Twist (mathematics)

Short description: Differential geometry term

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon (X,U) be composed of space curve X=X(s), where s is the arc length of X, and U=U(s) the a unit normal vector, perpendicular at each point to X. Since the ribbon (X,U) has edges X and X′=X+εU, the twist (or total twist number) Tw measures the average winding of the edge curve X′ around and along the axial curve X. According to Love (1944) twist is defined by

Tw=12π∫(U×dUds)⋅dXdsds,

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∈[0,1) and intrinsic twist N∈ℤ as

Tw=12π∫τds+[Θ]X2π=T+N,

where τ=τ(s) is the torsion of the space curve X, and [Θ]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), the torsion τ becomes singular. The total torsion T jumps by ±1 and the total angle N simultaneously makes an equal and opposite jump of ∓1 (Moffatt & Ricca 1992) and Tw remains continuous. This behavior has many important consequences for energy considerations in many fields of science (Ricca 1997, 2005; Goriely 2006).

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.

## See also

* Twist (screw theory)
* Twist (rational trigonometry)
* Twisted sheaf

## 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.
* Goriely, A. (2006) Twisted elastic rings and the rediscoveries of Michell’s instability. J Elasticity 84, 281-299.
* 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 Calugareanu invariant. Proc. R. Soc. London A 439, 411-429. Also in: (1995) Knots and Applications (ed. L.H. Kauffman), pp. 251-269. World Scientific.
* Ricca, R.L. (1997) Evolution and inflexional instability of twisted magnetic flux tubes. Solar Physics 172, 241-248.
* Ricca, R.L. (2005) Inflexional disequilibrium of magnetic flux tubes. Fluid Dynamics Research 36, 319-332.

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Twist_(mathematics)&oldid=3907972"