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Darboux Vector

The rotation Vector of the Trihedron of a curve with Curvature $\kappa\not=0$ when a point moves along a curve with unit Speed. It is given by

{\bf D}=\tau{\bf T}+\kappa{\bf B},
\end{displaymath} (1)

where $\tau$ is the Torsion, T the Tangent Vector, and B the Binormal Vector. The Darboux vector field satisfies
$\displaystyle \dot{\bf T}$ $\textstyle =$ $\displaystyle {\bf D}\times {\bf T}$ (2)
$\displaystyle \dot{\bf N}$ $\textstyle =$ $\displaystyle {\bf D}\times {\bf N}$ (3)
$\displaystyle \dot{\bf B}$ $\textstyle =$ $\displaystyle {\bf D}\times {\bf B}.$ (4)

See also Binormal Vector, Curvature, Tangent Vector, Torsion (Differential Geometry)


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 151, 1993.

© 1996-9 Eric W. Weisstein