# line of curvature

A line $\gamma $ on a surface $S$ is a line of curvature of $S$, if in every point of $\gamma $ one of the principal sections has common tangent^{} with $\gamma $.

By the parent entry (http://planetmath.org/NormalCurvatures), a surface $F(x,y,z)=0$, where $F$ has continuous^{} first and partial derivatives^{}, has two distinct families of lines of curvature, which families are orthogonal^{} (http://planetmath.org/ConvexAngle) to each other.

For example, the meridian curves and the circles of latitude are the two families of the lines of curvature on a surface of revolution.

On a developable surface^{}, the other family of its curvature lines consists of the generatrices of the surface.

A necessary and sufficient condition for that the surface normals of a surface $S$ set along a curve $c$ on $S$ would form a developable surface, is that $c$ is a line of curvature of $S$.

Title | line of curvature |
---|---|

Canonical name | LineOfCurvature |

Date of creation | 2013-03-22 18:08:44 |

Last modified on | 2013-03-22 18:08:44 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 53A05 |

Classification | msc 26B05 |

Classification | msc 26A24 |

Synonym | curvature line |

Related topic | TiltCurve |