Twist (differential geometry)

{{Short description|Differential geometry term}}

{{For|twists of curves in algebraic geometry|twists of elliptic curves}}

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

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

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 $T\; \backslash in\; [0,1)$ and intrinsic twist $N\; \backslash in\; \backslash mathbb\{Z\}$ as

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

where $\backslash tau=\backslash tau(s)$ is the torsion of the space curve $X$, and $\backslash left[\; \backslash Theta\; \backslash right]\_X$ 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), the torsion $\backslash tau$ becomes singular. The total torsion $T$ jumps by $\backslash pm\; 1$ and the total angle $N$ simultaneously makes an equal and opposite jump of $\backslash mp\; 1$ (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.