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Twisted Curve

A twisted curve is a smooth space curve with nowhere-vanishing curvature and torsion (Costa 1990, Khan 2024). This convention excludes plane curves and permits the Frenet formulas to be used along the entire curve. For a regular parameterization X(t), the condition is

det(X^'(t),X^('')(t),X^(''')(t))!=0,

where det denotes the determinant. The torsion is

tau(t)=(det(X^'(t),X^('')(t),X^(''')(t)))/(||X^'(t)×X^('')(t)||^2).

A helix with nonzero radius and nonzero pitch is an example. Nonplanarity alone is weaker than this condition. A nonplanar curve may have zero torsion at isolated points.

See also

Curvature, Frenet Formulas, Space Curve, Torsion

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References

Costa, S. I. R. "On Closed Twisted Curves." Proc. Amer. Math. Soc. 109, 205-214, 1990. https://doi.org/10.1090/S0002-9939-1990-0993746-1.do Carmo, M. P. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, 1976.Khan, G. "Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow." Bull. Austral. Math. Soc. 109, 571-580, 2024. https://doi.org/10.1017/S0004972723000643.

Cite this as:

Weisstein, Eric W. "Twisted Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TwistedCurve.html

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