View source for Darboux vector

The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302101.png" /> of the instantaneous axis of rotation around which the natural trihedral of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302102.png" /> is rotating during the uniform movement of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302103.png" /> along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302104.png" />. The Darboux vector lies in the rectifying plane of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302105.png" /> and is expressed in terms of the principal normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302106.png" /> and the tangent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d0302108.png" /> by the formula

<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/d/d030/d030210/d0302109.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021011.png" /> are the curvature and the torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021013.png" /> is the angle between the Darboux vector and the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021014.png" />. The [[Frénet formulas|Frénet formulas]] may be written with the aid of the Darboux vector as follows:

<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/d/d030/d030210/d03021015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021016.png" /> is the [[Binormal|binormal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030210/d03021017.png" />.

G. Darboux [[#References|[1]]] was the first to point out the geometric significance of the Darboux vector for the natural trihedral of a space curve.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)  pp. 1–18</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of the theory of surfaces in a tensor setting" , '''1''' , Moscow-Leningrad  (1947)  (In Russian)</TD></TR></table>



====Comments====
The natural trihedral (a name used by S. Sternberg [[#References|[a1]]]) is commonly called Frénet frame (also [[Frénet trihedron|Frénet trihedron]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>