How to calculate Frenet-Serret equations of the helix
$$\gamma : \Bbb R \to \ \Bbb R^3$$
$$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), \left(\frac{s}{\sqrt 2}\right)\right)$$
I know the following info about Frenet-Serret equations:
$$\frac{\mathrm{d}}{\mathrm{d}s} \begin{bmatrix} t \\ n \\ b \end{bmatrix} = \begin{bmatrix} 0 & \kappa & 0 \\ – \kappa & 0 & \tau \\ 0 & -\tau & 0 \end{bmatrix}\begin{bmatrix} t \\ n \\ b \end{bmatrix}$$
Best Answer
Just compute $\kappa=\dfrac{\|\dot\gamma\times\ddot\gamma\|}{\|\dot\gamma\|^3}$ and $\tau=\dfrac{\det(\dot\gamma,\ddot\gamma,\dddot\gamma)}{\|\dot\gamma\times\ddot\gamma\|^2}$ as usual.
Edit: $\|\dot\gamma\||=1$ is not needed, just $\|\dot\gamma\|\neq0$.