Twist (mathematics)

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon be composed of space curve , where is the arc length of , and the a unit normal vector, perpendicular at each point to . Since the ribbon has edges and , the twist (or total twist number) measures the average winding of the edge curve around and along the axial 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), the torsion becomes singular. The total torsion jumps by and the total angle simultaneously makes an equal and opposite jump of (Moffatt & Ricca 1992) and 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 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.