# torsion (space curve)

Let $I\subset R$ be an interval^{} and let $\gamma :I\to {\mathbb{R}}^{3}$ be a
parameterized space curve, assumed to be
regular^{} (http://planetmath.org/SpaceCurve) and free of points of inflection. We
interpret $\gamma (t)$ as the trajectory of a particle moving through
3-dimensional space. Let $T(t),N(t),B(t)$ denote the corresponding
moving trihedron. The speed of this particle is given by $\parallel {\gamma}^{\prime}(t)\parallel $.

In order for a moving particle to escape the osculating plane, it is
necessary for the particle to “roll” along the axis of its tangent
vector^{}, thereby lifting the normal acceleration vector out of the
osculating plane. The “rate of roll”, that is to say the rate at
which the osculating plane rotates about the tangent vector, is given
by $B(t)\cdot {N}^{\prime}(t)$; it is a number that depends on the
speed of the particle. The rate of roll relative to the particle’s
speed is the quantity

$$\tau (t)=\frac{B(t)\cdot {N}^{\prime}(t)}{\parallel {\gamma}^{\prime}(t)\parallel}=\frac{({\gamma}^{\prime}(t)\times {\gamma}^{\prime \prime}(t))\cdot {\gamma}^{\prime \prime \prime}(t)}{{\parallel {\gamma}^{\prime}(t)\times {\gamma}^{\prime \prime}(t)\parallel}^{2}},$$ |

called the torsion^{} of the curve, a quantity
that is invariant with respect to reparameterization. The torsion
$\tau (t)$ is, therefore, a measure of an intrinsic property of the
oriented space curve, another real number that can be covariantly
assigned to the point $\gamma (t)$.

Title | torsion (space curve) |
---|---|

Canonical name | TorsionspaceCurve |

Date of creation | 2013-03-22 12:15:05 |

Last modified on | 2013-03-22 12:15:05 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 9 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 14H50 |

Synonym | torsion |

Related topic | SpaceCurve |

Related topic | CurvatureOfACurve |