Twist (mathematics)

In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve ${\displaystyle X=X(s)}$, where ${\displaystyle s}$ is the arc length of ${\displaystyle X}$ and ${\displaystyle U=U(s)}$ a unit vector perpendicular at each point to ${\displaystyle X}$. Since the ribbon ${\displaystyle (X,U)}$ has edges ${\displaystyle X}$ and ${\displaystyle X'=X+\varepsilon U}$ the twist (or total twist number) ${\displaystyle Tw}$ measures the average winding of the curve ${\displaystyle X'}$ around and along the curve ${\displaystyle X}$. According to Love (1944) twist is defined by

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