# Higher-Order Curvatures of Plane and Space Parametrized Curves
## 3 Higher-Order Curvatures from the Iteration of the Frenet Equation  
This short note defines new curvature functions for \(C\) assumed to be parametrized by arc length; hence, we have a smooth function \(\theta:I\rightarrow\mathbb{R}\) such that \(r^{\prime}(s)=e^{i\theta(s)}\). It follows that the curvature is generated by this _structural angle function_ through:  
\[k(s)=\theta^{\prime}(s). \tag{10}\]  
Let us fix also an integer \(n\in\mathbb{N}^{*}:=\{1,2,3,....\}\)  
**Definition 1**. : _The \(n\)-order Frenet equation of \(C\) is:_  
\[\frac{d^{n}}{ds^{n}}\mathcal{F}(s)=K_{p}^{n}(s)\mathcal{F}(s)+K_{o}^{n}(s)J \mathcal{F}(s). \tag{11}\]  
_with \(K_{p}^{n}\) the parallel \(n\)-curvature and \(K_{o}^{n}\) the orthogonal \(n\)-curvature of \(C\)._  
We note that the new curvature functions \(K_{p,o}^{n}\) are smooth and from (9) it results the first ones are \(K_{p}^{1}=0\) and \(K_{o}^{1}=-k\). More precisely, these curvatures are obtained through a recurrence process expressed by the following theorem. **Theorem 1**. : _The higher-order curvatures are provided by the following recurrence relation in which \(I_{2}\) is the \(2\times 2\) unit matrix:_  
\[\left( \begin{array}{c}K_{p}^{n+1}\\ K_{o}^{n+1}\end{array} \right)=\frac{d}{ds}\left( \begin{array}{c}K_{p}^{n}\\ K_{o}^{n}\end{array} \right)-kI\left( \begin{array}{c}K_{p}^{n}\\ K_{o}^{n}\end{array} \right)=\left( \begin{array}{c}I_{2}\frac{d}{ds}-kI\end{array}  \right)\left( \begin{array}{c}K_{p}^{n}\\ K_{o}^{n}\end{array} \right). \tag{12}\]  
_Working in the complex algebra \(\mathbb{C}\) with the sequence of smooth functions \(z_{n}(s)=K_{p}^{n}(s)+iK_{o}^{n}(s)\), the recurrence \(z_{n+1}=z_{n}^{\prime}-kiz_{n}\) follows._  
This immediately yields the first two higher-order curvatures:  
\[K_{p}^{2}=-k^{2}\leq 0,\quad K_{o}^{2}=-k^{\prime}=-\theta^{\prime\prime}, \tag{13}\]  
\[K_{p}^{3}=-3 kk^{\prime},\quad K_{o}^{3}=k^{3}-k^{\prime\prime}=(\theta^{ \prime})^{3}-\theta^{\prime\prime\prime} \tag{14}\]  
and then the \(o2\)-inflection points of \(C\) (i.e., the zeros of \(K_{o}^{2}\)) are exactly its vertices. Note that the unit circle \(S^{1}\) has an almost periodicity of order three: \(K_{p}^{3}=0=K_{p^{\prime}}^{1}\)\(K_{o}^{3}=1=-K_{o}^{1}\). **Example 1**. : _Since the particular case of a vanishing curvature deserves attention, we note that \(K_{o}^{3}=0\) if and only if \(k(s)=\frac{\sqrt{2}}{s}\) for \(s\in(0,+\infty)\); equivalently, \(\theta(s)=\sqrt{2}\ln s\). It results in the curve:_  
\[C:r(s)=\frac{s}{3}\Big{(}\cos(\sqrt{2}\ln s)+\sqrt{2}\sin(\sqrt{2}\ln s),\sin (\sqrt{2}\ln s)-\sqrt{2}\cos(\sqrt{2}\ln s)\Big{)} \tag{15}\]  
_and the reparametrization \(s=e^{\frac{t}{\sqrt{2}}}\) yields:_  
\[C:r(t)=\frac{e^{\frac{t}{\sqrt{2}}}}{3}(\cos t+\sqrt{2}\sin t,\sin t-\sqrt{2} \cos t),\quad r^{\prime}(t)=\frac{1}{\sqrt{2}}e^{\left(\frac{1}{\sqrt{2}}+i \right)t}. \tag{16}\]  
_We note in passing that the minimal polynomial of the complex number \(\frac{1}{\sqrt{2}}+i\) is \(P(X)=4X^{4}+4X^{2}+9\). Recall that a logarithmic spiral has the polar parametrization \(\rho_{R,\alpha}(t)=Re^{at}\) with \(R>0\) and \(\alpha>0\) as its generators. It follows that the derivative curve \(C^{\prime}\) is the logarithmic spiral with \(R=\alpha=\frac{1}{\sqrt{2}}\)._  
**Remark 1**.