# Higher-Order Curvatures of Plane and Space Parametrized Curves  
[PERSON]  
1 Faculty of Mathematics, University \"Alexandru Ioan Cuza\", 700506 Iasi, Romania; [EMAIL_ADDRESS] Faculty of Mathematics, University \"Alexandru Ioan Cuza\", 700506 Iasi, Romania; mcrasm@uaic. The ambient setting \(\mathbb{R}^{2}\) is a Euclidean vector space with respect to the canonical inner product:  
\[\langle u,v\rangle=u^{1}v^{1}+u^{2}v^{2},\quad u=(u^{1},u^{2})\in\mathbb{R}^{2},v =(v^{1},v^{2})\in\mathbb{R}^{2},\quad 0\leq\|u\|^{2}=\langle u,u\rangle. \tag{2}\]  
The infinitesimal generator of the rotations in \(\mathbb{R}^{2}=\mathbb{C}\) is the linear vector field, called _angular_:  
\[\xi(u):=-u^{2}\frac{\partial}{\partial u^{1}}+u^{1}\frac{\partial}{\partial u^ {2}},\quad\xi(u)=i\cdot u=i\cdot(u^{1}+iu^{2}),\quad i=\sqrt{-1}. \tag{3}\]  
It is a complete vector field with integral curves the circles \(\mathcal{C}(O,r)\):  
\[\left\{ \begin{array}{l}\gamma^{\xi}_{u_{0}}(t)=(u_{0}^{1}\cos t-u_{0}^{2} \sin t,u_{0}^{1}\sin t+u_{0}^{2}\cos t)=R(t)\cdot\left(\begin{array}{cc}u_{ 0}^{1}\\ u_{0}^{2}\end{array}\right),\quad t\in\mathbb{R},\\ r=\|u_{0}\|=\|(u_{0}^{1},u_{0}^{2})\|,\quad R(t):=\left(\begin{array}{cc} \cos t&-\sin t\\ \sin t&\cos t\end{array}\right)\in SO(2)=S^{1}\end{array} \right. \tag{4}\]  
and since the rotations \(R(t)\) are isometries of the Riemannian metric \(g_{can}=dx^{2}+dy^{2}=|dz|^{2}\), it follows that \(\xi\) is a Killing vector field of the Riemannian manifold \((\mathbb{R}^{2},g_{can})\). The first integrals of \(\xi\) are the Gaussian functions, i.e., multiples of the square norm: \(f_{\alpha}(x,y)=\alpha(x^{2}+y^{2})\), \(\alpha\in\mathbb{R}\). For an arbitrary vector field \(X=A(x,y)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}\), its Lie bracket with \(\xi\) is:  
\[[X,\xi]=(yA_{x}-xA_{y}-B)\frac{\partial}{\partial x}+(A+yB_{x}-xB_{y})\frac{ \partial}{\partial y}, \tag{5}\]  
where the subscript denotes the variable corresponding to the partial derivative. For example, \(\xi\) commutes with _the radial_ (or Euler) vector field \(E(x,y)=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\), which is also a linear complete vector field having as integral curves the homotheties \(\gamma^{E}_{u_{0}}(t)=e^{t}u_{0}\) for all \(t\in\mathbb{R}\). The Frenet apparatus of the curve \(C\) is provided by ([4]):  
\[\left\{ \begin{array}{l}T(t)=\frac{r^{\prime}(t)}{\|P^{\prime}(t)\|}\in S^{1 },\quad N(t)=i\cdot T(t)=\frac{1}{\|P^{\prime}(t)\|}(-y^{\prime}(t),x^{\prime }(t))\in S^{1},\\ k(t)=\frac{1}{\|P^{\prime}(t)\|}(\gamma^{T}(t),N(t))=\frac{1}{\|P^{\prime}(t)\| ^{2}}\langle r^{\prime\prime}(t),ir^{\prime}(t)\rangle=\frac{1}{\|P^{\prime}(t )\|^{2}}[x^{\prime}(t)y^{\prime\prime}(t)-y^{\prime}(t)x^{\prime\prime}(t)]. \end{array} \right. \tag{6}\]  
Hence, if \(C\) is naturally parametrized (or parametrized by arc length), i.e., \(\|r^{\prime}(s)\|=1\) for all \(s\in I=(0,L\leq+\infty)\), then \(r^{\prime\prime}(s)=k(s)ir^{\prime}(s)\). In a complex approach based on \(z(t)=x(t)+iy(t)\in\mathbb{C}=\mathbb{R}^{2}\), we have:  
\[\left\{ \begin{array}{l}k(t)=\frac{1}{|z^{\prime}(t)|}Im(z^{\prime}(t)\cdot z ^{\prime\prime}(t))=\frac{1}{|z^{\prime}(t)|}Im\Big{(}\frac{z^{\prime\prime}(t )}{z^{\prime}(t)}\Big{)}=\frac{1}{|z^{\prime}(t)|}Im\Big{[}\frac{d}{dt}(\ln z^ {\prime}(t))\Big{]}\in\mathbb{R},\\ Re(z^{\prime}(t)\cdot z^{\prime\prime}(t))=\frac{1}{2}\frac{d}{dt}\|r^{\prime}(t )\|^{2},\quad f_{\alpha}(z)=\alpha|z|^{2}.\end{array} \right. \tag{7}\]  
The multiplication with the complex unit \(i\) corresponds to the rotation \(J:=R\big{(}\frac{\pi}{2}\big{)}\); we have also:  
\[\frac{d}{dt}R(t)=R\Big{(}t+\frac{\pi}{2}\Big{)}=R(t)R\Big{(}\frac{\pi}{2} \Big{)}=R\Big{(}\frac{\pi}{2}\Big{)}R(t), \tag{8}\]  
and the Frenet equations can be unified by means of the column matrix \(\mathcal{F}(t)=\left( \begin{array}{c}T\\ N\end{array}