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)
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where is the unit normal vector and is the unit binormal vector. Written explicitly in terms of a parameterized vector function ,
(2)
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(3)
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(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 .