Twist (mathematics)

In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve $X=X(s)$ , where $s$ is the arc length of $X$ and $U=U(s)$ a unit vector perpendicular at each point to $X$ . Since the ribbon $(X,U)$ has edges $X$ and $X'=X+\varepsilon U$ the twist (or total twist number) $Tw$ measures the average winding of the curve $X'$ around and along the curve $X$ . According to Love (1944) twist is defined by

Tw = \dfrac{1}{2\pi} \int \left( \dfrac{dU}{ds} \times U \right) \cdot \dfrac{dX}{ds} ds \; ,

where $dX/ds$ is the unit tangent vector to $X$ . The total twist number $Tw$ can be decomposed (Moffatt & Ricca 1992) into normalized total torsion $T\in [0,1)$ and intrinsic twist $N\in \mathbb {Z}$ as

Tw = \dfrac{1}{2\pi} \int \tau \; ds + \dfrac{\left[ \Theta \right]_X}{2\pi} = T+N \; ,

where $\tau=\tau(s)$ is the torsion of the space curve $X$ , and $\left[ \Theta \right]_X$ denotes the total rotation angle of $U$ along $X$ . Neither $N$ nor $Tw$ are independent of the ribbon field $U$ . Instead, only the normalized torsion $T$ is an invariant of the curve $X$ (Banchoff & White 1975).

When the ribbon is deformed so as to pass through an inflectional state (i.e. $X$ has a point of inflection) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) and $Tw$ remains continuous. This behavior has many important consequences for energy considerations in many fields of science.

Together with the writhe $Wr$ of $X$ , twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula $Lk = Wr + Tw$ in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.

References

Banchoff, T.F. & White, J.H. (1975) The behavior of the total twist and self-linking number of a closed space curve under inversions. Math. Scand. 36, 254–262.
Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity. Dover, 4th Ed., New York.
Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411–429.

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Twist (mathematics)

See also

References