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 tau is positive for a right-handed curve, and negative for a left-handed curve. A curve with curvature kappa!=0 is planar iff tau=0.

The torsion can be defined by


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


(Gray 1997, p. 192), where |abc| 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

Bundle Torsion, Curvature, Group Torsion, Lancret Equation, Radius of Curvature, Radius of Torsion, Torsion Number, Torsion Tensor, Total Curvature

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Gray, A. "Drawing Space Curves with Assigned Curvature." §10.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 222-224, 1997.Kreyszig, E. "Torsion." §14 in Differential Geometry. New York: Dover, pp. 37-40, 1991.

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Cite this as:

Weisstein, Eric W. "Torsion." From MathWorld--A Wolfram Web Resource.

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