Torsion (Differential Geometry)

The rate of change of the Osculating Plane of a Space Curve. The torsion $\tau$ is Positive for a right-handed curve, and Negative for a left-handed curve. A curve with Curvature $\kappa\not=0$ is planar Iff $\tau=0$.

The torsion can be defined by

$$\tau\equiv -{\bf N}\cdot {\bf B}',$$

where ${\bf N}$ is the unit Normal Vector and ${\bf B}$ is the unit Binormal Vector. Written explicitly in terms of a parameterized Vector Function ${\bf x}$,

$$\tau={\vert\dot{\bf x}\,\ddot{\bf x}\,\raise6pt\hbox{.}\mker... ...6pt\hbox{.}\mkern-2mu\raise6pt\hbox{.}\mkern-11mu{\bf x}\vert,$$

where $\vert{\bf a}\,{\bf b}\,{\bf c}\vert$ denotes a Scalar Triple Product and $\rho$ is the Radius of Curvature. The quantity $1/\tau$ is called the Radius of Torsion and is denoted $\sigma$ or $\phi$.

See also Curvature, Radius of Curvature, Radius of Torsion

References

Gray, A. ``Drawing Space Curves with Assigned Curvature.'' §7.8 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 145-147, 1993.

Kreyszig, E. ``Torsion.'' §14 in Differential Geometry. New York: Dover, pp. 37-40, 1991.