# Difference between revisions of "Torsion of a curve"

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{{dablink|For other notions of torsion, see [[Torsion (disambiguation)|Torsion]].}} | {{dablink|For other notions of torsion, see [[Torsion (disambiguation)|Torsion]].}} | ||

− | In the elementary [[differential geometry of curves]] in [[three dimensions]], the '''torsion''' of a [[curve]] measures how sharply it is twisting. Taken together, | + | In the elementary [[differential geometry of curves]] in [[three dimensions]], the '''torsion''' of a [[curve]] measures how sharply it is twisting out of the plane of curvature. Taken together, |

the [[curvature]] and the torsion of a space curve are analogous to the [[curvature]] of a plane curve. For example, they are coefficients in the system of [[differential equation]]s for the [[Frenet frame]] given by the [[Frenet–Serret formulas]]. | the [[curvature]] and the torsion of a space curve are analogous to the [[curvature]] of a plane curve. For example, they are coefficients in the system of [[differential equation]]s for the [[Frenet frame]] given by the [[Frenet–Serret formulas]]. | ||

== Definition == | == Definition == | ||

− | [[ | + | [[File:Torus-Knoten uebereinander Animated.gif|thumb|upright|Animation of the torsion and the corresponding rotation of the binormal vector]] |

− | Let ''C'' be a space curve parametrized by [[arc length]] <math>s</math> and with the [[unit tangent vector]] '''t'''. If the [[curvature]] <math>\kappa</math> of ''C'' at a certain point is not zero then the | + | Let ''C'' be a [[space curve]] parametrized by [[arc length]] <math>s</math> and with the [[unit tangent vector]] '''t'''. If the [[curvature]] <math>\kappa</math> of ''C'' at a certain point is not zero then the [[principal normal vector]] and the [[binormal vector]] at that point are the unit vectors |

: <math> \mathbf{n}=\frac{\mathbf{t}'}{\kappa}, \quad \mathbf{b}=\mathbf{t}\times\mathbf{n},</math> | : <math> \mathbf{n}=\frac{\mathbf{t}'}{\kappa}, \quad \mathbf{b}=\mathbf{t}\times\mathbf{n},</math> | ||

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* A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. | * A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. | ||

− | * The curvature and the torsion of a [[helix]] are constant. Conversely, any space curve with constant non-zero curvature and constant torsion is a helix. The torsion is | + | * The curvature and the torsion of a [[helix]] are constant. Conversely, any space curve with constant non-zero curvature and constant torsion is a helix. The torsion is negative for a right-handed<ref>http://encyclopedia2.thefreedictionary.com/right-handed+curve</ref> helix and is positive for a left-handed one. |

== Alternative description == | == Alternative description == | ||

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: <math> \tau = \frac{x'''(y'z''-y''z') + y'''(x''z'-x'z'') + z'''(x'y''-x''y')}{(y'z''-y''z')^2 + (x''z'-x'z'')^2 + (x'y''-x''y')^2}.</math> | : <math> \tau = \frac{x'''(y'z''-y''z') + y'''(x''z'-x'z'') + z'''(x'y''-x''y')}{(y'z''-y''z')^2 + (x''z'-x'z'')^2 + (x'y''-x''y')^2}.</math> | ||

+ | |||

+ | ==Notes== | ||

+ | {{Reflist}} | ||

==References== | ==References== | ||

− | *Andrew | + | <references /> |

+ | * {{ citation | last1 = Pressley | first1 = Andrew | title = Elementary Differential Geometry | publisher = Springer Undergraduate Mathematics Series, [[Springer-Verlag]] | year = 2001 | isbn = 1-85233-152-6 }} | ||

== External links == | == External links == | ||

− | *{{ | + | *{{Commons category|Illustrations for curvature and torsion of curves|Graphical illustrations of the torsion of space curves}} |

− | *[http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw Create your own animated illustrations of the torsion] ([[ | + | *[http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw Create your own animated illustrations of the torsion] ([[Maple (software)|Maple]]-Worksheet) |

{{curvature}} | {{curvature}} | ||

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[[Category:Curvature (mathematics)]] | [[Category:Curvature (mathematics)]] | ||

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[[ru:Дифференциальная геометрия кривых#Кручение]] | [[ru:Дифференциальная геометрия кривых#Кручение]] | ||

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## Revision as of 06:35, 23 October 2013

Template:Dablink
In the elementary differential geometry of curves in three dimensions, the **torsion** of a curve measures how sharply it is twisting out of the plane of curvature. Taken together,
the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

## Definition

Let *C* be a space curve parametrized by arc length and with the unit tangent vector **t**. If the curvature of *C* at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors

where the prime denotes the derivative of the vector with respect to the parameter . The **torsion** measures the speed of rotation of the binormal vector at the given point. It is found from the equation

which means

*Remark*: The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a by-product of the historical development of the subject.

The **radius of torsion**, often denoted by σ, is defined as

**Geometric relevance:** The torsion measures the turnaround of the binormal vector. The larger the torsion is, the faster rotates the binormal vector around the axis given by the tangent vector (graphical illustrations).
In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.

## Properties

- A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane.
- The curvature and the torsion of a helix are constant. Conversely, any space curve with constant non-zero curvature and constant torsion is a helix. The torsion is negative for a right-handed
^{[1]}helix and is positive for a left-handed one.

## Alternative description

Let **r** = **r**(*t*) be the parametric equation of a space curve. Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically, **r**(*t*) is a three times differentiable function of *t* with values in **R**^{3} and the vectors

are linearly independent.

Then the torsion can be computed from the following formula:

Here the primes denote the derivatives with respect to *t* and the cross denotes the cross product. For *r* = (*x*, *y*, *z*), the formula in components is

## Notes

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}