# Higher-Order Curvatures of Plane and Space Parametrized Curves
## 3 Higher-Order Curvatures from the Iteration of the Frenet Equation
: _More generally, due to the presence of the third derivative of \(\theta\) in the expression of \(K_{o}^{3}\), we recall its Schwarzian derivative ([5]):_  
\[S_{\theta}=\frac{\theta^{\prime\prime\prime}}{\theta^{\prime}}-\frac{3}{2}{ \left(\frac{\theta^{\prime\prime}}{\theta^{\prime}}\right)}^{2}=\frac{\theta^{ \prime\prime\prime}}{k}-\frac{3}{2}{\left(\frac{k^{\prime}}{k}\right)}^{2} \tag{17}\]  
_and then we obtain a new formula for \(K_{o}^{3}\):_  
\[K_{o}^{3}=k{\left(k^{2}-\frac{3}{2}{\left(\frac{k^{\prime}}{k}\right)}^{2}-S_{ \theta}\right)}. \tag{18}\]  
The next pair of curvatures is:  
\[K_{p}^{4}=-4 kk^{\prime\prime}-3(k^{\prime})^{2}+k^{4},\quad K_{o}^{4}=6k^{2}k^ {\prime}-k^{\prime\prime\prime}=k^{\prime}{\left(6k^{2}-\frac{3}{2}{\left( \frac{k^{\prime\prime}}{k^{\prime}}\right)}^{2}-S_{k}\right)} \tag{19}\]  
and then the problem of a vanishing \(K_{o}^{4}\) becomes the following example. **Example 2**. : _We have \(K_{o}^{4}=0\) if and only if \(k(s)=\frac{1}{s}\), having also \(S_{k}=0\); equivalently, \(\theta=\ln s\) which yields the curve:_  
\[C:r(s)=\frac{s}{2}{\left(\sin(\ln s)+\cos(\ln s),\sin(\ln s)-\cos(\ln s)\right)} \tag{20}\]  
_and the reparametrization \(s=e^{t}\) gives the curve:_  
\[C:r(t)=\frac{e^{t}}{2}{\left(\sin t+\cos t,\sin t-\cos t\right)},\quad r^{ \prime}(t)=e^{(1+i)t}. \tag{21}\]  
_Now, the derivative curve \(C^{\prime}\) is the logarithmic spiral with \(R=\alpha=1\)._  
**Example 3**. : _For the well-known catenary curve:_  
\[C:r(s)={\left(\ln(s+\sqrt{1+s^{2}}),\sqrt{1+s^{2}}\right)}.\quad\theta(s)= \arctan s \tag{22}\]  
_we have:_  
\[K_{p}^{3}=-3 kk^{\prime},\quad K_{o}^{3}=k^{3}-k^{\prime\prime}=(\theta^{ \prime})^{3}-\theta^{\prime\prime\prime} \tag{14}\]0  
_and then:_  
\[K_{p}^{3}=-3 kk^{\prime},\quad K_{o}^{3}=k^{3}-k^{\prime\prime}=(\theta^{ \prime})^{3}-\theta^{\prime\prime\prime} \tag{14}\]1  
_There exists only one \(o3\)-inflection point \(P\in C\) corresponding to \(s=\frac{1}{\sqrt{2}}\) which gives the point \(P{\left(\frac{\ln(2+\sqrt{3})}{2},\sqrt{\frac{3}{2}}\right)}\in C\)._  
**Example 4**. : _Fix \(\alpha\in\mathbb{R}^{*}\). The \(\alpha\)-clothoid, also called the Cornu spiral, is the curve:_  
\[K_{p}^{3}=-3 kk^{\prime},\quad K_{o}^{3}=k^{3}-k^{\prime\prime}=(\theta^{ \prime})^{3}-\theta^{\prime\prime\prime} \tag{14}\]2  
_having the linear curvature \(k(s)=\alpha s\). Then,_  
\[\left\{ \begin{array}{l}K_{p}^{2}(s)=-\alpha^{2}s^{2}\leq 0,\quad K_{o}^{2}=- \alpha^{2}<0,\\ K_{p}^{3}(s)=-3\alpha^{2}s,K_{o}^{3}(s)=\alpha^{3}s^{3},K_{p}^{4}(s)=\alpha^{2 }(\alpha^{2}s^{4}-3),K_{o}^{4}(s)=6\alpha^{3}s^{2}\end{array} \right. \tag{26}\]_and there exists a \(p4\)-inflection point \(s=\frac{4\sqrt[4]{3}}{\sqrt{|\alpha|}}\)._  
**4. Further Studies**  
**3.1** An important tool in dynamics described by curves is the Fermi-Walker derivative. Let \(\mathcal{X}_{C}\) be the set of vector fields along the curve \(C\). Then, the Fermi-Walker derivative is the map ([6]) \(\
abla^{FW}_{C}:\mathcal{X}_{C}\to\mathcal{X}_{C}\):  
\[\
abla^{FW}_{C}(X):=\frac{d}{dt}X+\|r^{\prime}(\cdot)\|k[\langle X,N\rangle T -\langle X,T\rangle N]=\frac{d}{dt}X+\|r^{\prime}(\cdot)\|k[X^{\flat}(N)T-X^{ \flat}(T)N] \tag{27}\]  
with \(X^{\flat}\) the differential \(1\)-form dual to \(X\) with respect to the Euclidean metric. In a matrix form, we can express this as follows:  
\[\
abla^{FW}_{C}(\cdot)=\frac{d}{dt}-\|r^{\prime}\|k\left| \begin{array}{cc}( \cdot)^{\flat}(T)&(\cdot)^{\flat}(N)\\ T&N\end{array} \right|=\frac{d}{dt}+\|r^{\prime}\|k\left| \begin{array}{cc}T&( \cdot)^{\flat}(T)\\ N&(\cdot)^{\flat}(N)\end{array} \right| \tag{28}\]  
For example, the Frenet frame is Fermi-Walker parallel:  
\[\
abla^{FW}_{C}(\mathcal{F})=\left( \begin{array}{c}0\\ 0\end{array} \right)\!. \tag{29}\]  
Using the new functions, we define two sequences of Fermi-Walker derivatives for naturally parametrized plane curves:  
\[\left\{