Darboux vector
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
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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:
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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.
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) |
Darboux vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_vector&oldid=46584