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Difference between revisions of "Frénet formulas"

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Formulas that express the derivatives of the unit vectors of the tangent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416901.png" />, the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416902.png" /> and the [[Binormal|binormal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416903.png" /> to a regular curve with respect to the [[Natural parameter|natural parameter]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416904.png" /> in terms of these same vectors and the values of the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416905.png" /> and torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416906.png" /> of the curve:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416907.png" /></td> </tr></table>
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$$\tau_x'=k_1\nu,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416908.png" /></td> </tr></table>
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$$\nu_s'=-k_1\tau-k_2\beta,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041690/f0416909.png" /></td> </tr></table>
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$$\beta_s'=k_2\nu.$$
  
 
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
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article