In differential geometry, the twist of a ribbon is its rate of axial rotation.
be composed of a space curve,
is the arc length of
a unit normal vector, perpendicular at each point to
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
is the torsion of the space curve
denotes the total rotation angle of
are independent of the ribbon field
Instead, only the normalized torsion
(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
The total torsion
and the total angle
simultaneously makes an equal and opposite jump of
This behavior has many important consequences for energy considerations in many fields of science (Ricca 1997, 2005; Goriely 2006).
, 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.