Twist (mathematics)
In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon be composed of space curve , where is the arc length of , and the a unit normal vector, perpendicular at each point to . Since the ribbon has edges and , the twist (or total twist number) measures the average winding of the edge curve around and along the axial curve . According to Love (1944) twist is defined by
where is the unit tangent vector to . The total twist number can be decomposed (Moffatt & Ricca 1992) into normalized total torsion and intrinsic twist as
where is the torsion of the space curve , and denotes the total rotation angle of along . Neither nor are independent of the ribbon field . Instead, only the normalized torsion is an invariant of the curve (Banchoff & White 1975).
When the ribbon is deformed so as to pass through an inflectional state (i.e. has a point of inflection), the torsion becomes singular. The total torsion jumps by and the total angle simultaneously makes an equal and opposite jump of (Moffatt & Ricca 1992) and 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 of , twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.
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References[edit]
- 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.