# Darboux vector

The vector $\pmb\delta$ of the instantaneous axis of rotation around which the natural trihedral of a curve $L$ is rotating during the uniform movement of a point $M$ along the curve $L$. The Darboux vector lies in the rectifying plane of the curve $L$ and is expressed in terms of the principal normal $\mathbf n$ and the tangent $\mathbf t$ of $L$ by the formula

$$\pmb\delta = \sqrt {\tau ^ {2} + \sigma ^ {2} } ( \mathbf t \cos \theta + \mathbf n \sin \theta ) ,$$

where $\tau$ and $\sigma$ are the curvature and the torsion of $L$ and $\theta$ is the angle between the Darboux vector and the tangent to $L$. The Frénet formulas may be written with the aid of the Darboux vector as follows:

$$\dot{\mathbf t} = [ \pmb\delta , \mathbf t ] ,\ \ \dot{\mathbf n} = [ \pmb\delta , \mathbf n ] ,\ \ \dot{\mathbf b} = [ \pmb\delta , \mathbf b ] ,$$

where $\mathbf b$ is the binormal of $L$.

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

#### References

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