Torsion

The torsion of a space curve, sometimes also called the "second curvature" (Kreyszig 1991, p. 47), is the rate of change of the curve's osculating plane. 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

(1)

where is the unit normal vector and is the unit binormal vector. Written explicitly in terms of a parameterized vector function ,

			(2)
			(3)

(Gray 1997, p. 192), where denotes a scalar triple product and is the radius of curvature.

The quantity is called the radius of torsion and is denoted or .