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 
,
(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 
.
See also
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.
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Cite this as:
Weisstein, Eric W. "Torsion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Torsion.html