Alcune formule:

\[ \begin{aligned} s(T) & = \int _{t_0}^ T \left\Vert \dfrac {dP}{dt}\right\Vert \, dt \iff \dfrac {ds}{dt} = \left\Vert \frac{dP}{dt}\right\Vert \\ \vt & = \dfrac {dP}{ds} = \dfrac {\frac{dP}{dt}}{\left\Vert \frac{dP}{dt}\right\Vert } \\ \boldsymbol {\kappa }& = \dfrac {d^2P}{ds^2} = \kappa \vn \\ \kappa & = \dfrac { \left\Vert \frac{dP}{dt} \times \frac{d^2P}{dt^2}\right\Vert } { \left\Vert \frac{dP}{dt}\right\Vert ^3 }\\ \vb & = \vt \times \vn = \dfrac { \frac{dP}{dt} \times \frac{d^2P}{dt^2}}{ \left\Vert \frac{dP}{dt} \times \frac{d^2P}{dt^2}\right\Vert } \implies \vn = \vb \times \vt = \dfrac {( \frac{dP}{dt} \times \frac{d^2P}{dt^2} ) \times \frac{dP}{dt} }{ \left\Vert \frac{dP}{dt} \times \frac{d^2P}{dt^2}\right\Vert \left\Vert \frac{dP}{dt}\right\Vert } \\ \tau & = \dfrac { \frac{dP}{dt} \times \frac{d^2P}{dt^2} \cdot \frac{d^3P}{dt^2}}{ \left\Vert \frac{dP}{dt} \times \frac{d^2P}{dt^2}\right\Vert ^2 } \\ & \left\{ \begin{aligned} \dfrac {d\vt }{ds} & = \kappa \vn \\ \dfrac {d\vn }{ds} & = -\kappa \vt + \tau \vb \\ \dfrac {d\vb }{ds} & = -\tau \vn \end{aligned}\right. \iff \dfrac {d}{ds} \begin{bmatrix} \vt \\ \vn \\ \vb \end{bmatrix} = \begin{bmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{bmatrix}\begin{bmatrix} \vt \\ \vn \\ \vb \end{bmatrix}\end{aligned} \]