the hyperbolic
spinor representations of the adapted framed frame along the spacelike framed
curve are defined as follows:

$\displaystyle\breve{\nu}_{1}+j\breve{\nu}_{2}$
$\displaystyle=\Upsilon^{t}\sigma\Upsilon,$ (36) $\displaystyle\mu$
$\displaystyle=-\widehat{\Upsilon}^{t}\sigma\Upsilon,$ (37)

where $\overline{\Upsilon}^{t}\Upsilon=1$.

Also, the following single spinor equations with respect to the adapted frame
for spacelike framed curves can be seen:

  1. 1.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\Upsilon$ represents
the triad $\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$. Then, we have
(according to the Bishop type frame):

$\cfrac{d\Upsilon}{ds}=-\cfrac{1}{2}\left(\breve{l}_{2}+j\breve{l}_{3}\right)\widehat{\Upsilon}.$
(38)

  2. 2.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curve and the hyperbolic spinor $\Upsilon$ represents
the triad $\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$. Then, we have
(according to the Frenet-Serret type frame):

$\cfrac{d\Upsilon}{ds}=\cfrac{1}{2}\left(jq\Upsilon-p\widehat{\Upsilon}\right).$
(39)

Now, in the following Theorem 5, let us construct the spinor relations between
adapted frame and general frame of the spacelike framed curves:

######  Theorem 5.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The following
spinor relations are given:

$\displaystyle\phi^{t}\sigma\phi$
$\displaystyle=e^{j\Theta}\left(\Upsilon^{t}\sigma\Upsilon\right),$ (40)
$\displaystyle\mu$ $\displaystyle=\mu,$

where $\mu=\breve{\mu}$ and the hyperbolic angle between the vectors $\nu_{1}$
and $\breve{\nu}_{1}$ is $\Theta$.

###### Proof.

By using the equation (14), we have:

$\displaystyle\nu_{1}+j\nu_{2}$
$\displaystyle=\left(\cosh{\Theta}\breve{\nu}_{1}+\sinh{\Theta}\breve{\nu}_{2}\right)+j\left(\sinh{\Theta}\breve{\nu}_{1}+\cosh{\Theta}\breve{\nu}_{2}\right)$
$\displaystyle=\left(\breve{\nu}_{1}+j\breve{\nu}_{2}\right)\cosh{\Theta}+j\left(\breve{\nu}_{1}+j\breve{\nu}_{2}\right)\sinh{\Theta}$
$\displaystyle=\left(\breve{\nu}_{1}+j\breve{\nu}_{2}\right)\left(\cosh\Theta+j\sinh\Theta\right)$
$\displaystyle=e^{j\Theta}\left(\breve{\nu}_{1}+j\breve{\nu}_{2}\right).$ (41)

Then, via the equation (24) and (36), we have the desired result. ∎

######  Theorem 6.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The following
relation between these spinors $\phi$ and $\Upsilon$ holds:

$\phi=\pm e^{j\frac{\Theta}{2}}\Upsilon.$ (42)

###### Proof.

Via the equations (8) and (41), we have:

$\displaystyle\phi^{t}\sigma\phi$
$\displaystyle=\left(\phi^{2}_{1}-\phi^{2}_{2},j\left(\phi^{2}_{1}+\phi^{2}_{2}\right),-2\phi_{1}\phi_{2}\right),$
$\displaystyle\Upsilon^{t}\sigma\Upsilon$
$\displaystyle=\left(\Upsilon^{2}_{1}-\Upsilon^{2}_{2},j\left(\Upsilon^{2}_{1}+\Upsilon^{2}_{2}\right),-2\Upsilon_{1}\Upsilon_{2}\right),$

and then we get $\phi^{2}_{1}=e^{j\Theta}\Upsilon^{2}_{1}$ and
$\phi^{2}_{2}=e^{j\Theta}\Upsilon^{2}_{2}$. Thus, $\phi_{1}=\pm
e^{j\frac{\Theta}{2}}\phi_{1}$ and $\phi_{2}=\pm
e^{j\frac{\Theta}{2}}\phi_{2}$ can be obtained. The spinor $\phi$ and $-\phi$
correspond to the triad $\\{\nu_{1},\nu_{2},\mu\\}$, and the spinor $\Upsilon$
and $-\Upsilon$ correspond to the triad
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$. Hence, we can write $\phi=\pm
e^{j\frac{\Theta}{2}}\Upsilon$. ∎

######  Corollary 2.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. Then, the angle
between the spinors $\phi$ and $\Upsilon$ is $\Theta/2$.

######  Theorem 7.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represents the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The following
relation between the spinors $\phi$ and $\Upsilon$ is satisfied:

$\widehat{\phi}=\pm e^{-j\frac{\Theta}{2}}\widehat{\Upsilon}.$ (43)

###### Proof.

By taking the mate of both sides of the equation (42), we have
$\widehat{\phi}=\widehat{\pm e^{j\frac{\Theta}{2}}\Upsilon}.$ With the help of
the equation (6), we get $\widehat{\phi}=\pm
e^{-j\frac{\Theta}{2}}\widehat{\Upsilon}$. ∎

######  Corollary 3.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. While the spinor
$\phi$ makes a rotation with the angle $\Theta/2$ to the spinor $\Upsilon$,
the spinor $\widehat{\phi}$ makes an opposite rotation with the same angle to
the spinor $\widehat{\Upsilon}$.

######  Theorem 8.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The derivative of
the spinor $\phi$ can be written by using the curvatures with respect to the
adapted spacelike framed frame (according to the Bishop type frame).

$\cfrac{d\phi}{ds}=\cfrac{1}{2}\left[j\Theta^{\prime}\phi-\left(\breve{l}_{2}+j\breve{l}_{3}\right)e^{j\Theta}\widehat{\phi}\right].$
(44)

###### Proof.

From the equations (26), (15) and (17), we have:

$\displaystyle\cfrac{d\phi}{ds}$
$\displaystyle=\cfrac{1}{2}\left[jl_{1}\phi-\left(l_{2}+jl_{3}\right)\widehat{\phi}\right]$
$\displaystyle=\cfrac{1}{2}\left[j\Theta^{\prime}\phi-\left(\left(\cosh{\Theta}\breve{l}_{2}+\sinh{\Theta}\breve{l}_{3}\right)+j\left(\sinh{\Theta}\breve{l}_{2}+\cosh{\Theta}\breve{l}_{3}\right)\right)\widehat{\phi}\right]$
$\displaystyle=\cfrac{1}{2}\left[j\Theta^{\prime}\phi-\left(\breve{l}_{2}+j\breve{l}_{3}\right)e^{j\Theta}\widehat{\phi}\right].$

∎

######  Corollary 4.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The angle between
$\cfrac{d\phi}{ds}$ and $\widehat{\phi}$ is $\Theta$ on condition that
$\Theta^{\prime}=0$.

######  Theorem 9.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. Then the following
equation is satisfied:

$\cfrac{d\phi}{ds}=\pm\cfrac{1}{2}e^{j\frac{\Theta}{2}}\left[j\Theta^{\prime}\Upsilon-\left(\breve{l}_{2}+j\breve{l}_{3}\right)\widehat{\Upsilon}\right].$

###### Proof.

By using the equation (42), (43) and (44), we have the desired result. ∎

######  Corollary 5.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The angle between
$\cfrac{d\phi}{ds}$ and $\widehat{\Upsilon}$ is $\Theta/2$ on condition that
$\Theta^{\prime}=0$.

######  Theorem 10.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curves and the hyperbolic spinor $\phi$ represents the
triad $\\{\nu_{1},\nu_{2},\mu\\}$. The derivative of the spinor $\phi$ can be
written by using the curvatures with respect to the adapted spacelike framed
frame (according to the Frenet-Serret type frame):

$\cfrac{d\phi}{ds}=\cfrac{1}{2}\left[j\left(q+\Theta^{\prime}\right)\phi+pe^{j\Theta}\widehat{\phi}\right].$
(45)

###### Proof.

From the equations (26), (18), (20), we have:

$\displaystyle\cfrac{d\phi}{ds}$
$\displaystyle=\cfrac{1}{2}\left[jl_{1}\phi-\left(l_{2}+jl_{3}\right)\widehat{\phi}\right]$
$\displaystyle=\cfrac{1}{2}\left[j\left(q+\Theta^{\prime}\right)\phi-\left(p\cosh\Theta+jp\sinh\Theta\right)\widehat{\phi}\right]$
$\displaystyle=\cfrac{1}{2}\left[j\left(q+\Theta^{\prime}\right)\phi-
pe^{j\Theta}\widehat{\phi}\right].$

∎

######  Corollary 6.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. The angle between
$\cfrac{d\phi}{ds}$ and $\widehat{\phi}$ is $\Theta$ on condition that
$\Theta^{\prime}=0$.

######  Theorem 11.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curves and the hyperbolic spinor $\Upsilon$ represents
the triad $\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$. The derivative of the
spinor $\Upsilon$ can be written by using the curvatures with respect to the
adapted spacelike framed frame (according to the Bishop type frame):

$\cfrac{d\Upsilon}{ds}=-\cfrac{1}{2}\left(l_{2}\cosh\Theta-
l_{3}\sinh\Theta+j\left(-l_{2}\sinh\Theta+l_{3}\cosh\Theta\right)\right)\widehat{\Upsilon}.$

######  Theorem 12.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a spacelike framed curves and the hyperbolic spinor $\Upsilon$ represents
the triad $\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$. The derivative of the
spinor $\Upsilon$ can be written by using the curvatures with respect to the
adapted spacelike framed frame (according to the Frenet-Serret type frame):

$\cfrac{d\Upsilon}{ds}=\cfrac{1}{2}\left(j\left(-\Theta^{\prime}+l_{1}\right)\Upsilon-||\mu^{\prime}||\widehat{\Upsilon}\right).$

######  Theorem 13.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right),\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be spacelike framed curves and the hyperbolic spinors $\phi$ and $\Upsilon$
represent the triads $\\{\nu_{1},\nu_{2},\mu\\}$ and
$\\{\breve{\nu}_{1},\breve{\nu}_{2},\mu\\}$, respectively. Then the following
equation is satisfied:

$\cfrac{d\phi}{ds}=\pm\cfrac{1}{2}e^{j\frac{\Theta}{2}}\left[j\left(q+\Theta^{\prime}\right)\Upsilon-p\widehat{\Upsilon}\right].$

###### Proof.

By using the equation (42), (43) and (45), we have the desired result. ∎

###  4.2 Hyperbolic spinor representations of timelike framed curves

In this section, we obtain the hyperbolic spinors representations of timelike
framed curves in the three-dimensional Minkowski space $\mathbb{R}_{1}^{3}$.

One should note that in this part of this study, we do not give the proofs for
the sake of brevity, since all proofs can be shown using the methods used in
the previous subpart.

######  Definition 5.

Let
$\left(\gamma,{\nu}_{1},{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{{\mu},\nu_{1},{\nu_{2}}\\}$. Then, the hyperbolic spinor
representations of the timelike framed curve are written as follows:

$\displaystyle\mu+j\nu_{1}$ $\displaystyle=\psi^{t}\sigma\psi,$
$\displaystyle\nu_{2}$ $\displaystyle=-\widehat{\psi}^{t}\sigma\psi,$

where $\overline{\psi}^{t}\psi=1$.

######  Theorem 14.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{\mu,\nu_{1},\nu_{2}\\}$. The single spinor equation that includes
the curvatures of timelike framed curve is written as:

$\cfrac{d\psi}{ds}=\cfrac{1}{2}\left[jl_{2}\psi-\left(l_{3}+jl_{1}\right)\widehat{\psi}\right].$

Similar to the previous section, we do not give hyperbolic spinor
representations of the adapted frame of the timelike framed curve because of
the fact that all theorems and corollaries in this part with respect to the
hyperbolic spinor representations of the adapted frame of the timelike framed
curve can be constructed easily. However, we want to present the following
definition:

######  Definition 6.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\Upsilon$ corresponds to
the triad $\\{{\mu},\breve{\nu}_{1},\breve{\nu}_{2}\\}$. Then, the hyperbolic
spinor representations of the adapted frame along the timelike framed curve
are given as follows:

$\displaystyle\mu+j\breve{\nu}_{1}$ $\displaystyle=\eta^{t}\sigma\eta,$
$\displaystyle\breve{\nu}_{2}$ $\displaystyle=-\widehat{\eta}^{t}\sigma\eta,$

where $\overline{\eta}^{t}\eta=1$.

In addition to these, the following single spinor equations with respect to
the adapted frame along the curve timelike framed curve can be given:

  1. 1.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\eta$ represents the
triad $\\{{\mu},\breve{\nu}_{1},\breve{\nu}_{2}\\}$. Then, we get (according
to the Bishop type frame):

$\cfrac{d\eta}{ds}=\cfrac{1}{2}\left(j\breve{l}_{2}\eta-\breve{l}_{3}\widehat{\eta}\right).$
(46)

  2. 2.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\eta$ represents the
triad $\\{{\mu},\breve{\nu}_{1},\breve{\nu}_{2}\\}$. Then, we have (according
to the Frenet-Serret type frame):

$\cfrac{d\eta}{ds}=\cfrac{j}{2}\left(p\eta-q\widehat{\eta}\right).$ (47)

######  Theorem 15.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{\mu,\nu_{1},\nu_{2}\\}$. The hyperbolic spinor representations of
timelike framed vectors are given as:

$\displaystyle{\mu}$
$\displaystyle=\cfrac{1}{2}\left(\psi^{t}\sigma\psi-{\widehat{\psi}^{t}\sigma\widehat{\psi}}\right),$
$\displaystyle\nu_{1}$
$\displaystyle=\cfrac{j}{2}\left(\psi^{t}\sigma\psi+\widehat{\psi}^{t}\sigma\widehat{\psi}\right),$
$\displaystyle{\nu_{2}}$ $\displaystyle=-\widehat{\psi}^{t}\sigma\psi.$

######  Corollary 7.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{\mu,\nu_{1},\nu_{2}\\}$. Then the hyperbolic spinor components for
timelike framed vectors are given as:

$\displaystyle\mu=$
$\displaystyle\cfrac{1}{2}\left(\psi^{2}_{1}-\psi^{2}_{2}+\overline{\psi}^{2}_{1}-\overline{\psi}^{2}_{2},j\left(\psi^{2}_{1}+\psi^{2}_{2}-\overline{\psi}^{2}_{1}-\overline{\psi}^{2}_{2}\right),-2\left(\psi_{1}\psi_{2}+\overline{\psi}_{1}\overline{\psi}_{2}\right)\right),$
$\displaystyle\nu_{1}=$
$\displaystyle\cfrac{j}{2}\left(\psi^{2}_{1}-\psi^{2}_{2}-\overline{\psi}^{2}_{1}+\overline{\psi}^{2}_{2},j\left(\psi^{2}_{1}+\psi^{2}_{2}+\overline{\psi}^{2}_{1}+\overline{\psi}^{2}_{2}\right),2\left(\overline{\psi}_{1}\overline{\psi}_{2}-\psi_{1}\psi_{2}\right)\right),\vspace{+2mm}$
$\displaystyle\nu_{2}=$
$\displaystyle\left(\psi_{1}\overline{\psi}_{2}+\overline{\psi}_{1}\psi_{2},j\left(\psi_{1}\overline{\psi}_{2}-\overline{\psi}_{1}\psi_{2}\right),|\psi_{1}|^{2}-|\psi_{2}|^{2}\right).$

######  Theorem 16.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{\mu,\nu_{1},\nu_{2}\\}$. The derivative of the spinor $\psi$ can be
written by using the curvatures with respect to the adapted frame of the
timelike framed curve (according to the Bishop type frame):

$\cfrac{d\psi}{ds}=\cfrac{1}{2}\left[j\left(\cos\Theta\breve{l}_{2}+\sin\Theta\breve{l}_{3}\right)\psi-\left(\left(-\sin\Theta\breve{l}_{2}+\cos\theta\breve{l}_{3}\right)+j\Theta^{\prime}\right)\widehat{\psi}\right].$

######  Theorem 17.

Let
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\psi$ corresponds to the
triad $\\{\mu,\nu_{1},\nu_{2}\\}$. The derivative of the spinor $\psi$ can be
written by using the curvatures with respect to the adapted frame of the
timelike framed curve (according to the Frenet-Serret type frame):

$\cfrac{d\psi}{ds}=\cfrac{1}{2}\left[jp\cos\Theta\psi-\left(-p\sin\Theta+j\left(q+\Theta^{\prime}\right)\right)\widehat{\psi}\right].$

######  Theorem 18.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\eta$ corresponds to the
triad $\\{\mu,\breve{\nu}_{1},\breve{\nu}_{2}\\}$. The derivative of the
spinor $\eta$ (according to the Bishop type frame) can be expressed by using
the curvatures with respect to the general frame of the timelike framed curve
as follows:

$\cfrac{d\eta}{ds}=\cfrac{1}{2}\left[j\left(l_{2}\cos\Theta-
l_{3}\sin\Theta\right)\eta-\left(l_{2}\sin\Theta+l_{3}\cos\Theta\right)\widehat{\eta}\right].$

######  Theorem 19.

Let
$\left(\gamma,\breve{\nu}_{1},\breve{\nu}_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
be a timelike framed curve and the hyperbolic spinor $\eta$ corresponds to the
triad $\\{\mu,\breve{\nu}_{1},\breve{\nu}_{2}\\}$. The derivative of the
spinor $\eta$ (according to the Frenet-Serret-type) can be given by using the
curvatures with respect to the general frame of the timelike framed curve as
follows:

$\cfrac{d\eta}{ds}=\cfrac{j}{2}\left(||\mu^{\prime}||\eta-\left(-\Theta^{\prime}+l_{1}\right)\widehat{\eta}\right).$

##  5 Examples

In this section, we construct two examples with respect to the hyperbolic
spinor representations of both spacelike and timelike framed curves.

######  Example 1.

Let us take spacelike curve $\gamma$ in $\mathbb{R}_{1}^{3}$ which is defined
by

$\gamma(s)=\left(\cfrac{s^{3}}{3},\cfrac{s^{4}}{4}+\cfrac{s^{3}}{3},\cfrac{s^{5}}{5}+\cfrac{s^{3}}{3}\right).$
(48)

The singular point of $\gamma(s)$ is $s=0$. Then, we have:

$\displaystyle\nu_{1}(s)$
$\displaystyle=\cfrac{1}{\sqrt{\left(s^{2}+1\right)^{2}+\left(s+1\right)^{2}}}\left(0,s^{2}+1,-s-1\right),$
$\displaystyle\nu_{2}(s)$
$\displaystyle=\cfrac{1}{\sqrt{\left(s^{4}+3s^{2}+2s+1\right)\left(s^{4}+3s^{2}+2s+2\right)}}\left(\left(s^{2}+1\right)^{2}+\left(s+1\right)^{2},s+1,s^{2}+1\right),$
$\displaystyle\mu(s)$
$\displaystyle=\cfrac{-1}{\sqrt{s^{4}+3s^{2}+2s+1}}\left(1,s+1,s^{2}+1\right),$
$\displaystyle\delta(s)$ $\displaystyle=1.$

We can easily say that
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{1}$
is a spacelike framed curve. In the following Figure 1, we can examine the
spacelike framed curve given in the equation (48).

Figure 1: The spacelike framed curve $\gamma(s)$

Then, by straightforward calculations, we have the curvature of the spacelike
framed curve as follows:

$\displaystyle l_{1}(s)$
$\displaystyle=\cfrac{s\left(s^{4}+2s^{3}+2s^{2}+6s-3\right)}{\left(s^{4}+3s^{2}+2s+2\right)^{3/2}\sqrt{s^{8}+6s^{6}+4s^{5}+12s^{4}+12s^{3}+13s^{2}+6s+2}},$
$\displaystyle l_{2}(s)$
$\displaystyle=-\cfrac{s^{2}+2s-1}{\sqrt{\left(s^{4}+3s^{2}+2s+1\right)\left(s^{4}+3s^{2}+2s+2\right)}},$
$\displaystyle l_{3}(s)$
$\displaystyle=\cfrac{2s^{3}+3s+1}{\sqrt{\left(s^{4}+3s^{2}+2s+1\right)\left(s^{8}+6s^{6}+4s^{5}+12s^{4}+12s^{3}+13s^{2}+6s+2\right)}},$
$\displaystyle\alpha(s)$ $\displaystyle=-s^{2}\sqrt{s^{4}+3s^{2}+2s+1}.$

In addition to these, the following hyperbolic spinor components can be given:

$\displaystyle\phi_{1}$
$\displaystyle=\pm\cfrac{\sqrt{s+1+j\left(\left(s^{2}+1\right)^{2}+\left(s+1\right)^{2}+\left(s^{2}+1\right)\sqrt{s^{4}+3s^{2}+2s+2}\right)}}{\sqrt[4]{4\left(s^{4}+3s^{2}+2s+1\right)\left(s^{4}+3s^{2}+2s+2\right)}},$
$\displaystyle\phi_{2}$
$\displaystyle=\pm\cfrac{\sqrt{s+1+j\left(-\left(s^{2}+1\right)^{2}-\left(s+1\right)^{2}+\left(s^{2}+1\right)\sqrt{s^{4}+3s^{2}+2s+2}\right)}}{\sqrt[4]{4\left(s^{4}+3s^{2}+2s+1\right)\left(s^{4}+3s^{2}+2s+2\right)}}.$

By using the equation (26), we have:

$\cfrac{d\phi}{ds}=\cfrac{1}{2}\left[\begin{array}[]{rl}&j\left(\cfrac{s\left(s^{4}+2s^{3}+2s^{2}+6s-3\right)}{\left(s^{4}+3s^{2}+2s+2\right)^{3/2}\sqrt{s^{8}+6s^{6}+4s^{5}+12s^{4}+12s^{3}+13s^{2}+6s+2}}\right)\phi\\\
-&\left(\begin{array}[]{rl}&-\cfrac{s^{2}+2s-1}{\sqrt{\left(s^{4}+3s^{2}+2s+1\right)\left(s^{4}+3s^{2}+2s+2\right)}}\\\
&+j\left(\cfrac{2s^{3}+3s+1}{\sqrt{\left(s^{4}+3s^{2}+2s+1\right)\left(s^{8}+6s^{6}+4s^{5}+12s^{4}+12s^{3}+13s^{2}+6s+2\right)}}\right)\end{array}\right)\widehat{\phi}\end{array}\right].$

######  Example 2.

Let us take timelike curve $\gamma$ in $\mathbb{R}_{1}^{3}$ which is defined
by

$\gamma(s)=\left(2\left(s-3\right)\sinh{s}-2\cosh{s},-2\left(s-3\right)\cosh{s}+2\sinh{s},\cfrac{s^{2}}{2}-3s\right).$
(49)

The singular point of $\gamma(s)$ is $s=0$. Also, we have:

$\displaystyle\nu_{1}(s)$ $\displaystyle=\left(\sinh s,-\cosh s,0\right),$
$\displaystyle\nu_{2}(s)$ $\displaystyle=\left(-\cfrac{1}{\sqrt{3}}\cosh
s,\cfrac{1}{\sqrt{3}}\sinh s,-\cfrac{2}{{\sqrt{3}}}\right),$
$\displaystyle\mu(s)$ $\displaystyle=\left(\cfrac{2}{\sqrt{3}}\cosh
s,-\cfrac{2}{\sqrt{3}}\sinh s,\cfrac{1}{\sqrt{3}}\right).$

In that case, we can easily say that
$\left(\gamma,\nu_{1},\nu_{2}\right):I\rightarrow\mathbb{R}_{1}^{3}\times\Delta_{2}$
is a timelike framed curve. In the following Figure 2, we can see the timelike
framed curve presented in the equation (49).

Figure 2: The timelike framed curve $\gamma(s)$

Then, by straightforward calculations, we have the curvature of the timelike
framed curve as follows:

$\displaystyle p(s)$ $\displaystyle=\cfrac{2}{\sqrt{3}},$ $\displaystyle q(s)$
$\displaystyle=\cfrac{1}{\sqrt{3}},$ $\displaystyle\alpha(s)$
$\displaystyle=\sqrt{3}\left(s-3\right).$

Additionally, the following hyperbolic spinor components can be given:

$\displaystyle\eta_{1}$
$\displaystyle=\pm\cfrac{1}{\sqrt{2}}\sqrt{\left(-1+\cfrac{2}{\sqrt{3}}\right)e^{-js}},$
$\displaystyle\eta_{2}$
$\displaystyle=\pm\cfrac{1}{\sqrt{2}}\sqrt{\left(-1-\cfrac{2}{\sqrt{3}}\right)e^{js}}.$

With the help of the equation (47), we obtain:

$\cfrac{d\eta}{ds}=\cfrac{j}{2}\left(\cfrac{2}{\sqrt{3}}\eta-\cfrac{1}{\sqrt{3}}\widehat{\eta}\right).$

##  6 Conclusions

The main purpose of this study is to investigate the hyperbolic spinor
representations of not only spacelike framed curves but also timelike framed
curves in Minkowski 3-space. In order to make the theorems and materials we
found in this study more comprehensive, and to contribute to the literature,
first, adapted frames for spacelike and timelike framed curves were obtained.
We found hyperbolic spinor representations for non-null framed curves and some
relations using both the existing information in the literature and the newly
produced adapted frames. Also, we obtained the hyperbolic spinor equations of
vectors of non-null framed curves, and then components of these vectors. We
also construct some geometric interpretations and results with respect to
these new hyperbolic spinor representations. Then, we construct numerical
examples in order to construct given theorems and results with some useful
figures.

We strongly believe that this study will make a great contribution to the
existing literature for both classical differential geometry, singularity
theory and also spinor structure, and we hope that it will shed light on
future studies.

We plan that in the future study, we investigate the hyperbolic spinor
representations of spacelike and timelike framed curves in de Sitter 2-space
in detail.

## Acknowledgement

First, our deceased co-author Assist. Prof. Dr. Mehmet Güner for his
contributions to education, science, and all his students. Also, the authors
would like to thank Prof. Dr. Murat Tosun for his valuable suggestions and
support. Moreover, the authors would like to thank editors and anonymous
referees for their invaluable comments and careful reading.

## Declarations

Funding Not applicable. Conflict of interest Not applicable. Ethics approval
Not applicable. Consent to participate Not applicable. Consent for publication
Not applicable. Availability of data and materials Consent for publication.
Code availability Consent for publication. Authors’ contributions All authors
contributed equally.

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