differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the twist of a ''

ribbon A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic mater ...

'' is its rate of

axial rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rot ...

. Let a ribbon

(X,U)

be composted of

space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...

X=X(s)

, where

s

is the

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

X

, and

U=U(s)

the a unit

normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...

, perpendicular at each point to

X

. Since the ribbon

(X,U)

has edges

X

and

X'=X+\varepsilon U

, the twist (or ''total twist number'')

Tw

measures the average

winding An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil (spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in de ...

of the edge curve

X'

around and along the axial curve

X

. According to Love (1944) twist is defined by :

Tw = \dfrac \int \left( U \times \dfrac \right) \cdot \dfrac 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

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bio ...

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 In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

), the torsion

\tau

becomes singular. The total torsion

T

jumps by

\pm 1

and the total angle

N

simultaneously makes an equal and opposite jump of

\mp 1

(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). Together with the

writhe In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...

Wr

X

, twist is a geometric quantity that plays an important role in the application of the

Călugăreanu–White–Fuller formula In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by (X,U) includes a curve X given by a three-dimensional vector X(s), depending continuou ...

Lk = Wr + Tw

topological fluid dynamics Topological ideas are relevant to fluid dynamics (including magnetohydrodynamics) at the kinematic level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particula ...

(for its close relation to

kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...

and

magnetic helicity In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scal ...

of a vector field),

physical knot theory Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally cons ...

, and

structural complexity In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves t ...

analysis.

References

*Banchoff, T.F. & White, J.H. (1975) The behavior of the total twist and self-linking number of a closed space curve under inversions. ''Math. Scand.'' 36, 254–262. *Goriely, A. (2006) Twisted elastic rings and the rediscoveries of Michell’s instability. ''J Elasticity'' 84, 281-299. * Love, A.E.H. (1944
''A Treatise on the Mathematical Theory of Elasticity''
Dover, 4th Ed., New York. * Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Calugareanu invariant. ''Proc. R. Soc. London A'' 439, 411-429. Also in: (1995) Knots and Applications (ed. L.H. Kauffman), pp. 251-269. World Scientific. * Ricca, R.L. (1997) Evolution and inflexional instability of twisted magnetic flux tubes. ''Solar Physics'' 172, 241-248. * Ricca, R.L. (2005) Inflexional disequilibrium of magnetic flux tubes. ''Fluid Dynamics Research'' 36, 319-332. Differential geometry Topology

See also

References