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# 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 .

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

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## References

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.

Torsion

## Cite this as:

Weisstein, Eric W. "Torsion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Torsion.html