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\begin{document}

\title{\textbf{Serret-Frenet Formulas for Octonionic Curves}}
\author{\"{O}zcan BEKTA\c{S} \and Salim Y\"{U}CE}
\date{}
\maketitle

\begin{abstract}
In this paper, we define spatial octonionic curves (SOC) in $\mathbb{R}^{7}$
and octonionic curves (OC) in $\mathbb{R}^{8}$ by using octonions. Firstly,
we determine Serret-Frenet equations, and curvatures of the SROC in $\mathbb{%
R}^{7}$. Then, Serret-Frenet equations for the OC in $\mathbb{R}^{8}$ are
calculated with the help of Serret-Frenet equations of SOC in $\mathbb{R}^{7}
$.
\end{abstract}

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\textbf{Keywords:} Serret-Frenet formula, spatial octonionic curve,
octonionic curve, curvature, torsion.

\textbf{M.S.C. 2010:} 53A04. 14H50. 11R52.

\section{Introduction}

Octonions and quaternions are important subjects in differential geometry.
Quaternionic curves play an important role in differential geometry. Spatial
quaternion set (if the real part of quaternion is equal to zero, then the
quaternion is called spatial quaternion) is isomorphic to Euclidean $3$%
-space. Moreover, the set of all quaternions is isomorph to Euclidean $4$%
-space. For that reason, Bharathi and Nagaraj studied the differential
geometry of a smooth curve in Euclidean $4$-space $\mathbb{R}^{4}$ [2]. The
elements of $\mathbb{R}^{4}$ are identified with the quaternions. The
Serret-Frenet apparatus for the quaternionic curves were determined by the
Serret-Frenet apparatus for a main curve in $\mathbb{R}^{3}$ which is
embedded in $\mathbb{R}^{4}$ \ [2]. Then, various papers were published by
using the quaternionic curves in $\mathbb{R}^{3}$ and $\mathbb{R}^{4}$. For
example, Karadag, Gunes and Sivridag determined the Serret-Frenet formulas
for dual quaternion valued functions of a single real variable [10]. The
quaternion valued functions, and quaternionic inclined curves were studied
in the semi-Euclidean space by Coken and Tuna [5] and [14].

Octonions were defined independently by J. Thomas Graves and A. Cayley. The
set of the octonions $\mathbb{O}$ are expressed as $\mathbb{O}=\{\mathbb{A}%
_{0}e_{0}+\underset{i=1}{\overset{7}{\sum }}\mathbb{A}_{i}e_{i};~\mathbb{A}%
_{i}\in \mathbb{R}\}$, where $e_{0}=+1$, $e_{i}^{2}=-1$, $%
e_{0}e_{i}~=e_{i}e_{0}=e_{i}$, $(\forall i=1,2,...,7)$, $e_{i}e_{j}=-\delta
_{ij}e_{0}+\varepsilon _{ijk}e_{k}$, $~\delta _{ij}$ is Kronecker delta, $%
\varepsilon _{ijk}~$is completely antisimetric tensor, $\left( ijk\right)
=\left( 123\right) ,~\left( 145\right) ,~\left( 176\right) ,~\left(
246\right) ,~\left( 257\right) ,~\left( 347\right) ,~\left( 365\right) $
[8]. There has been many investigations related to octonions. Various
authors studied octonions in analysis, physics, and differential geometry.
Spatial octonion set (if the real part of octonion is equal to zero, then
the real octonion is called the spatial octonion [15]. These octonions
create the spatial octonion set and denoted by $\mathbb{O}_{S}$) is isomorph
to Euclidean $7$-space, and the set of octonions is isomorphic to Euclidean $%
8$-space. In other words, $\mathbb{O}$ $=$ $\mathbb{R}$ $\oplus $ $\mathbb{R}%
^{7}$ [12]. \ The points in $\mathbb{R}^{8}~$can be represented by the
octonions [1]. Thus, we can ask ourselves the following questions:

Can we define the octonionic curves in $\mathbb{R}^{7}$and $\mathbb{R}^{8}$,
and find the Serret-Frenet apparatus for the octonionic curve $\beta _{%
\mathbb{O}}:I\subset \mathbb{R}\rightarrow \mathbb{O}$, $\beta _{\mathbb{O}%
}\left( s\right) =\underset{i=0}{\overset{7}{\sum }}\gamma _{i}\left(
s\right) e_{i}=\gamma _{0}\left( s\right) e_{0}+\gamma _{\mathbb{O}}\left(
s\right) ~$by using the Serret-Frenet apparatus for a main spatial
octonionic curve,  $\gamma _{O}:I\subset \mathbb{R}\rightarrow \mathbb{O}_{S}
$, $\gamma _{\mathbb{O}}\left( s\right) =\underset{i=1}{\overset{7}{\sum }}%
\gamma _{i}\left( s\right) e_{i}$ in $\mathbb{R}^{7}\ $which is embedded in $%
\mathbb{R}^{8}$? In this paper, we will answer the above questions. We
transfer the concept of the quaternionic curve [2] in Euclidean $4$-space to
the concept of the octonionic curve in Euclidean $8$-space by using the real
octonions (We are dealing with real octonions in this paper and henceworth
use just word octonions).

Our study is prepared as follows. In preliminaries part, we give fundamental
definitions, properties, and informations about the octonion algebras. In
section 3, we introduce the spatial octonionic curves (SOC) in $\mathbb{R}%
^{7}$, and the octonionic curves (OC) in $\mathbb{R}^{8}$. Then, we find the
Serret-Frenet apparatus for SOC in $\mathbb{R}^{7}$. By using the
Serret-Frenet apparatus for SOC in $\mathbb{R}^{7}$, we compute the
Serret-Frenet apparatus for OC in $\mathbb{R}^{8}$.

\section{Preliminaries}

In this section, we denote the set of spatial octonions with $\mathbb{O}_{S}$%
, and the octonions set with $\mathbb{O}$. Let us first give some
fundamental notions of the octonions. The real octonion $\mathbb{A}$ is
defined by

\begin{equation*}
\mathbb{A}=\mathbb{A}_{0}e_{0}+\underset{i=1}{\overset{7}{\sum }}\mathbb{A}%
_{i}e_{i}\text{,}
\end{equation*}%
where $\mathbb{A}_{i}~$are the real numbers components of the octonions, $%
e_{i}~\left( i=1,2,\ldots ,7\right) $ are the unit octonions basis elements,
and $e_{0}=+1$ is the scalar element. The multiplication rules of the unit
octonions basis elements are given by 
\begin{eqnarray*}
&&\mathbf{Table~1.}\text{The Multiplication Table of Unit Octonions Basis
Elements } \\
&&~~~\ ~~~~~~~%
\begin{tabular}{c|cccccccc}
$\times $ & $e_{0}$ & $\ \ e_{1}$ & $\ \ e_{2}$ & $\ \ e_{3}$ & $\ \ e_{4}$
& $\ \ e_{5}$ & $\ \ e_{6}$ & $\ \ e_{7}$ \\ \hline
$e_{0}$ & $e_{0}$ & $\ \ e_{1}$ & $\ \ e_{2}$ & $\ \ e_{3}$ & $\ \ e_{4}$ & $%
\ \ e_{5}$ & $\ \ e_{6}$ & $\ \ e_{7}$ \\ 
$e_{1}$ & $e_{1}$ & $-e_{0}$ & $\ \ e_{3}$ & $-e_{2}$ & $\ \ e_{5}$ & $%
-e_{4} $ & $-e_{7}$ & $\ \ e_{6}$ \\ 
$e_{2}$ & $e_{2}$ & $-e_{3}$ & $-e_{0}$ & $\ \ e_{1}$ & $\ \ e_{6}$ & $\ \
e_{7}$ & $-e_{4}$ & $-e_{5}$ \\ 
$e_{3}$ & $e_{3}$ & $\ \ e_{2}$ & $-e_{1}$ & $-e_{0}$ & $\ \ e_{7}$ & $%
-e_{6} $ & $\ \ e_{5}$ & $-e_{4}$ \\ 
$e_{4}$ & $e_{4}$ & $-e_{5}$ & $-e_{6}$ & $-e_{7}$ & $-e_{0}$ & $\ \ e_{1}$
& $\ \ e_{2}$ & $\ \ e_{3}$ \\ 
$e_{5}$ & $e_{5}$ & $\ \ e_{4}$ & $-e_{7}$ & $\ \ e_{6}$ & $-e_{1}$ & $%
-e_{0} $ & $-e_{3}$ & $\ \ e_{2}$ \\ 
$e_{6}$ & $e_{6}$ & $\ \ e_{7}$ & $\ \ e_{4}$ & $-e_{5}$ & $-e_{2}$ & $\ \
e_{3}$ & $-e_{0}$ & $-e_{1}$ \\ 
$e_{7}$ & $e_{7}$ & $-e_{6}$ & $\ \ e_{5}$ & $\ \ e_{4}$ & $-e_{3}$ & $%
-e_{2} $ & $\ \ e_{1}$ & $-e_{0}$%
\end{tabular}%
\text{.}
\end{eqnarray*}%
The addition, the scalar multiplication, and the octonion multiplication are
the operations of the set of the octonions. The sum of two octonions is
defined by

\begin{eqnarray*}
\mathbb{A}\pm \mathbb{B} &=&\underset{i=0}{\overset{7}{\sum }}\left( \mathbb{%
A}_{i}\pm \mathbb{B}_{i}\right) e_{i} \\
&=&\left( \mathbb{A}_{0}e_{0}+\mathbb{A}_{1}e_{1}+\mathbb{A}_{2}e_{2}+%
\mathbb{A}_{3}e_{3}+\mathbb{A}_{4}e_{4}+\mathbb{A}_{5}e_{5}+\mathbb{A}%
_{6}e_{6}+\mathbb{A}_{7}e_{7}\right) \\
&&\pm \left( \mathbb{B}_{0}e_{0}+\mathbb{B}_{1}e_{1}+\mathbb{B}_{2}e_{2}+%
\mathbb{B}_{3}e_{3}+\mathbb{B}_{4}e_{4}+\mathbb{B}_{5}e_{5}+\mathbb{B}%
_{6}e_{6}+\mathbb{B}_{7}e_{7}\right) \text{.}
\end{eqnarray*}%
$\overline{\mathbb{A}}$ is called conjugate of the octonion ~$\mathbb{A}$,
and conjugate of the octonion is defined by

\begin{eqnarray*}
\overline{\mathbb{A}} &=&\mathbb{A}_{0}e_{0}-\mathbb{A}_{1}e_{1}-\mathbb{A}%
_{2}e_{2}-\mathbb{A}_{3}e_{3}-\mathbb{A}_{4}e_{4}-\mathbb{A}_{5}e_{5}-%
\mathbb{A}_{6}e_{6}-\mathbb{A}_{7}e_{7} \\
&=&\mathbb{A}_{0}e_{0}-\underset{i=1}{\overset{7}{\sum }}\mathbb{A}_{i}e_{i}%
\text{,}
\end{eqnarray*}%
where $\overline{e_{0}}$ $=$ $e_{0}$ and $\overline{e_{j}}$ $=$ $%
-e_{j}~\left( j=1,\ldots ,7\right) $ [7]. The octonion $\mathbb{A}$ has real
part and vectorial part. So, the octonion $\mathbb{A}~$is decomposed with
respect to its real $\left( S_{\mathbb{A}}\right) $, and vectorial $\left(
V_{\mathbb{A}}\right) $ parts as follows:

\begin{equation*}
S_{\mathbb{A}}=\frac{1}{2}\left( \mathbb{A}+\overline{\mathbb{A}}\right) =%
\mathbb{A}_{0}e_{0},~~~~V_{\mathbb{A}}=\frac{1}{2}\left( \mathbb{A}-%
\overline{\mathbb{A}}\right) =\underset{i=1}{\overset{7}{\sum }}\mathbb{A}%
_{i}e_{i}\text{.}~
\end{equation*}%
Let us denote octonions with a real number $\left( S_{\mathbb{A}}=\mathbb{A}%
_{0}\right) $ in $\mathbb{R}$, and a vector $\left( V_{\mathbb{A}}=\underset{%
i=1}{\overset{7}{\sum }}\mathbb{A}_{i}e_{i}\right) $ in $\mathbb{R}^{7}$.
So, the octonion is given by

\begin{equation*}
\mathbb{A}=S_{\mathbb{A}}+V_{\mathbb{A}}\text{.}
\end{equation*}%
The multiplication of two octonions is defined by 
\begin{equation}
\mathbb{A}\times \mathbb{B}=S_{\mathbb{A}}S_{\mathbb{B}}-g\left( V_{\mathbb{A%
}},V_{\mathbb{B}}\right) +S_{\mathbb{A}}V_{\mathbb{B}}+S_{\mathbb{B}}V_{%
\mathbb{A}}+V_{\mathbb{A}}\wedge V_{\mathbb{B}}\text{,}  \label{2.1}
\end{equation}%
where $\forall \mathbb{A},\mathbb{B}\in \mathbb{O}$. We use the inner and
the cross products in $\mathbb{R}^{7}$ in above equations [11]. The
symmetric, non-degenerate real-valued bilinear form $g$ is introduced by

\begin{equation*}
g:\mathbb{O}\times \mathbb{O}\rightarrow \mathbb{R},~g\left( \mathbb{A},%
\mathbb{B}\right) =\frac{1}{2}\left( \mathbb{A}\times \overline{\mathbb{B}}+%
\mathbb{B}\times \overline{\mathbb{A}}\right) \text{,}
\end{equation*}%
where $\mathbb{A},\mathbb{B}\in \mathbb{O}$. $g$ is determined with the help
of the real octonionic multiplication. $g$ is called the octonionic inner
product. Thus, $g\left( \mathbb{A},\mathbb{B}\right) =\underset{i=0}{\overset%
{7}{\sum }}\mathbb{A}_{i}\mathbb{B}_{i}$ [3,6]. If$\mathbb{~A}+\overline{%
\mathbb{A}}=0$, then the octonion $\mathbb{A}$ is called the spatial
octonion. The spatial octonion set is represented by

\begin{equation*}
\mathbb{O}_{S}=\{\underset{i=1}{\overset{7}{\sum }}\mathbb{A}_{i}e_{i};~%
\mathbb{A}_{i}\in \mathbb{R}\}\text{,}
\end{equation*}%
where $e_{i}^{2}=-1$, $e_{i}e_{j}=-\delta _{ij}e_{0}+\varepsilon _{ijk}e_{k}$%
, $(i,j,k=1,2,...,7)$, $(i\neq j\neq k,~i\neq 0,~j\neq 0,~k\neq 0)$. The
spatial octonion set is isomorphic to $\mathbb{R}^{7}$.

The vector product of two vectors is only defined in $3$-dimensional
Euclidean space, $\mathbb{R}^{3}$ and $7$-dimensional Euclidean space, $%
\mathbb{R}^{7}$[7]. We express the vector product in $\mathbb{R}^{7}$ as
follows. Let $\mathbb{A}$ and $\mathbb{B~}$be the spatial octonions. Then,
the vector product in $\mathbb{R}^{7}$ is defined by

\begin{equation*}
\mathbb{A}\wedge \mathbb{B}=\mathbb{A\times B}+\left\langle \mathbb{A},%
\mathbb{B}\right\rangle \text{,}
\end{equation*}%
where $\mathbb{AB}$ $=\left( \mathbb{A}_{1},\mathbb{A}_{2},\mathbb{A}_{3},%
\mathbb{A}_{4},\mathbb{A}_{5},\mathbb{A}_{6},\mathbb{A}_{7}\right) \left( 
\mathbb{B}_{1},\mathbb{B}_{2},\mathbb{B}_{3},\mathbb{B}_{4},\mathbb{B}_{5},%
\mathbb{B}_{6},\mathbb{B}_{7}\right) $ is defined by 
\begin{eqnarray*}
\mathbb{AB} &=&(\mathbb{A}_{2}\mathbb{B}_{3}-\mathbb{A}_{3}\mathbb{B}_{2}+%
\mathbb{A}_{4}\mathbb{B}_{5}-\mathbb{A}_{5}\mathbb{B}_{4}+\mathbb{A}_{7}%
\mathbb{B}_{6}-\mathbb{A}_{6}\mathbb{B}_{7}, \\
&&\mathbb{A}_{3}\mathbb{B}_{1}-\mathbb{A}_{1}\mathbb{B}_{3}+\mathbb{A}_{4}%
\mathbb{B}_{6}-\mathbb{A}_{6}\mathbb{B}_{4}+\mathbb{A}_{5}\mathbb{B}_{7}-%
\mathbb{A}_{7}\mathbb{B}_{5}, \\
&&\mathbb{A}_{1}\mathbb{B}_{2}-\mathbb{A}_{2}\mathbb{B}_{1}+\mathbb{A}_{4}%
\mathbb{B}_{7}-\mathbb{A}_{7}\mathbb{B}_{4}+\mathbb{A}_{6}\mathbb{B}_{5}-%
\mathbb{A}_{5}\mathbb{B}_{6}, \\
&&\mathbb{A}_{5}\mathbb{B}_{1}-\mathbb{A}_{1}\mathbb{B}_{5}+\mathbb{A}_{6}%
\mathbb{B}_{2}-\mathbb{A}_{2}\mathbb{B}_{6}+\mathbb{A}_{7}\mathbb{B}_{3}-%
\mathbb{A}_{3}\mathbb{B}_{7}, \\
&&\mathbb{A}_{1}\mathbb{B}_{4}-\mathbb{A}_{4}\mathbb{B}_{1}+\mathbb{A}_{3}%
\mathbb{B}_{6}-\mathbb{A}_{6}\mathbb{B}_{3}+\mathbb{A}_{7}\mathbb{B}_{2}-%
\mathbb{A}_{2}\mathbb{B}_{7}, \\
&&\mathbb{A}_{1}\mathbb{B}_{7}-\mathbb{A}_{7}\mathbb{B}_{1}+\mathbb{A}_{2}%
\mathbb{B}_{4}-\mathbb{A}_{4}\mathbb{B}_{2}+\mathbb{A}_{5}\mathbb{B}_{3}-%
\mathbb{A}_{3}\mathbb{B}_{5}, \\
&&\mathbb{A}_{2}\mathbb{B}_{5}-\mathbb{A}_{5}\mathbb{B}_{2}+\mathbb{A}_{3}%
\mathbb{B}_{4}-\mathbb{A}_{4}\mathbb{B}_{3}+\mathbb{A}_{6}\mathbb{B}_{1}-%
\mathbb{A}_{1}\mathbb{B}_{6})\text{,}
\end{eqnarray*}%
and $\left\langle \mathbb{A},\mathbb{B}\right\rangle =\underset{i=1}{\overset%
{7}{\sum }}\mathbb{A}_{i}\mathbb{B}_{i}~$is standart inner product in $%
\mathbb{R}^{7}~$[4,7]. Moreover, this vector product is given by [7,13] for
all $\mathbb{A}=\underset{i=1}{\overset{7}{\sum }}\mathbb{A}_{i}e_{i}=\left( 
\mathbb{A}_{i}\right) $, $1\leq i\leq 7$ and $\mathbb{B}=\underset{i=1}{%
\overset{7}{\sum }}\mathbb{B}_{i}e_{i}=\left( \mathbb{B}_{i}\right) $, $%
1\leq i\leq 7$. The vector product in $\mathbb{R}^{7}~$satisfies the
following properties:

$i)~$\textbf{Distributive property: }$\mathbb{A}\wedge \left( \mathbb{B}+%
\mathbb{C}\right) =\mathbb{A}\wedge \mathbb{B}+\mathbb{A}\wedge \mathbb{C}$,$%
~$

$ii)~$\textbf{The vector product of the spatial octonion with itself is zero}%
, $\mathbb{A}\wedge \mathbb{A}=0$,

$iii)~$\textbf{Alternating property: }$\mathbb{A}\wedge \mathbb{B}=-\mathbb{B%
}\wedge \mathbb{A}$, $\ $

$iv)~\left\langle \mathbb{A},\mathbb{A}\wedge \mathbb{B}\right\rangle =0$,

$v)~$\textbf{The norm of the vector product of two spatial octonions: }$%
\left\Vert \mathbb{A}\wedge \mathbb{B}\right\Vert =\left\Vert \mathbb{A}%
\right\Vert \left\Vert \mathbb{B}\right\Vert sin\theta $, $\ $

$vi)$\textbf{Mixed scalar product: }$\left\langle \mathbb{A}\wedge \mathbb{B}%
,\mathbb{C}\right\rangle =\left\langle \mathbb{B}\wedge \mathbb{C},\mathbb{A}%
\right\rangle =\left\langle \mathbb{C}\wedge \mathbb{A},\mathbf{\ }\mathbb{B}%
\right\rangle $,

$vii\mathbf{)~}\mathbb{A}\wedge \left( \mathbb{A}\wedge \mathbb{B}\right)
=\left\langle \mathbb{A},\mathbb{B}\right\rangle \mathbf{\ }\mathbb{A}%
-\left\langle \mathbb{A},\mathbb{A}\right\rangle \mathbb{B}$.

The norm of \ the octonion $\mathbb{A}$ is denoted by

\begin{equation*}
\left\Vert \mathbb{A}\right\Vert =\sqrt{\mathbb{A}\times \overline{\mathbb{A}%
}}=\sqrt{\underset{i=0}{\overset{7}{\sum }}\mathbb{A}_{i}^{2}}\text{.}
\end{equation*}%
If $\left\Vert \mathbb{A}_{0}\right\Vert $ $=1$, then $\mathbb{A}_{0}$ is
called the unit octonion.

In differential geometry, curve theory is a developed subject of study. The
planar curve in $2$-dimensional Euclidean space $\mathbb{R}^{2}$, the space
curve in $3$-dimensional Euclidean space $\mathbb{R}^{3}$, and the space
curve in $n$-dimensional Euclidean space $\mathbb{R}^{n}$ were defined.
Since $\mathbb{R}^{2}~$is corresponding to~$%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$, then the planar curves in $\mathbb{R}^{2}$ were studied with respect to
complex numbers. Accordingly, since $\mathbb{R}^{3}$ and $\mathbb{R}^{4}$
are corresponding to~the spatial quaternion set $\mathbb{H}_{S}$ and the
real quaternion set $\mathbb{H}$, respectively, then the space curves in $%
\mathbb{R}^{3}$ and $\mathbb{R}^{4}$ were studied in terms of quaternions.
The main question of our paper is: Can we study the space curves in $\mathbb{%
R}^{7}$ and $\mathbb{R}^{8}$ by using the octonions? To answer this
question, we have to know strong evidence. Our main evidence is that $%
\mathbb{R}^{7}$ and $~\mathbb{R}^{8}$ are corresponding to~$\mathbb{O}_{S}$
and $\mathbb{O}$, respectively. In this paper, we use the Serret-Frenet
formulas for the well known space curve in $\mathbb{R}^{7}$ and $\mathbb{R}%
^{8}~$(We know that space curves were defined in $\mathbb{R}^{n}$. So, if we
take $n=7$, $8$, we get the $\mathbb{R}^{7}$ and $\mathbb{R}^{8}~$).

\textbf{The Serret Frenet frame and curvatures in }$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{8}$\textbf{:} Let $\Gamma :I\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{8}$ be an unit speed space curve in $\mathbb{R}^{8}$ and $\left\{ \mathbf{U%
}_{j}\right\} $, $1\leq j\leq 8$ be the Serret Frenet $8$-frame related to $%
\Gamma $. The Serret-Frenet formulas for the curve $\Gamma :I\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{8}$ are given by

\begin{equation}
\begin{array}{l}
\mathbf{U}_{1}^{^{\prime }}\left( s\right) =k_{1}\left( s\right) \mathbf{U}_{%
\mathbf{2}}\left( s\right) \\ 
\mathbf{U}_{\mathbf{m}}^{^{\prime }}\left( s\right) =-k_{m-1}\left( s\right) 
\mathbf{U}_{m-1}\left( s\right) +k_{m}\left( s\right) \mathbf{U}_{m\mathbf{+1%
}}\left( s\right) ,~~2\leq m\leq 7 \\ 
\mathbf{U}_{\mathbf{8}}^{^{\prime }}\left( s\right) =-k_{7}\left( s\right) 
\mathbf{U}_{\mathbf{7}}\left( s\right) \text{.}%
\end{array}
\label{2.2}
\end{equation}%
On the other hand, $\mathbf{U}_{j}\left( s\right) =\frac{\mathbf{E}%
_{j}\left( s\right) }{\left\Vert \mathbf{E}_{j}\left( s\right) \right\Vert }$%
, $k_{j}\left( s\right) =\left\langle \mathbf{U}_{j}^{^{\prime }}\left(
s\right) ,\mathbf{U}_{j+1}\left( s\right) \right\rangle =\frac{\left\Vert 
\mathbf{E}_{j}+1\left( s\right) \right\Vert }{\left\Vert \mathbf{E}%
_{j}\left( s\right) \right\Vert }$ for $1\leq j\leq 8$ [9].

In this paper, we compute the above Eq. (2.2) by the help of the octonions
for SOC in $\mathbb{R}^{7}$. Then, by using the Serret-Frenet apparatus for
SOC in $\mathbb{R}^{7}$, we compute the Serret-Frenet apparatus for OC in $%
\mathbb{R}^{8}$.

\section{Serret-Frenet Formulas For Octonionic Curves in the Euclidean Space}

Let $\gamma :I\subset \mathbb{R}\rightarrow $ $\mathbb{R}^{7}$ be a space
curve and $\left\{ \mathbf{V}_{j}\right\} $, $0\leq j\leq 6~$be the Frenet
frame of $\gamma $ in the Euclidean $7$-space $\mathbb{R}^{7}$. Now, we are
going to take the spatial octonions instead of all the Frenet elements.
Thus, Frenet elements $\mathbf{V}_{j}~$can be written as a spatial octonion
which is defined by

\begin{equation*}
\mathbf{V}_{j}=\underset{i=0}{\overset{6}{\sum }}v_{j}e_{j}.
\end{equation*}

\textbf{Definition 3.1 }Let $\mathbb{R}^{7}$ characterize the Euclidean $7$%
-space with octonionic metric $g$ \ and $\mathbb{O}_{S}=\left\{ \gamma _{%
\mathbb{O}}\in \mathbb{O}\text{ \ }\left\vert \text{ }\gamma _{\mathbb{O}}+%
\overset{-}{\gamma _{\mathbb{O}}}=0\right. \right\} $ show the spatial
octonion set. $\mathbb{R}^{7}$ is identified with the set of the spatial
octonion. The curve $\gamma _{\mathbb{O}}:I\subset \mathbb{R}\rightarrow 
\mathbb{O}_{S}$, $\gamma _{\mathbb{O}}\left( s\right) =\underset{i=1}{%
\overset{7}{\sum }}\gamma _{i}\left( s\right) e_{i}~$is called the spatial
octonionic curve (SOC).

\textbf{Definition 3.2 }If\textbf{\ }the norm of the first derivative of the
SOC is equal $1$, then SOC is called unit speed spatial octonionic curves
(USSOC)$.$

\begin{theorem}
Let~$\gamma _{\mathbb{O}}:I\subset \mathbb{R}\rightarrow \mathbb{O}_{S}~$be
an USSOC and $\mathbf{V}_{0}\left( s\right) =$ $\gamma _{\mathbb{O}%
}^{^{\prime }}\left( s\right) =\underset{i=1}{\overset{7}{\sum }}\gamma
_{i}^{^{\prime }}\left( s\right) e_{i}$ be unit tangent vector of $\gamma $.
Then, the following equations are provided
\end{theorem}

\textbf{i) }$g\left( \mathbf{V}_{0}\mathbf{,V}_{0}^{^{\prime }}\right) =0$,

\textbf{ii) }$\mathbf{V}_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}%
\mathbf{~\ }$is a spatial octonion.

\begin{proof}
Let \bigskip $\gamma _{\mathbb{O}}:I\subset \mathbb{R}\rightarrow \mathbb{O}%
_{S}$, $\gamma _{\mathbb{O}}\left( s\right) =\underset{i=1}{\overset{7}{\sum 
}}\gamma _{i}\left( s\right) e_{i}$~\ be an USSROC. Since ~$\mathbf{V}_{0}=$ 
$\gamma _{\mathbb{O}}^{^{\prime }}\left( s\right) =\underset{i=1}{\overset{7}%
{\sum }}\gamma _{i}^{^{\prime }}\left( s\right) e_{i}~$has unit length~(in
other words, $\left\Vert \mathbf{V}_{0}\left( s\right) \right\Vert =1~$for
all $s)$, we get $\left\Vert \mathbf{V}_{0}\right\Vert ^{2}=g\left( \mathbf{V%
}_{0},\mathbf{V}_{0}\right) =\mathbf{V}_{0}\mathbf{\times }\overline{\mathbf{%
V}_{0}}=1.~$Thus, differentiating with respect to $s$ gives $\mathbf{V}%
_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+\mathbf{V}_{0}\mathbf{%
\times }\left( \overline{\mathbf{V}_{0}}\right) ^{^{\prime }}=0$. Since $%
\mathbf{V}_{0}=$ $\gamma ^{^{\prime }}\left( s\right) =\underset{i=1}{%
\overset{7}{\sum }}\gamma _{i}^{^{\prime }}\left( s\right) e_{i}$, we may
write $\overline{\mathbf{V}_{0}}=-\underset{i=1}{\overset{7}{\sum }}\gamma
_{i}^{^{\prime }}e_{i}$ and $\left( \overline{\mathbf{V}_{0}}\right)
^{^{\prime }}=-\underset{i=1}{\overset{7}{\sum }}\gamma _{i}^{^{\prime
\prime }}e_{i}$. So, we have $\left( \overline{\mathbf{V}_{0}}\right)
^{^{\prime }}=\overline{\mathbf{V}_{0}^{^{\prime }}}.~$Substituting the
statement $\left( \overline{\mathbf{V}_{0}}\right) ^{^{\prime }}=\overline{%
\mathbf{V}_{0}^{^{\prime }}}$ into $\mathbf{V}_{0}\times \overline{\mathbf{V}%
_{0}}+\mathbf{V}_{0}\mathbf{\times }\left( \overline{\mathbf{V}_{0}}\right)
^{^{\prime }}$, we obtain 
\begin{equation}
\mathbf{V}_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+\mathbf{V}%
_{0}\times \overline{\mathbf{V}_{0}^{^{\prime }}}=0\text{.}  \label{3.1}
\end{equation}

\textbf{i)}If we multiply on both sides of (3.1) by $\frac{1}{2}$\textbf{\ }%
and\textbf{\ }put in order it, then we get\textbf{\ }$\ g\left( \mathbf{V}%
_{0}\mathbf{,V}_{0}^{^{\prime }}\right) =\frac{1}{2}\left[ \mathbf{V}%
_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+\mathbf{V}_{0}\times 
\overline{\mathbf{V}_{0}^{^{\prime }}}\right] =0$. Thus, $\mathbf{V}%
_{0}^{^{\prime }}$ is orthogonal to $\mathbf{V}_{0}\mathbf{.~}$

\textbf{ii) }By using the definition of the conjugate of the octonion into
(3.1), we get $\mathbf{V}_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+%
\mathbf{V}_{0}\times \overline{\mathbf{V}_{0}^{^{\prime }}}=\mathbf{V}%
_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+\overline{\overline{%
\mathbf{V}_{0}}}\times \overline{\mathbf{V}_{0}^{^{\prime }}}=\mathbf{V}%
_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}+\overline{\left( \mathbf{V}%
_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}\right) }=0$.$~$Finally, $%
\mathbf{V}_{0}^{^{\prime }}\times \overline{\mathbf{V}_{0}}\mathbf{~\ }$is a
spatial octonion.
\end{proof}

\textbf{Note 3.1 }Let~$\gamma _{\mathbb{O}}~$be an USSOC and $\left\{ 
\mathbf{V}_{j}\right\} $, $0\leq j\leq 6~$be the Frenet frame of SOC in $%
\mathbb{R}^{7}$. The Frenet elements of USSOC are given by

\begin{equation*}
\mathbf{V}_{i}\times \mathbf{V}_{j}=\left\langle \mathbf{V}_{i},\mathbf{V}%
_{j}\right\rangle +\mathbf{\mathbf{V}_{i}\wedge \mathbf{V}_{j}}=\mathbf{%
\mathbf{V}_{k}}\text{,}
\end{equation*}

where $ijk=012,034,065,135,146,236,254$. On the other hand, we get following
table.

\begin{eqnarray*}
&&\mathbf{Table~1.}\text{The Multiplication Table of Unit Octonion Basis
Elements } \\
&&~~~\ ~~~~~~~%
\begin{tabular}{c|ccccccc}
$\times $ & $\ \ \ \mathbf{V}_{0}$ & $\ \ \mathbf{V}_{\mathbf{1}}$ & $\ \ 
\mathbf{V}_{\mathbf{2}}$ & $\ \ \mathbf{V}_{\mathbf{3}}$ & $\ \ \ \mathbf{V}%
_{\mathbf{4}}$ & $\ ~\mathbf{V}_{\mathbf{5}}$ & $\ \ \mathbf{V}_{\mathbf{6}}$
\\ \hline
$\mathbf{V}_{0}$ & $-1$ & $\ \ \mathbf{V}_{\mathbf{2}}$ & $-\mathbf{V}_{%
\mathbf{1}}$ & $\ \ \mathbf{V}_{\mathbf{4}}$ & $-\mathbf{V}_{\mathbf{3}}$ & $%
-\mathbf{V}_{\mathbf{6}}$ & $\ \ \mathbf{V}_{\mathbf{5}}$ \\ 
$\mathbf{V}_{\mathbf{1}}$ & $-\mathbf{V}_{\mathbf{2}}$ & $-1$ & $\ \ \ 
\mathbf{V}_{0}$ & $\ \ \mathbf{V}_{\mathbf{5}}$ & $\ \ \mathbf{V}_{\mathbf{6}%
}$ & $-\mathbf{V}_{\mathbf{3}}$ & $-\mathbf{V}_{\mathbf{4}}$ \\ 
$\mathbf{V}_{\mathbf{2}}$ & $~~\mathbf{V}_{\mathbf{1}}$ & $-\mathbf{V}_{0}$
& $-1$ & $\ \ \mathbf{V}_{\mathbf{6}}$ & $-\mathbf{V}_{\mathbf{5}}$ & $\ ~%
\mathbf{V}_{\mathbf{4}}$ & $-\mathbf{V}_{\mathbf{3}}$ \\ 
$\mathbf{V}_{\mathbf{3}}$ & $-\mathbf{V}_{\mathbf{4}}$ & $-\mathbf{V}_{%
\mathbf{5}}$ & $-\mathbf{V}_{\mathbf{6}}$ & $-1$ & $\ \ \mathbf{V}_{0}$ & $~$%
\ $\mathbf{V}_{\mathbf{1}}$ & $\ \ \mathbf{V}_{\mathbf{2}}$ \\ 
$\mathbf{V}_{\mathbf{4}}$ & $\ \ \mathbf{V}_{\mathbf{3}}$ & $-\mathbf{V}_{%
\mathbf{6}}$ & $\ \ \mathbf{V}_{\mathbf{5}}$ & $-\mathbf{V}_{0}$ & $-1$ & $-%
\mathbf{V}_{\mathbf{2}}$ & $\ ~\mathbf{V}_{\mathbf{1}}$ \\ 
$\mathbf{V}_{\mathbf{5}}$ & $\ \ \mathbf{V}_{\mathbf{6}}$ & $\ \ \mathbf{V}_{%
\mathbf{3}}$ & $-\mathbf{V}_{\mathbf{4}}$ & $-\mathbf{V}_{\mathbf{1}}$ & $\ 
\mathbf{V}_{\mathbf{2}}$ & $-1$ & $-\mathbf{V}_{0}$ \\ 
$\mathbf{V}_{\mathbf{6}}$ & $-\mathbf{V}_{\mathbf{5}}$ & $\ \ \mathbf{V}_{%
\mathbf{4}}$ & $\ \ \mathbf{V}_{\mathbf{3}}$ & $-\mathbf{V}_{\mathbf{2}}$ & $%
-\mathbf{V}_{\mathbf{1}}$ & $\ \ \mathbf{V}_{0}$ & $-1$%
\end{tabular}%
\end{eqnarray*}

\begin{theorem}
Let~$\gamma _{\mathbb{O}}~$be an USSOC and $\left\{ \mathbf{V}_{j}\right\} $%
, $0\leq j\leq 6~$be the Frenet frame of USSOC in $\mathbb{R}^{7}$. Then the
Frenet equations are obtained by%
\begin{equation}
\begin{array}{l}
\mathbf{V}_{0}^{^{\prime }}\left( s\right) =k_{1}\left( s\right) \mathbf{V}_{%
\mathbf{1}}\left( s\right) \\ 
\mathbf{V}_{\mathbf{m}}^{^{\prime }}\left( s\right) =-k_{m}\left( s\right) 
\mathbf{V}_{m-1}\left( s\right) +k_{m+1}\left( s\right) \mathbf{V}_{m\mathbf{%
+1}}\left( s\right) \\ 
\mathbf{V}_{\mathbf{6}}^{^{\prime }}\left( s\right) =-k_{6}\left( s\right) 
\mathbf{V}_{\mathbf{5}}\left( s\right) \text{,}%
\end{array}
\label{3.2}
\end{equation}

where $k_{i},1\leq i\leq 6$, $1\leq m\leq 5$ curvature functions. Eq. (3.2)
is called Serret-Frenet formulae for the USSOC.
\end{theorem}

\begin{proof}
Since$\ \mathbf{V}_{0}^{^{\prime }}=\underset{i=1}{\overset{7}{\sum }}\gamma
_{i}^{^{\prime \prime }}e_{i}\mathbf{~}$is a spatial octonion.$\mathbf{~}$

Since $\mathbf{V}_{0}^{^{\prime }}$ is a spatial octonion, we describe the
spatial octonion $\mathbf{V}_{\mathbf{1}}$ and the nonnegative scalar
function $k_{1}$ as follows:%
\begin{equation}
\mathbf{V}_{0}^{^{\prime }}=k_{1}\mathbf{V}_{\mathbf{1}}\text{.}  \label{3.3}
\end{equation}

\textbf{I. }$\mathbf{V}_{\mathbf{1}}^{^{\prime }}\left( s\right) +k_{1}%
\mathbf{V}_{0}\left( s\right) ~$is orthogonal to $\mathbf{V}_{\mathbf{0}}$
and $\mathbf{V}_{\mathbf{1}}.$From the definition of Frenet frame of SOC, we
get 
\begin{equation}
g\left( \mathbf{V}_{\mathbf{0}},\mathbf{V}_{\mathbf{1}}\right) =\frac{1}{2}%
\left[ \mathbf{V}_{\mathbf{0}}\times \overline{\mathbf{V}_{\mathbf{1}}}+%
\mathbf{V}_{\mathbf{1}}\times \overline{\mathbf{V}_{\mathbf{0}}}\right] =0%
\text{.}  \label{3.4}
\end{equation}

Hence, the following equation is obtained from (3.4) by differentiating with
respect to $s$ 
\begin{equation*}
\frac{1}{2}\left[ \mathbf{V}_{\mathbf{0}}^{^{\prime }}\times \overline{%
\mathbf{V}_{\mathbf{1}}}+\mathbf{V}_{\mathbf{0}}\mathbf{\times }\overline{%
\mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}}}+\mathbf{V_{\mathbf{1}%
}^{^{\prime }}\times \overline{\mathbf{V}_{\mathbf{0}}}+V}_{\mathbf{1}%
}\times \overline{\mathbf{V}_{\mathbf{0}}^{^{\prime }}}\right] =0\text{.}
\end{equation*}

Thus, we have%
\begin{equation*}
\frac{1}{2}\left[ \mathbf{V}_{\mathbf{0}}^{^{\prime }}\times \overline{%
\mathbf{V}_{\mathbf{1}}}+\mathbf{V}_{\mathbf{1}}\times \overline{\mathbf{V}_{%
\mathbf{0}}^{^{\prime }}}\right] +\frac{1}{2}\left[ \mathbf{V}_{\mathbf{0}%
}^{^{\prime }}\mathbf{\times }\overline{\mathbf{\mathbf{V}_{\mathbf{1}%
}^{^{\prime }}}}+\mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}\times 
\overline{\mathbf{V}_{\mathbf{0}}^{^{\prime }}}}\right] =0\text{.}
\end{equation*}

By using the octonionic inner product, then we may write%
\begin{equation*}
g\left( \mathbf{V}_{\mathbf{0}}^{^{\prime }},\mathbf{V}_{\mathbf{1}}\right)
+g\left( \mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}},\mathbf{V}_{\mathbf{0}%
}\right) =0\Longrightarrow g\left( \mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime
}}},\mathbf{V}_{\mathbf{0}}\right) =-k_{1}\text{.}
\end{equation*}

Differentiating on both sides of the expression $g\left( \mathbf{V}_{\mathbf{%
1}},\mathbf{V}_{\mathbf{1}}\right) =\mathbf{V}_{\mathbf{1}}\times \overline{%
\mathbf{V}_{\mathbf{1}}}=1~$with respect to $s$, we get%
\begin{equation*}
\mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}}\times \overline{\mathbf{V}_{%
\mathbf{1}}}+\mathbf{V}_{\mathbf{1}}\times \overline{\mathbf{\mathbf{V}_{%
\mathbf{1}}^{^{\prime }}}}=0\Longrightarrow \frac{1}{2}\left[ \mathbf{%
\mathbf{V}_{\mathbf{1}}^{^{\prime }}\times }\overline{\mathbf{V}_{\mathbf{1}}%
}+\mathbf{V}_{\mathbf{1}}\mathbf{\times }\overline{\mathbf{\mathbf{V}_{%
\mathbf{1}}^{^{\prime }}}}\right] =\left\langle \mathbf{V}_{\mathbf{1}},%
\mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}}\right\rangle =0\text{.}
\end{equation*}

If we use the last two equations, then we obtain%
\begin{equation*}
g\left( \mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}}+k_{1}\mathbf{V}_{%
\mathbf{0}}\mathbf{,V}_{\mathbf{0}}\right) =g\left( \mathbf{\mathbf{V}_{%
\mathbf{1}}^{^{\prime }},V}_{\mathbf{0}}\right) +k_{1}g\left( \mathbf{V}_{%
\mathbf{0}}\mathbf{,V}_{\mathbf{0}}\right) =-k_{1}+k_{1}=0\text{,}
\end{equation*}

and thus%
\begin{equation*}
g\left( \mathbf{\mathbf{V}_{\mathbf{1}}^{^{\prime }}}+k_{1}\mathbf{V}_{%
\mathbf{0}}\mathbf{,V}_{\mathbf{1}}\right) =g\left( \mathbf{\mathbf{V}_{%
\mathbf{1}}^{^{\prime }},V}_{\mathbf{1}}\right) +k_{1}g\left( \mathbf{V}_{%
\mathbf{0}}\mathbf{,V}_{\mathbf{1}}\right) =0\text{.}
\end{equation*}

Hence, $\mathbf{V}_{\mathbf{1}}^{^{\prime }}\left( s\right) +k_{1}\mathbf{V}%
_{\mathbf{0}}\left( s\right) ~$is orthogonal to $\mathbf{V}_{\mathbf{0}}$
and $\mathbf{V}_{\mathbf{1}}$, and $\left( \mathbf{V}_{\mathbf{0}}\times 
\mathbf{V}_{\mathbf{1}}\right) ~$is parallel to~$\mathbf{V}_{\mathbf{1}%
}^{^{\prime }}\left( s\right) +k_{1}\mathbf{V}_{\mathbf{0}}\left( s\right) $%
. Thus, we have 
\begin{equation*}
\mathbf{V}_{\mathbf{1}}^{^{\prime }}\left( s\right) +k_{1}\mathbf{V}_{%
\mathbf{0}}\left( s\right) =\lambda \left( \mathbf{V}_{\mathbf{0}}\times 
\mathbf{V}_{\mathbf{1}}\right) \text{.}
\end{equation*}

Since $\mathbf{V}_{\mathbf{0}}\times \mathbf{V}_{\mathbf{1}}=\mathbf{V}_{%
\mathbf{2}}$, we get 
\begin{equation*}
g\left( \mathbf{V}_{\mathbf{1}}^{^{\prime }}+k_{1}\mathbf{V}_{\mathbf{0}},%
\mathbf{V}_{\mathbf{2}}\right) =g\left( \lambda \mathbf{V}_{\mathbf{2}},%
\mathbf{V}_{\mathbf{2}}\right) \text{,}
\end{equation*}%
\begin{equation*}
g\left( \mathbf{V}_{\mathbf{1}}^{^{\prime }},\mathbf{V}_{\mathbf{2}}\right)
+k_{1}g\left( \mathbf{V}_{\mathbf{0}},\mathbf{V}_{\mathbf{2}}\right)
=\lambda g\left( \mathbf{V}_{\mathbf{2}},\mathbf{V}_{\mathbf{2}}\right) 
\text{,}
\end{equation*}%
\begin{equation*}
\lambda =g\left( \mathbf{V}_{\mathbf{1}}^{^{\prime }},\mathbf{V}_{\mathbf{2}%
}\right) =k_{2}.
\end{equation*}

Finally, the following equation is obtained by 
\begin{equation}
\mathbf{V}_{\mathbf{1}}^{^{\prime }}\left( s\right) =-k_{1}\mathbf{V}_{%
\mathbf{0}}\left( s\right) +k_{2}\mathbf{V}_{\mathbf{2}}\left( s\right) 
\text{.}  \label{3.5}
\end{equation}

The following cases \textbf{II, III, IV, V }can likewise be proved using the
techniques of the proof of \textbf{I.}

\textbf{II.} Since $\mathbf{V}_{\mathbf{2}}^{^{\prime }}\left( s\right)
+k_{2}\mathbf{V}_{\mathbf{1}}\left( s\right) ~$is orthogonal to $\mathbf{V}_{%
\mathbf{1}}$ and $\mathbf{-V}_{\mathbf{5}}$, then $\left( \mathbf{\ V}_{%
\mathbf{1}}\times \left( -\mathbf{V}_{\mathbf{5}}\right) \right) ~$is
parallel to~$\mathbf{V}_{\mathbf{2}}^{^{\prime }}\left( s\right) +k_{2}%
\mathbf{V}_{\mathbf{1}}\left( s\right) $. Thus, we have%
\begin{equation}
\mathbf{V}_{\mathbf{2}}^{^{\prime }}\left( s\right) =-k_{2}\mathbf{V}_{%
\mathbf{1}}\left( s\right) +k_{3}\mathbf{V}_{\mathbf{3}}\left( s\right) 
\text{.}  \label{3.6}
\end{equation}

\textbf{III.} Since $\mathbf{V}_{\mathbf{3}}^{^{\prime }}\left( s\right)
+k_{3}\mathbf{V}_{\mathbf{2}}\left( s\right) ~$is orthogonal to $\mathbf{V}_{%
\mathbf{2}}$ and $\mathbf{V}_{\mathbf{5}}$, then $\left( \mathbf{V}_{\mathbf{%
2}}\times \mathbf{V}_{\mathbf{5}}\right) ~$is parallel to~$\mathbf{V}_{%
\mathbf{3}}^{^{\prime }}\left( s\right) +k_{3}\mathbf{V}_{\mathbf{2}}\left(
s\right) $. Thus, we get%
\begin{equation}
\mathbf{V}_{\mathbf{3}}^{^{\prime }}\left( s\right) =-k_{3}\mathbf{V}_{%
\mathbf{2}}\left( s\right) +k_{4}\mathbf{V}_{\mathbf{4}}\left( s\right) 
\text{.}  \label{3.7}
\end{equation}

\textbf{IV.} Since $\mathbf{V}_{\mathbf{4}}^{^{\prime }}\left( s\right)
+k_{4}\mathbf{V}_{\mathbf{3}}\left( s\right) ~$is orthogonal to $\mathbf{V}_{%
\mathbf{3}}$ and $\mathbf{-V}_{\mathbf{1}},$then $\left( \mathbf{V}_{\mathbf{%
2}}\times \left( \mathbf{-V}_{\mathbf{1}}\right) \right) ~$is parallel to~$%
\mathbf{V}_{\mathbf{4}}^{^{\prime }}\left( s\right) +k_{4}\mathbf{V}_{%
\mathbf{3}}\left( s\right) $. Thus, we obtain%
\begin{equation}
\mathbf{V}_{\mathbf{4}}^{^{\prime }}\left( s\right) =-k_{4}\mathbf{V}_{%
\mathbf{3}}\left( s\right) +k_{5}\mathbf{V}_{\mathbf{5}}\left( s\right)
\label{3.8}
\end{equation}

\textbf{V.} Since $\mathbf{V}_{\mathbf{4}}^{^{\prime }}\left( s\right) +k_{4}%
\mathbf{V}_{\mathbf{3}}\left( s\right) ~$is orthogonal to $\mathbf{V}_{%
\mathbf{3}}$ and $\mathbf{-V}_{\mathbf{1}},$then $\left( \mathbf{V}_{\mathbf{%
2}}\times \left( \mathbf{-V}_{\mathbf{1}}\right) \right) ~$is parallel to~$%
\mathbf{V}_{\mathbf{4}}^{^{\prime }}\left( s\right) +k_{4}\mathbf{V}_{%
\mathbf{3}}\left( s\right) $. Thus, we have%
\begin{equation}
\mathbf{V}_{\mathbf{5}}^{^{\prime }}\left( s\right) =-k_{5}\mathbf{V}_{%
\mathbf{3}}\left( s\right) +k_{6}\mathbf{V}_{\mathbf{5}}\left( s\right)
\label{3.9}
\end{equation}

\textbf{VI.} Differentiating on both sides of the expression $\mathbf{V}_{%
\mathbf{5}}=\mathbf{V}_{\mathbf{1}}\wedge \mathbf{V}_{3}~$with respect to $s$%
, we get 
\begin{equation*}
\mathbf{V}_{\mathbf{5}}^{^{\prime }}=\mathbf{V}_{\mathbf{1}}^{^{\prime
}}\wedge \mathbf{V}_{3}+\mathbf{V}_{\mathbf{1}}\wedge \mathbf{V}_{\mathbf{3}%
}^{^{\prime }}\text{.}
\end{equation*}%
Hence, from the last equations we obtain%
\begin{eqnarray*}
g\left( \mathbf{V}_{\mathbf{5}}^{^{\prime }},\mathbf{V}_{\mathbf{6}}\right)
&=&g\left( \mathbf{V}_{\mathbf{1}}^{^{\prime }}\wedge \mathbf{V}_{3}+\mathbf{%
V}_{\mathbf{1}}\wedge \mathbf{V}_{\mathbf{3}}^{^{\prime }},\mathbf{V}_{%
\mathbf{2}}\wedge \mathbf{V}_{3}\right) \\
&=&g\left( \mathbf{V}_{\mathbf{1}}^{^{\prime }}\wedge \mathbf{V}_{3},\mathbf{%
V}_{\mathbf{2}}\wedge \mathbf{V}_{3}\right) +g\left( \mathbf{V}_{\mathbf{1}%
}\wedge \mathbf{V}_{\mathbf{3}}^{^{\prime }},\mathbf{V}_{\mathbf{2}}\wedge 
\mathbf{V}_{3}\right) \text{.}
\end{eqnarray*}%
If we use properties $vi$\textbf{\ }in the section 2, we may write%
\begin{equation*}
g\left( \mathbf{V}_{\mathbf{5}}^{^{\prime }},\mathbf{V}_{\mathbf{6}}\right)
=g\left( \mathbf{V}_{3}\wedge \left( \mathbf{V}_{\mathbf{\ 2}}\wedge \mathbf{%
V}_{3}\right) ,\mathbf{V}_{\mathbf{1}}^{^{\prime }}\right) +g\left( \mathbf{V%
}_{\mathbf{3}}^{^{\prime }}\wedge \left( \mathbf{V}_{\mathbf{\ 2}}\wedge 
\mathbf{V}_{3}\right) ,\mathbf{V}_{1}\right) \text{.}
\end{equation*}%
Thus, from the properties $iii$\textbf{\ }and $vii$\textbf{\ }we have,%
\begin{eqnarray*}
g\left( \mathbf{V}_{\mathbf{5}}^{^{\prime }},\mathbf{V}_{\mathbf{6}}\right)
&=&g\left( \mathbf{V}_{3}\wedge \left( -\mathbf{V}_{\mathbf{\ 3}}\wedge 
\mathbf{V}_{2}\right) ,\mathbf{V}_{\mathbf{1}}^{^{\prime }}\right) +g\left( 
\mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{\ 6}},\mathbf{%
V}_{1}\right) \\
&=&-g\left( \mathbf{V}_{3}\wedge \left( \mathbf{V}_{\mathbf{\ 3}}\wedge 
\mathbf{V}_{2}\right) ,\mathbf{V}_{\mathbf{1}}^{^{\prime }}\right) +g\left( 
\mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{\ 6}},\mathbf{%
V}_{1}\right) \\
&=&-g\left( \left\langle \mathbf{V}_{3},\mathbf{V}_{2}\right\rangle \mathbf{V%
}_{\mathbf{\ 3}}-\left\langle \mathbf{V}_{3},\mathbf{V}_{3}\right\rangle 
\mathbf{V}_{2},\mathbf{V}_{\mathbf{1}}^{^{\prime }}\right) +g\left( \mathbf{V%
}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{\ 6}},\mathbf{V}%
_{1}\right) \\
&=&g\left( \mathbf{V}_{2},\mathbf{V}_{\mathbf{1}}^{^{\prime }}\right)
+g\left( \mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{6}},%
\mathbf{V}_{1}\right) \\
&=&k_{2}+g\left( \mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{%
\mathbf{6}},\mathbf{V}_{1}\right) \text{.}
\end{eqnarray*}%
Let us calculate%
\begin{eqnarray*}
g\left( \mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{6}},%
\mathbf{V}_{1}\right) &=&g\left( \left( -k_{3}\mathbf{V}_{\mathbf{2}}+k_{4}%
\mathbf{V}_{\mathbf{4}}\right) \wedge \mathbf{V}_{\mathbf{6}},\mathbf{V}%
_{1}\right) \\
&=&-k_{3}g\left( \mathbf{V}_{\mathbf{2}}\wedge \mathbf{V}_{\mathbf{6}},%
\mathbf{V}_{1}\right) +k_{4}g\left( \mathbf{V}_{\mathbf{4}}\wedge \mathbf{V}%
_{\mathbf{6}},\mathbf{V}_{1}\right) \\
&=&-k_{3}g\left( \left( -\mathbf{V}_{\mathbf{3}}\right) ,\mathbf{V}%
_{1}\right) +k_{4}g\left( \mathbf{V}_{1},\mathbf{V}_{1}\right)
\end{eqnarray*}%
\begin{equation}
g\left( \mathbf{V}_{\mathbf{3}}^{^{\prime }}\wedge \mathbf{V}_{\mathbf{6}},%
\mathbf{V}_{1}\right) =k_{4}\text{,}  \label{3.10}
\end{equation}

and thus%
\begin{equation}
k_{6}=g\left( \mathbf{V}_{\mathbf{5}}^{^{\prime }},\mathbf{V}_{\mathbf{6}%
}\right) =k_{2}+k_{4}\text{.}  \label{3.11}
\end{equation}%
Differentiating on both sides of the expression $\mathbf{V}_{\mathbf{6}}=%
\mathbf{V}_{\mathbf{2}}\times \mathbf{V}_{3}~$with respect to $s$, we get%
\begin{equation*}
\mathbf{V}_{\mathbf{6}}^{^{\prime }}=\mathbf{V}_{\mathbf{2}}^{^{\prime
}}\times \mathbf{V}_{3}+\mathbf{V}_{2}\times \mathbf{V}_{\mathbf{3}%
}^{^{\prime }}
\end{equation*}%
Finally, from the Eqs. (3.6), (3.7), (3.10) and (3.11) we get 
\begin{eqnarray*}
\mathbf{V}_{\mathbf{6}}^{^{\prime }} &=&\mathbf{V}_{\mathbf{2}}^{^{\prime
}}\times \mathbf{V}_{3}+\mathbf{V}_{2}\times \mathbf{V}_{\mathbf{3}%
}^{^{\prime }} \\
&=&\left( -k_{2}\mathbf{V}_{\mathbf{1}}+k_{3}\mathbf{V}_{\mathbf{3}}\right)
\times \mathbf{V}_{3}+\mathbf{V}_{2}\times \mathbf{V}_{\mathbf{3}}^{^{\prime
}} \\
&=&-k_{2}\left( \mathbf{V}_{\mathbf{1}}\times \mathbf{V}_{3}\right)
+k_{3}\left( \mathbf{V}_{3}\times \mathbf{V}_{3}\right) +\mathbf{V}%
_{2}\times \mathbf{V}_{\mathbf{3}}^{^{\prime }} \\
&=&-k_{2}\mathbf{V}_{\mathbf{5}}-k_{3}+\mathbf{V}_{2}\times \left( -k_{3}%
\mathbf{V}_{2}+k_{4}\mathbf{V}_{\mathbf{4}}\right) \\
&=&-k_{2}\mathbf{V}_{\mathbf{5}}-k_{3}-k_{3}\left( \mathbf{V}_{2}\times 
\mathbf{V}_{2}\right) +k_{4}\left( \mathbf{V}_{2}\times \mathbf{V}_{\mathbf{4%
}}\right) \\
&=&-k_{2}\mathbf{V}_{\mathbf{5}}-k_{3}+k_{3}-k_{4}\mathbf{V}_{5} \\
&=&-\left( k_{2}+k_{4}\right) \mathbf{V}_{5} \\
&=&-k_{6}\mathbf{V}_{5}
\end{eqnarray*}
\end{proof}

\textbf{Corollary 4.1 }Serret-Frenet formulae for the USSROC can be written
in matrix notation as%
\begin{equation}
\left[ 
\begin{array}{c}
\mathbf{V}_{\mathbf{0}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{1}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{2}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{3}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{4}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{5}}^{^{\prime }} \\ 
\mathbf{V}_{\mathbf{6}}^{^{\prime }}%
\end{array}%
\right] =\left[ 
\begin{array}{ccccccc}
0 & k_{1} & 0 & 0 & 0 & 0 & 0 \\ 
-k_{1} & 0 & k_{2} & 0 & 0 & 0 & 0 \\ 
0 & -k_{2} & 0 & k_{3} & 0 & 0 & 0 \\ 
0 & 0 & -k_{3} & 0 & k_{4} & 0 & 0 \\ 
0 & 0 & 0 & -k_{4} & 0 & k_{5} & 0 \\ 
0 & 0 & 0 & 0 & -k_{5} & 0 & k_{6} \\ 
0 & 0 & 0 & 0 & 0 & -k_{6} & 0%
\end{array}%
\right] \left[ 
\begin{array}{c}
\mathbf{V}_{0} \\ 
\mathbf{V}_{\mathbf{1}} \\ 
\mathbf{V}_{\mathbf{2}} \\ 
\mathbf{V}_{\mathbf{3}} \\ 
\mathbf{V}_{\mathbf{4}} \\ 
\mathbf{V}_{\mathbf{5}} \\ 
\mathbf{V}_{\mathbf{6}}%
\end{array}%
\right] \text{.}  \label{3.12}
\end{equation}

Let us now by using the Serret-Frenet apparatus for SOC, we compute the
Serret-Frenet apparatus for OC.

Let $\beta :I\subset \mathbb{R}\rightarrow $ $\mathbb{R}^{8}$ be a space
curve and $\left\{ \mathbf{W}_{j}\right\} $, $0\leq j\leq 7~$be the Frenet
frame of $\beta $ in the the Euclidean $8$-space, $\mathbb{R}^{8}$. Now, we
are going to take the octonions instead of all the Frenet elements. Thus,
Frenet elements $\mathbf{W}_{j}~$can be written as an octonion which is
defined by

\begin{equation*}
\mathbf{W}_{j}=\underset{i=0}{\overset{7}{\sum }}w_{j}e_{j}.
\end{equation*}

\textbf{Definition 3.3 }Let $\mathbb{R}^{8}$ characterize the Euclidean $8$%
-space with octonionic metric $g$ \ and $\mathbb{R}^{8}$ is identified with
the set of the octonion. The curve $\beta _{\mathbb{O}}:I\subset \mathbb{R}%
\rightarrow \mathbb{O}$, $\beta _{\mathbb{O}}\left( s\right) =\underset{i=0}{%
\overset{7}{\sum }}\gamma _{i}\left( s\right) e_{i}~$is called octonionic
curve (OC). Note that the vector part of $\beta _{\mathbb{O}}$ is same to
SOC $\gamma _{\mathbb{O}}$ in $\mathbb{O}_{S}$.

\textbf{Definition 3.4 }If\textbf{\ }the norm of the first derivative of the
OC is equal $1$, then OC is called the unit speed octonionic curves (USOC).

\begin{theorem}
Let~$\beta _{\mathbb{O}}:I\subset \mathbb{R}\rightarrow \mathbb{O}~$be an
USOC~and $\mathbf{W}_{0}\left( s\right) =$ $\beta ^{^{\prime }}\left(
s\right) =\underset{i=0}{\overset{7}{\sum }}\gamma _{i}^{^{\prime }}\left(
s\right) e_{i}$ be unit tangent vector of $\beta $. Then, $\mathbf{W}%
_{0}^{^{\prime }}$ is orthogonal to $\mathbf{W}_{0}$.
\end{theorem}

\begin{proof}
Let \bigskip $\beta _{\mathbb{O}}:I\subset \mathbb{R}\rightarrow \mathbb{O}$%
, $\beta _{\mathbb{O}}\left( s\right) =\underset{i=1}{\overset{7}{\sum }}%
\gamma _{i}\left( s\right) e_{i}$~\ be an USOC. Since the tangent ~$\mathbf{W%
}_{0}=$ $\beta ^{^{\prime }}\left( s\right) =\underset{i=0}{\overset{7}{\sum 
}}\gamma _{i}^{^{\prime }}\left( s\right) e_{i}~$has unit length~(in other
words, $\left\Vert \mathbf{W}_{0}\right\Vert =1~$for all $s)$, we get%
\begin{equation*}
\left\Vert \mathbf{W}_{0}\right\Vert ^{2}=g\left( \mathbf{W}_{0},\mathbf{W}%
_{0}\right) =\mathbf{W}_{0}\mathbf{\times }\overline{\mathbf{W}_{0}}=1\text{.%
}
\end{equation*}

Thus, differentiating with respect to $s$ gives 
\begin{equation}
\mathbf{W}_{0}^{^{\prime }}\times \overline{\mathbf{W}_{0}}+\mathbf{W}%
_{0}\times \left( \overline{\mathbf{W}_{0}}\right) ^{^{\prime }}=0\text{.}
\label{3.13}
\end{equation}

Since $\mathbf{W}_{0}=\underset{i=0}{\overset{7}{\sum }}\gamma
_{i}^{^{\prime }}\left( s\right) e_{i}$, we may write $\overline{\mathbf{W}%
_{0}}=\gamma _{0}^{^{\prime }}-\underset{i=1}{\overset{7}{\sum }}\gamma
_{i}^{^{\prime }}e_{i}$ and so $\left( \overline{\mathbf{W}_{0}}\right)
^{^{\prime }}=\gamma _{0}^{^{\prime \prime }}-\underset{i=1}{\overset{7}{%
\sum }}\gamma _{i}^{^{\prime \prime }}e_{i}$. So, we have $\left( \overline{%
\mathbf{W}_{0}}\right) ^{^{\prime }}=\overline{\mathbf{W}_{0}^{^{\prime }}}$.%
$~$Substituting the statement $\left( \overline{\mathbf{W}_{0}}\right)
^{^{\prime }}=\overline{\mathbf{W}_{0}^{^{\prime }}}$ into the Eq. (3.13),
we obtain

\begin{equation*}
\mathbf{W}_{0}^{^{\prime }}\times \overline{\mathbf{W}_{0}}+\mathbf{W}%
_{0}\times \overline{\mathbf{W}_{0}^{^{\prime }}}=0\text{.}
\end{equation*}

In this case, $\mathbf{W}_{0}^{^{\prime }}$ is orthogonal to $\mathbf{W}_{0}%
\mathbf{.}$
\end{proof}

\begin{theorem}
Let~$\beta _{\mathbb{O}}~$be an USOC and $\left\{ \mathbf{W}_{j}\right\} $, $%
0\leq j\leq 7~$be the Frenet frame of USOC in $\mathbb{R}^{8}.$~Then Frenet
equations are obtained by 
\begin{equation}
\begin{array}{l}
\mathbf{W}_{0}^{^{\prime }}\left( s\right) =K\left( s\right) \mathbf{W}_{%
\mathbf{1}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{1}}^{^{\prime }}\left( s\right) =-K\left( s\right) 
\mathbf{W}_{0}\left( s\right) +k_{1}\left( s\right) \mathbf{W}_{\mathbf{2}%
}\left( s\right) \\ 
\mathbf{W}_{\mathbf{2}}^{^{\prime }}\left( s\right) =-k_{1}\left( s\right) 
\mathbf{W}_{\mathbf{1}}\left( s\right) +\left( k_{2}-K\right) \left(
s\right) \mathbf{W}_{\mathbf{3}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{3}}^{^{\prime }}\left( s\right) =-\left( k_{2}-K\right)
\left( s\right) \mathbf{W}_{\mathbf{2}}\left( s\right) +k_{3}\left( s\right) 
\mathbf{W}_{\mathbf{4}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{4}}^{^{\prime }}\left( s\right) =-k_{3}\left( s\right) 
\mathbf{W}_{\mathbf{3}}\left( s\right) +\left( k_{4}-K\right) \left(
s\right) \mathbf{W}_{\mathbf{5}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{5}}^{^{\prime }}\left( s\right) =-\left( k_{4}-K\right)
\left( s\right) \mathbf{W}_{\mathbf{4}}\left( s\right) +k_{5}\left( s\right) 
\mathbf{W}_{\mathbf{6}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{6}}^{^{\prime }}\left( s\right) =-k_{5}\left( s\right) 
\mathbf{W}_{\mathbf{5}}\left( s\right) +\left( k_{6}+K\right) \left(
s\right) \mathbf{W}_{\mathbf{7}}\left( s\right) \\ 
\mathbf{W}_{\mathbf{7}}^{^{\prime }}\left( s\right) =-\left( k_{6}+K\right)
\left( s\right) \mathbf{W}_{\mathbf{6}}\left( s\right) \text{,}%
\end{array}
\label{3.14}
\end{equation}%
where $\mathbf{W}_{\mathbf{1}}=\mathbf{V}_{0}\mathbf{\times W}_{0}$\textbf{,~%
}$\mathbf{W}_{\mathbf{2}}=\mathbf{V}_{\mathbf{1}}\mathbf{\times W}_{0}$%
\textbf{,~}$\mathbf{W}_{\mathbf{3}}=\mathbf{V}_{\mathbf{2}}\mathbf{\times W}%
_{0}$\textbf{, }$\mathbf{W}_{\mathbf{4}}=\mathbf{V}_{\mathbf{3}}\mathbf{\
\times W}_{0}$\textbf{,~}$\mathbf{W}_{\mathbf{5}}=\mathbf{V}_{\mathbf{4}}%
\mathbf{\times W}_{0}$\textbf{,~}$\mathbf{\mathbf{W}_{\mathbf{6}}=V_{\mathbf{%
5}}\mathbf{\times }W}_{0}$, $\mathbf{W}_{\mathbf{7}}=\mathbf{V}_{\mathbf{6}}%
\mathbf{\times W}_{0}$\textbf{,}$\mathbf{~}K=\left\Vert \mathbf{W}%
_{0}^{^{\prime }}\left( s\right) \right\Vert .$
\end{theorem}

\begin{proof}
Let us assume that%
\begin{equation}
\mathbf{W}_{0}^{^{\prime }}=K\mathbf{N}_{\mathbf{1}},~K=\left\Vert \mathbf{W}%
_{0}^{^{\prime }}\left( s\right) \right\Vert ,~\left\Vert \mathbf{N}_{%
\mathbf{1}}\right\Vert =1\text{.}  \label{3.15}
\end{equation}

Hence, substituting Eq. (3.15) into the $\mathbf{W}_{0}^{^{\prime }}\times 
\overline{\mathbf{W}_{0}}+\mathbf{W}_{0}\times \overline{\mathbf{W}%
_{0}^{^{\prime }}}=0$\textbf{,} we get%
\begin{eqnarray*}
\left( K\mathbf{N}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}\right)
+\left( \mathbf{W}_{0}\times \overline{K\mathbf{N}_{\mathbf{1}}}\right) &=&0
\\
K\left( \mathbf{N}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}+\mathbf{W}%
_{0}\times \overline{\mathbf{N}_{\mathbf{1}}}\right) &=&0\text{.}
\end{eqnarray*}

On the other hand, $\mathbf{W}_{\mathbf{1}}~$is orthogonal to $\mathbf{W}%
_{0} $. So, we have $g\left( \mathbf{W}_{0},\mathbf{W}_{\mathbf{1}}\right) =0
$. From the rules of conjugate of the octonion, we obtain%
\begin{equation*}
\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}+\mathbf{W}_{0}\times 
\overline{\mathbf{W}_{\mathbf{1}}}=\mathbf{W}_{\mathbf{1}}\times \overline{%
\mathbf{W}_{0}}+\overline{\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W}%
_{0}}}=0\text{.}
\end{equation*}

Thus, $\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}~$ is a
spatial octonion.

Since $\mathbf{V}_{0}$ and $\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W%
}_{0}}$ are the spatial octonions and they have unit magnitude, we may
choose $\mathbf{V}_{0}$ as follows:%
\begin{equation}
\mathbf{V}_{0}=~\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}
\label{3.16}
\end{equation}

On the other hand, we get%
\begin{eqnarray*}
\mathbf{V}_{0}\times \mathbf{W}_{0} &=&\left( \mathbf{W}_{\mathbf{1}}\times 
\overline{\mathbf{W}_{0}}\right) ~\times \mathbf{W}_{0} \\
&=&\mathbf{W}_{\mathbf{1}}\times \overline{\mathbf{W}_{0}}\times \mathbf{W}%
_{0} \\
&=&\mathbf{W}_{\mathbf{1}}\times \left\Vert \mathbf{W}_{0}\right\Vert ^{2}%
\text{,}
\end{eqnarray*}

and thus%
\begin{equation*}
\mathbf{W}_{\mathbf{1}}=\mathbf{V}_{0}\times \mathbf{W}_{0}.
\end{equation*}

By differentiating the last equation with respect to $s~$and using the
Eqs.(3.3) and (3.15) in the last equation, then we have%
\begin{eqnarray*}
\mathbf{N}_{\mathbf{1}}^{^{\prime }} &=&\mathbf{V}_{0}^{^{\prime }}\times 
\mathbf{W}_{0}\mathbf{+V}_{0}\times \mathbf{W}_{0}^{^{\prime }} \\
&=&k_{1}\mathbf{V}_{\mathbf{1}}\times \mathbf{W}_{0}\mathbf{+V}_{0}\times K%
\mathbf{W}_{\mathbf{1}} \\
&=&k_{1}\mathbf{W}_{\mathbf{2}}\mathbf{+}K\left( \mathbf{V}_{0}\times 
\mathbf{W}_{\mathbf{1}}\right) ,~\mathbf{W}_{\mathbf{2}}=\mathbf{V}_{\mathbf{%
1}}\times \mathbf{W}_{0} \\
&=&k_{1}\mathbf{W}_{\mathbf{2}}\mathbf{+}K\left( \mathbf{V}_{0}\times \left( 
\mathbf{V}_{0}\times \mathbf{W}_{0}\right) \right) \\
&=&-K\mathbf{W}_{0}\mathbf{+}k_{1}\mathbf{W}_{\mathbf{2}}\text{,}
\end{eqnarray*}

and thus%
\begin{equation}
\mathbf{W}_{\mathbf{1}}^{^{\prime }}=-K\mathbf{W}_{0}+k_{1}\mathbf{W}_{%
\mathbf{2}}\mathbf{,~}~\mathbf{W}_{\mathbf{2}}=\mathbf{V}_{\mathbf{1}}\times 
\mathbf{W}_{0}\text{\textbf{.}}  \label{3.17}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{2}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{2}}\right\Vert ^{2}=1$.

\textbf{ii) }$\mathbf{W}_{\mathbf{2}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$ and $\mathbf{W%
}_{\mathbf{2}}$ are mutually $g~$orthogonal since $\mathbf{V}_{\mathbf{0}}$
and $\mathbf{V}_{\mathbf{1}}$ are so.

By differentiating $\mathbf{W}_{\mathbf{2}}$ given by Eq. (3.17) and
substituting the Eqs. (3.5), (3.15) and (3.16) into this equation, we get%
\begin{equation}
\mathbf{W}_{\mathbf{2}}^{^{\prime }}=-k_{1}\mathbf{W}_{\mathbf{1}}\left(
s\right) +\left( k_{2}-K\right) \mathbf{W}_{\mathbf{3}},~\mathbf{W}_{\mathbf{%
3}}=\mathbf{V}_{\mathbf{2}}\times \mathbf{W}_{0}\mathbf{.}  \label{3.18}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{3}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{3}}\right\Vert ^{2}=1.$

\textbf{ii) }$\mathbf{W}_{\mathbf{3}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$,$\mathbf{W}_{%
\mathbf{2}}$,and $\mathbf{W}_{\mathbf{3}}$ are mutually $g~$orthogonal since 
$\mathbf{V}_{0}$,$\mathbf{V}_{\mathbf{1}}$ and $\mathbf{V}_{\mathbf{2}}$ are
so.

By differentiating $\mathbf{W}_{\mathbf{3}}$ given by Eq. (3.18) and
substituting the Eqs. (3.6), (3.15) ,(3.16) and (3.17) into this equation,
we get%
\begin{equation}
\mathbf{W}_{\mathbf{3}}^{^{\prime }}=-\left( k_{2}-K\right) \mathbf{W}_{%
\mathbf{2}}+k_{3}\mathbf{W}_{\mathbf{4}},~\mathbf{W}_{\mathbf{4}}=\mathbf{V}%
_{\mathbf{3}}\times \mathbf{W}_{0}\mathbf{.}  \label{3.19}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{4}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{4}}\right\Vert ^{2}=1.$

\textbf{ii) }$\mathbf{W}_{\mathbf{4}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$,$\mathbf{W}_{%
\mathbf{2}}$,$\mathbf{W}_{\mathbf{3}}$,and $\mathbf{W}_{\mathbf{4}}$ are
mutually $g~$orthogonal since $\mathbf{V}_{0}$,$\mathbf{V}_{\mathbf{1}}$,$%
\mathbf{V}_{\mathbf{2}}$ and $\mathbf{V}_{\mathbf{3}}$ are so.

By differentiating $\mathbf{W}_{\mathbf{4}}$ given by Eq. (3.19) and
substituting the Eqs. (3.8), (3.15) ,(3.17) and (3.18) into this equation,
we get%
\begin{equation}
\mathbf{W}_{\mathbf{4}}^{^{\prime }}=-k_{3}\mathbf{W}_{\mathbf{3}}+\left(
k_{4}-K\right) \mathbf{W}_{\mathbf{5}},~\mathbf{W}_{\mathbf{5}}=\mathbf{V}_{%
\mathbf{4}}\times \mathbf{W}_{0}\mathbf{.}  \label{3.20}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{5}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{5}}\right\Vert ^{2}=1.$

\textbf{ii) }$\mathbf{W}_{\mathbf{5}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$,$\mathbf{W}_{%
\mathbf{2}}$,$\mathbf{W}_{\mathbf{3}}\,$,$\mathbf{W}_{\mathbf{4}}$,and $%
\mathbf{W}_{\mathbf{5}}$ are mutually $g~$orthogonal since $\mathbf{V}_{0}$,$%
\mathbf{V}_{\mathbf{1}}$,$\mathbf{V}_{\mathbf{2}}$,$\mathbf{V}_{\mathbf{3}}$
and $\mathbf{V}_{\mathbf{4}}$ are so.

By differentiating $\mathbf{W}_{\mathbf{5}}$ given by Eq. (3.20) and
substituting the Eqs. (3.9), (3.15) ,(3.18) and (3.19) into this equation,
we get%
\begin{equation}
\mathbf{W}_{\mathbf{5}}^{^{\prime }}=-\left( k_{4}-K\right) \mathbf{W}_{%
\mathbf{4}}+k_{5}\mathbf{W}_{\mathbf{6}},~\mathbf{W}_{\mathbf{6}}=\mathbf{V}%
_{\mathbf{5}}\times \mathbf{W}_{0}\mathbf{.}  \label{3.21}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{6}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{6}}\right\Vert ^{2}=1.$

\textbf{ii) }$\mathbf{W}_{\mathbf{6}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$,$\mathbf{W}_{%
\mathbf{2}}$,$\mathbf{W}_{\mathbf{3}}\,$,$\mathbf{W}_{\mathbf{4}}$,$\mathbf{W%
}_{\mathbf{5}}$,and $\mathbf{W}_{\mathbf{6}}$ are mutually $g~$orthogonal
since $\mathbf{V}_{0}$,$\mathbf{V}_{\mathbf{1}}$,$\mathbf{V}_{\mathbf{2}}$,$%
\mathbf{V}_{\mathbf{3}}$,$\mathbf{V}_{\mathbf{4}}$ and $\mathbf{V}_{\mathbf{5%
}}$ are so.

By differentiating $\mathbf{W}_{\mathbf{6}}$ given by Eq. (3.21) and
substituting the Eqs. (3.10), (3.15) ,(3.19) and (3.20) into this equation,
we get%
\begin{equation}
\mathbf{W}_{\mathbf{6}}^{^{\prime }}=-k_{5}\mathbf{W}_{\mathbf{5}}+\left(
k_{6}+K\right) \mathbf{W}_{\mathbf{7}},~\mathbf{W}_{\mathbf{7}}=\mathbf{V}_{%
\mathbf{6}}\times \mathbf{W}_{0}\mathbf{.}  \label{3.22}
\end{equation}%
Let us give some informations about $\mathbf{W}_{\mathbf{7}}$ as follows:

\textbf{i)} $\left\Vert \mathbf{W}_{\mathbf{7}}\right\Vert ^{2}=1.$

\textbf{ii) }$\mathbf{W}_{\mathbf{7}}\left( s\right) ~$is a smooth octonion
function of $s$ and $\mathbf{W}_{0}$,$\mathbf{W}_{\mathbf{1}}$,$\mathbf{W}_{%
\mathbf{2}}$,$\mathbf{W}_{\mathbf{3}}\,$,$\mathbf{W}_{\mathbf{4}}$,$\mathbf{W%
}_{\mathbf{5}}$,$\mathbf{W}_{\mathbf{6}}$,and $\mathbf{W}_{\mathbf{7}}$ are
mutually $g~$orthogonal since $\mathbf{V}_{0}$,$\mathbf{V}_{\mathbf{1}}$,$%
\mathbf{V}_{\mathbf{2}}$,$\mathbf{V}_{\mathbf{3}}$,$\mathbf{V}_{\mathbf{4}}$,%
$\mathbf{V}_{\mathbf{5}}$ and $\mathbf{V}_{\mathbf{6}}$ are so.

Finally, by differentiating $\mathbf{W}_{\mathbf{7}}~$given by Eq. (3.22)
and substituting the Eqs., we get%
\begin{equation}
\mathbf{W}_{\mathbf{7}}^{^{\prime }}=-\left( k_{6}+K\right) \mathbf{W}_{%
\mathbf{6}}.  \label{3.23}
\end{equation}%
Serret-Frenet formulae for the USOC can be written in matrix notation as%
\begin{equation*}
\frac{d}{ds}{\tiny \left[ 
\begin{array}{c}
\mathbf{W}_{0} \\ 
\mathbf{W}_{\mathbf{1}} \\ 
\mathbf{W}_{\mathbf{2}} \\ 
\mathbf{W}_{\mathbf{3}} \\ 
\mathbf{W}_{\mathbf{4}} \\ 
\mathbf{W}_{\mathbf{5}} \\ 
\mathbf{W}_{\mathbf{6}} \\ 
\mathbf{W}_{\mathbf{7}}%
\end{array}%
\right] =\left[ 
\begin{array}{cccccccc}
0 & K & 0 & 0 & 0 & 0 & 0 & 0 \\ 
-K & 0 & k_{1} & 0 & 0 & 0 & 0 & 0 \\ 
0 & -k_{1} & 0 & \left( k_{2}-K\right) & 0 & 0 & 0 & 0 \\ 
0 & 0 & -\left( k_{2}-K\right) & 0 & k_{3} & 0 & 0 & 0 \\ 
0 & 0 & 0 & -k_{3} & 0 & \left( k_{4}-K\right) & 0 & 0 \\ 
0 & 0 & 0 & 0 & -\left( k_{4}-K\right) & 0 & k_{5} & 0 \\ 
0 & 0 & 0 & 0 & 0 & -k_{5} & 0 & \left( k_{6}+K\right) \\ 
0 & 0 & 0 & 0 & 0 & 0 & -\left( k_{6}+K\right) & 0%
\end{array}%
\right] \left[ 
\begin{array}{c}
\mathbf{W}_{0} \\ 
\mathbf{W}_{\mathbf{1}} \\ 
\mathbf{W}_{\mathbf{2}} \\ 
\mathbf{W}_{\mathbf{3}} \\ 
\mathbf{W}_{\mathbf{4}} \\ 
\mathbf{W}_{\mathbf{5}} \\ 
\mathbf{W}_{\mathbf{6}} \\ 
\mathbf{W}_{\mathbf{7}}%
\end{array}%
\right] }
\end{equation*}

This is the Serret Frenet formulae for USOC $\beta _{\mathbb{O}}$ in $%
\mathbb{R}^{8}.$

$\left\{ \mathbf{W}_{0}\text{,}\mathbf{W}_{\mathbf{1}}\text{,}\mathbf{W}_{%
\mathbf{2}}\text{,}\mathbf{W}_{\mathbf{3}}\text{,}\mathbf{W}_{\mathbf{4}}%
\text{,}\mathbf{W}_{\mathbf{5}}\text{,}\mathbf{W}_{\mathbf{6}}\text{,}%
\mathbf{W}_{\mathbf{7}}\text{,}K\text{,}k_{1}\text{,}\left( k_{2}-K\right) 
\text{,}k_{3}\text{,}\left( k_{4}-K\right) \text{,}k_{5}\text{,}\left(
k_{6}+K\right) \right\} ~$denotes Serret-Frenet apparatus for the USOC $%
\beta _{\mathbb{O}}~$in $\mathbb{R}^{8}.$
\end{proof}

\begin{conclusion}
We obtain the Serret-Frenet formulae and the Serret-Frenet apparatus for the
ROC $\beta _{\mathbb{O}}~$by making use of the Serret-Frenet formulae for a
SROC $\gamma _{\mathbb{O}}~$in $\mathbb{R}^{7}$. This curve is so chosen
that the unit tangent to it is $\mathbf{V}_{0}\left( s\right) $ is given by
(3.16). It should be noted here that the second curvature of $\beta _{%
\mathbb{O}}~$is first curvature of $\gamma _{\mathbb{O}}$, fourth curvature
of $\beta _{\mathbb{O}}~$is third curvature of $\gamma $, sixth curvature of 
$\beta _{\mathbb{O}}~$is fifth curvature of $\gamma $. Note that the third
curvature of $\beta _{\mathbb{O}}~$is $\left( k_{2}-K\right) $, where $%
k_{2}~ $is the second curvature of the curve $\gamma _{\mathbb{O}}~$and $K~$%
is the first curvature of $\beta _{\mathbb{O}}$, $~$fifth curvature of $%
\beta _{\mathbb{O}}~$is $\left( k_{4}-K\right) $, where $k_{4}~$is the
fourth curvature of the curve $\gamma _{\mathbb{O}}~$and $K~$is the first
curvature of $\beta _{\mathbb{O}}$, sixth~curvature of $\beta _{\mathbb{O}}~$%
is $\left( k_{6}+K\right) $, where $k_{6}~$is the sixth curvature of the
curve $\gamma _{\mathbb{O}}~$and $K~$is the first curvature of $\beta _{%
\mathbb{O}}$. Also, the sum of second curvature and fourth curvature of $%
\gamma _{\mathbb{O}}~$ is~sixth curvature of $\gamma _{\mathbb{O}}$. As can
be easliy seen, classical methods of elemantary differential geometry do
note give us the technique to determine the SROC $\gamma _{\mathbb{O}}~$in $%
\mathbb{R}^{7}~$corresponding to $\beta _{\mathbb{O}}~$ in $\mathbb{R}^{8}$.
Space curves in $7$ and $8$ Euclidean space are defined by with this study.
This paper is important from this point. In view of the quaternionic curves,
there are a lot of papers about quaternionic curves. After these defining
octonionic curves, a lot of paper about octonionic curves can be studied.
\end{conclusion}

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\"{O}.B.: Recep Tayyip Erdogan University, Faculty of Arts And Sciences,
Department of Mathematics, Rize, Turkey

Corresponding author.

E-mail: ozcan.bektas@erdogan.edu.tr.

\bigskip

S.Y.: Yildiz Technical University, Faculty of Arts And Sciences, Department
of Mathematics, Istanbul, Turkey

E-mail: sayuce@yildiz.edu.tr.

\bigskip

\end{document}
