Darboux vector

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The vector of the instantaneous axis of rotation around which the natural trihedral of a curve is rotating during the uniform movement of a point along the curve . The Darboux vector lies in the rectifying plane of the curve and is expressed in terms of the principal normal and the tangent of by the formula

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

where is the binormal of .

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


[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)


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


[a1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[a2] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
How to Cite This Entry:
Darboux vector. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article