# Meusnier’s theorem

Let $P$ be a point of a surface^{} $F(x,y,z)=0$ where $F$ is twice continuously differentiable in a neighbourhood of $P$. Set at $P$ a tangent^{} of the surface. At the point $P$, set through this tangent both the normal plane^{} and a skew plane forming the angle (http://planetmath.org/AngleBetweenTwoPlanes) $\omega $ with the normal plane. Let $\varrho $ be the radius of curvature^{} of the normal section^{} and ${\varrho}_{\omega}$ the radius of curvature of the inclined section.

Meusnier proved in 1779 that the equation

$${\varrho}_{\omega}=\varrho \mathrm{cos}\omega $$ |

between these radii of curvature^{} is valid.

One can obtain an illustrative interpretation for the Meusnier’s theorem, if one thinks the sphere with radius the radius $\varrho $ of curvature of the normal section and with centre the corresponding centre of curvature. Then the equation utters that the circle, which is intersected from the sphere by the inclined plane, is the circle of curvature of the intersection curve of this plane and the surface $F(x,y,z)=0.$

Title | Meusnier’s theorem |

Canonical name | MeusniersTheorem |

Date of creation | 2013-03-22 17:28:39 |

Last modified on | 2013-03-22 17:28:39 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 53A05 |

Classification | msc 26B05 |

Classification | msc 26A24 |

Synonym | theorem of Meusnier |

Related topic | EulersTheorem2 |

Related topic | ProjectionOfPoint |

Related topic | NormalCurvatures |

Defines | inclined section |