# normal section

Let $P$ be a point of a surface

$F(x,y,z)=0,$ | (1) |

where $F$ has the continuous^{} first and partial derivatives^{} in a neighbourhood of $P$. If one intersects the surface with a plane containing the surface normal at $P$, the intersection curve is called a normal section.

When the direction of the intersecting plane is varied, one gets different normal sections, and their curvatures^{} (http://planetmath.org/CurvaturePlaneCurve) at $P$, the so-called normal curvatures, vary having a minimum value ${\varkappa}_{1}$ and a maximum value ${\varkappa}_{2}$. The arithmetic mean^{} of ${\varkappa}_{1}$ and ${\varkappa}_{2}$ is called the mean curvature^{} of the surface at $P$.

By the suppositions on the function $F$, examining the normal curvatures can without loss of generality be to the following: Examine the curvature of the normal sections through the origin, the surface given in the form

$z=z(x,y),$ | (2) |

where $z(x,y)$ has the continuous first and partial derivatives in a neighbourhood of the origin and

$$z(0,\mathrm{\hspace{0.17em}0})={z}_{x}^{\prime}(0,\mathrm{\hspace{0.17em}0})={z}_{y}^{\prime}(0,\mathrm{\hspace{0.17em}0})=0.$$ |

Indeed, one can take a new rectangular coordinate system with $P$ the new origin and the normal at $P$ the new $z$-axis; then the new $xy$-plane coincides with the tangent plane^{} of the surface (1) at $P$. The equation (1) defines the function of (2).

Title | normal section |
---|---|

Canonical name | NormalSection |

Date of creation | 2013-03-22 17:26:25 |

Last modified on | 2013-03-22 17:26:25 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 53A05 |

Classification | msc 26A24 |

Classification | msc 26B05 |

Related topic | SecondFundamentalForm |

Related topic | DihedralAngle |

Defines | normal curvature |