Twist (mathematics)
Differential geometry term / From Wikipedia, the free encyclopedia
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
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.