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

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

where $\tau$ is the Torsion, T the Tangent Vector, and B the Binormal Vector. The Darboux vector field satisfies

$$\dot{\bf T} = {\bf D}\times {\bf T}$$ (2)

$$\dot{\bf N} = {\bf D}\times {\bf N}$$ (3)

$$\dot{\bf B} = {\bf D}\times {\bf B}.$$ (4)

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

References

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