# mean curvature (plane curve)

Let $\mathrm{\Gamma}$ be a piecewise ${C}^{1}$ planar curve.

The *total curvature ^{}*, ${\kappa}_{total}$ , of $\mathrm{\Gamma}$ is defined to be ${\int}_{\mathrm{\Gamma}}|\kappa (s)|\mathit{d}s$ where $\mathrm{\Gamma}$ is parameterized by arclength $s$
and $\kappa (s)$ is the curvature

^{}(http://planetmath.org/CurvatureOfACurve) of $\mathrm{\Gamma}$.

The *mean curvature ^{}* of $\mathrm{\Gamma}$ is defined to be the ratio of the total curvature to the length of $\mathrm{\Gamma}$ :

$$M(\mathrm{\Gamma})=\frac{{\kappa}_{total}(\mathrm{\Gamma})}{L(\mathrm{\Gamma})}$$ |

Title | mean curvature (plane curve) |
---|---|

Canonical name | MeanCurvatureplaneCurve |

Date of creation | 2013-03-22 15:31:16 |

Last modified on | 2013-03-22 15:31:16 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 11 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 53A04 |

Related topic | MeanCurvatureAtSurfacePoint |

Defines | total curvature |

Defines | mean curvature |