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Frénet formulas

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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).


Comments

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Frénet formulas. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9net_formulas&oldid=32754
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article