## Torsion (Differential Geometry)

The rate of change of the Osculating Plane of a Space Curve. The torsion is Positive for a right-handed curve, and Negative for a left-handed curve. A curve with Curvature is planar Iff .

The torsion can be defined by

where is the unit Normal Vector and is the unit Binormal Vector. Written explicitly in terms of a parameterized Vector Function ,

where denotes a Scalar Triple Product and is the Radius of Curvature. The quantity is called the Radius of Torsion and is denoted or .

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.