# Higher-Order Curvatures of Plane and Space Parametrized Curves
## 5 The Case of Three-Dimensional Curves  
Suppose now that the given curve is a space one, \(C\subset\mathbb{R}^{3}\), having the curvature function \(k=k(s)\) and the torsion function \(\tau=\tau(s)\). Consider the matrices:  
\[\left\{ \begin{array}{l}R_{1}(t)=\left(\begin{array}{ccc}1&0&0\\ 0&\cos t&-\sin\ t\\ 0&\sin t&\cos\ t\\ \end{array}\right)\in\{1\}\times SO(2),\\ R_{2}(t)=\left(\begin{array}{ccc}\cos t&0&\sin\ t\\ 0&1&0\\ \sin t&0&-\cos t\\ \end{array}\right)\in O^{-}(3):=O(3)\setminus SO(3)=\{\Gamma\in O(3);\det \Gamma=-1\},\\ R_{3}(t)=\left(\begin{array}{ccc}\cos t&-\sin t&0\\ \sin t&\cos t&0\\ 0&0&1\\ \end{array}\right)\in SO(2)\times\{1\}.\\ \end{array} \right. \tag{32}\]  
The Frenet frame  
\[\mathcal{F}(s):=\left( \begin{array}{c}T\\ N\\ B\end{array} \right)(s) \tag{33}\]  
satisfies the matrix Frenet equation  
\[\frac{d}{ds}\mathcal{F}(s)=-[\tau(s)R_{1}^{\prime}(0)+k(s)R_{3}^{\prime}(0)] \mathcal{F}(s), \tag{34}\]  
which recalls the expression of the associated Darboux vector field:  
\[\Omega(s):=\tau(s)T(s)+k(s)B(s). \tag{35}\]  
Deriving (34) we get:  
\[\frac{d^{2}}{ds^{2}}\mathcal{F}=[-\tau^{\prime}R_{1}^{\prime}(0)-k^{\prime}R_ {3}^{\prime}(0)+(\tau R_{1}^{\prime}(0)+kR_{3}^{\prime}(0))^{2}]\mathcal{F}. \tag{36}\]  
We have:  
\[[R_{1}^{\prime}(0)]^{2}=\left( \begin{array}{ccc}0&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{array} \right)\in\{0\}\times SO(2),[R_{3}^{\prime}(0)]^{2}=\left( \begin{array}{ccc}-1&0&0\\ 0&-1&0\\ 0&0&0\\ \end{array} \right)\in SO(2)\times\{0\}, \tag{37}\]  
\[R_{1}^{\prime}(0)R_{3}^{\prime}(0)=\left( \begin{array}{ccc}0&0&0\\ 0&0&0\\ 1&0&0\\ \end{array} \right),R_{3}^{\prime}(0)R_{1}^{\prime}(0)=\left( \begin{array}{ ccc}0&0&1\\ 0&0&0\\ 0&0&0\\ \end{array} \right),R_{1}^{\prime}(0)R_{3}^{\prime}(0)+R_{3}^{\prime}(0)R_{1}^{ \prime}(0)=R_{2}^{\prime}(0). \tag{38}\]  
The occurrence of \(R_{2}^{\prime}(0)\) in expression (36) and the fact that \(O^{-}(3)\) is not a subgroup in \(O(3)\) make the attempt to find some higher-order curvatures very difficult in the present manner, so we look for an alternative way. More precisely, we consider the middle Frenet equation \(N^{\prime}(s)=-k(s)T(s)+\tau(s)B(s)\) as the first rung of a ladder:  
\[\frac{d^{n}}{ds^{n}}N=R_{T}^{n}T+K_{N}^{n}N+K_{B}^{n}B,\quad n\geq 1,\quad K_{T} ^{1}=-k,\quad K_{N}^{1}=0,\quad K_{B}^{1}=\tau \tag{39}\]  
and then we get immediately the following theorem. Theorem 5.1: _The recurrence relation of the curvature's triple \((K_{T},K_{N},K_{B})\) is:_  
\[\left( \begin{array}{c}K_{T}^{n+1}\\ K_{N}^{n+1}\\ K_{B}^{n+1}\end{array} \right)=\frac{d}{ds}\left( \begin{array}{c}K_{T}^{n}\\ K_{N}^{n}\\ K_{B}^{n}\end{array} \right)-\left( \begin{array}{ccc}0&k&0\\ -k&0&\tau\\ 0&-\tau&0\\ \end{array} \right)\left( \begin{array}{c}K_{T}^{n}\\ K_{N}^{n}\\ K_{B}^{n}\end{array} \right). \tag{40}\]_Therefore, we have the first two higher-order triples:_  
\[\left\{ \begin{array}{l}(K_{T}^{2},K_{N^{\prime}}^{2},K_{B}^{2})=(-k^{\prime},-( k^{2}+\tau^{2}),\tau^{\prime}),\\ (K_{T^{\prime}}^{3},K_{N^{\prime}}^{3},K_{B}^{3})=(k(k^{2}+\tau^{2})-k^{\prime \prime},-3(kk^{\prime}+\tau\tau^{\prime}),\tau^{\prime\prime}-\tau(k^{2}+\tau^ {2})).\end{array} \right. \tag{41}\]  
**Remark 2**. : _(i) If \(C\) reduces to a plane curve, then \(\tau=0\) implies \(K_{T}^{1}=-k=K_{o}^{1}\), \(K_{T}^{2}=K_{o}^{2}\) and \(K_{T}^{3}=K_{o}^{3}\), which shows that this approach is the correct one. Moreover, \(K_{p}^{1}=K_{N}^{1}\), \(K_{p}^{2}=K_{N}^{2}\) and \(K_{p}^{3}=K_{N}^{3}\)._  
_(ii) Suppose that \(C\) is a Mannheim curve; then, according to relation (2) of [7], we have \(k=\lambda(k^{2}+\tau^{2})\) for a nonzero constant \(\lambda\). Then, the curvatures (41) become:_  
\[(K_{T}^{2},K_{N}^{2},K_{B}^{2})=\left(-k^{\prime},-\frac{k}{\lambda},\tau^{ \prime}\right),\quad(K_{T}^{3},K_{N}^{3},K_{B}^{3})=\left(\lambda k^{2}-k^{ \prime\prime},-\frac{3k^{\prime}}{2\lambda},\tau^{\prime\prime}-\lambda k\tau \right). \tag{42}\]  
**Definition 3**. : _Let \(\omega\in\mathbb{R}^{*}\) be a fixed number.