Difference between revisions of "Frénet formulas"
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| − | Formulas that express the derivatives of the unit vectors of the tangent | + | {{TEX|done}} |
| + | Formulas that express the derivatives of the unit vectors of the tangent $\tau$, the normal $\nu$ and the [[Binormal|binormal]] $\beta$ to a regular curve with respect to the [[Natural parameter|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). | They were obtained by F. Frénet (1847). | ||
Latest revision as of 09:55, 7 August 2014
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://encyclopediaofmath.org/index.php?title=Fr%C3%A9net_formulas&oldid=22467
Frénet formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9net_formulas&oldid=22467
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article