In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon ( X , U ) {\displaystyle (X,U)} be composed of a space curve, X = X ( s ) {\displaystyle X=X(s)} , where s {\displaystyle s} is the arc length of X {\displaystyle X} , and U = U ( s ) {\displaystyle U=U(s)} a unit normal vector, perpendicular at each point to ∂ X ( s ) ∂ s ( s ) {\displaystyle {\partial X(s) \over \partial s}(s)} . Since the ribbon ( X , U ) {\displaystyle (X,U)} has edges X {\displaystyle X} and X ′ = X + ε U {\displaystyle X'=X+\varepsilon U} , the twist (or total twist number) T w {\displaystyle Tw} measures the average winding of the edge curve X ′ {\displaystyle X'} around and along the axial curve X {\displaystyle X} . According to Love (1944) twist is defined by
T w = 1 2 π ∫ ( U × d U d s ) ⋅ d X d s d s , {\displaystyle Tw={\dfrac {1}{2\pi }}\int \left(U\times {\dfrac {dU}{ds}}\right)\cdot {\dfrac {dX}{ds}}ds\;,}
where d X / d s {\displaystyle dX/ds} is the unit tangent vector to X {\displaystyle X} . The total twist number T w {\displaystyle Tw} can be decomposed (Moffatt & Ricca 1992) into normalized total torsion T ∈ [ 0 , 1 ) {\displaystyle T\in [0,1)} and intrinsic twist N ∈ Z {\displaystyle N\in \mathbb {Z} } as
T w = 1 2 π ∫ τ d s + [ Θ ] X 2 π = T + N , {\displaystyle Tw={\dfrac {1}{2\pi }}\int \tau \;ds+{\dfrac {\left[\Theta \right]_{X}}{2\pi }}=T+N\;,}
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