Twist (mathematics)

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In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve , where is the arc length of and a unit vector perpendicular at each point to . Since the ribbon has edges and the twist (or total twist number) measures the average winding of the curve around and along the 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) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) 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 (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.

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