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)

Comments

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

References

[a1]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[a2]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)