# Frenet frame

Let $I\subset \mathbb{R}$ 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. The moving trihedron (also known as the Frenet
frame, the Frenet trihedron, the repère mobile, and the moving
frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined
and named as follows:

$T(t)$ | $={\displaystyle \frac{{\gamma}^{\prime}(t)}{\parallel {\gamma}^{\prime}(t)\parallel}},$ | the unit tangent; | ||

$N(t)$ | $={\displaystyle \frac{{T}^{\prime}(t)}{\parallel {T}^{\prime}(t)\parallel}},$ | the unit normal; | ||

$B(t)$ | $=T(t)\times N(t),$ | the unit binormal. |

A straightforward application of the chain rule^{} shows that these
definitions are covariant with respect to reparameterizations. Hence,
the above three vectors should be conceived as being attached to the
point $\gamma (t)$ of the oriented space curve, rather than being
functions of the parameter $t$.

Corresponding to the above vectors are 3 planes, passing through each
point $\gamma (t)$ of the space curve. The *osculating plane* at
the point $\gamma (t)$ is the plane spanned by $T(t)$ and $N(t)$; the
*normal plane ^{}* at $\gamma (t)$ is the plane spanned by $N(t)$ and
$B(t)$; the rectifying plane at $\gamma (t)$ is the plane spanned by
$T(t)$ and $B(t)$.

Title | Frenet frame |

Canonical name | FrenetFrame |

Date of creation | 2013-03-22 12:15:44 |

Last modified on | 2013-03-22 12:15:44 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 16 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 53A04 |

Synonym | moving trihedron |

Synonym | moving frame |

Synonym | repère mobile |

Synonym | Frenet trihedron |

Related topic | SpaceCurve |

Defines | osculating plane |

Defines | normal plane |

Defines | rectifying plane |

Defines | unit normal |

Defines | unit tangent |

Defines | binormal |