Twist (mathematics) explained

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon

(X,U)

be composed of a space curve,

X=X(s)

, where

s

is the arc length of

X

, and

U=U(s)

the a unit normal vector, perpendicular at each point to

X

. Since the ribbon

(X,U)

has edges

X

and

X'=X+\varepsilonU

, the twist (or total twist number)

Tw

measures the average winding of the edge curve

X'

around and along the axial curve

X

. According to Love (1944) twist is defined by

Tw=\dfrac{1}{2\pi}\int\left(U x \dfrac{dU}{ds}\right) ⋅ \dfrac{dX}{ds}ds ,

where

dX/ds

is the unit tangent vector to

X

.The total twist number

Tw

can be decomposed (Moffatt & Ricca 1992) into normalized total torsion

T\in[0,1)

and intrinsic twist

N\inZ

as

Tw=\dfrac{1}{2\pi}\int\tau ds+\dfrac{\left[\Theta\right]X}{2\pi}=T+N ,

where

\tau=\tau(s)

is the torsion of the space curve

X

, and

\left[\Theta\right]X

denotes the total rotation angle of

U

along

X

. Neither

N

nor

Tw

are independent of the ribbon field

U

. Instead, only the normalized torsion

T

is an invariant of the curve

X

(Banchoff & White 1975).

When the ribbon is deformed so as to pass through an inflectional state (i.e.

X

has a point of inflection), the torsion

\tau

becomes singular. The total torsion

T

jumps by

\pm1

and the total angle

N

simultaneously makes an equal and opposite jump of

\mp1

(Moffatt & Ricca 1992) and

Tw

remains continuous. This behavior has many important consequences for energy considerations in many fields of science (Ricca 1997, 2005; Goriely 2006).

Wr

of

X

, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula

Lk=Wr+Tw

in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.

See also

References

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