# expressions for curvature and torsion

For a regular^{} (http://planetmath.org/Curve), parameterized curve $\alpha :(a,b)\to {\mathbb{R}}^{3}$, not necessarily unit speed, the curvature^{} $\kappa (t)$ and torsion $\tau (t)$ are given, respectively, by

$\kappa (t)$ | $={\displaystyle \frac{\parallel {\alpha}^{\prime}(t)\times {\alpha}^{\prime \prime}(t)\parallel}{{\parallel {\alpha}^{\prime}(t)\parallel}^{3}}};$ | ||

$\tau (t)$ | $={\displaystyle \frac{({\alpha}^{\prime}(t)\times {\alpha}^{\prime \prime}(t))\cdot {\alpha}^{\prime \prime \prime}(t)}{{\parallel {\alpha}^{\prime}(t)\times {\alpha}^{\prime \prime}(t)\parallel}^{2}}}.$ |

## References

John McCleary, *Geometry ^{} from a Differentiable^{} Viewpoint*, Cambridge University Press, 1994.

Title | expressions for curvature and torsion |
---|---|

Canonical name | ExpressionsForCurvatureAndTorsion |

Date of creation | 2013-03-22 15:38:14 |

Last modified on | 2013-03-22 15:38:14 |

Owner | Simone (5904) |

Last modified by | Simone (5904) |

Numerical id | 7 |

Author | Simone (5904) |

Entry type | Theorem |

Classification | msc 53A04 |

Related topic | Torsion |

Related topic | CurvatureOfACurve |