In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve , where is the arc length of and a unit vector perpendicular at each point to . Since the ribbon has edges and the twist (or total twist number) measures the average winding of the curve around and along the 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) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) and remains continuous. This behavior has many important consequences for energy considerations in many fields of science.
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.
- 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.