Twist (differential geometry)

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon be composed of a space curve,, where is the arc length of, and a unit normal vector, perpendicular at each point to. Since the ribbon has edges and, the twist measures the average winding of the edge curve around and along the axial curve. According to Love twist is defined by
where is the unit tangent vector to.
The total twist number can be decomposed 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 .
When the ribbon is deformed so as to pass through an inflectional state, the torsion becomes singular. The total torsion jumps by and the total angle simultaneously makes an equal and opposite jump of 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, physical knot theory, and structural complexity analysis.