Frénet formulas

Formulas that express the derivatives of the unit vectors of the tangent $\tau$, the normal $\nu$ and the binormal $\beta$ to a regular curve with respect to the natural parameter $s$ in terms of these same vectors and the values of the curvature $k_1$ and torsion $k_2$ of the curve:

$$\tau_x'=k_1\nu,$$

$$\nu_s'=-k_1\tau-k_2\beta,$$

$$\beta_s'=k_2\nu.$$

They were obtained by F. Frénet (1847).

References

[a1]	C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4