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LINE TRACED ON A SURFACE

Lines (or curves) traced on a surface can be classified into three categories:

- the geometric lines, that only depend on the geometry of the surface:

Line | curvature | asymptotic | geodesic | pseudogeodesic | geodesic circle |

Definition (see the notations) | extreme normal curvature | zero normal curvature | - normal curvature = curvature of the curve
- minimal length |
normal curvature proportional to the curvature | constant geodesic curvature |

Differential condition | or |

- the topographic lines, based on a given vertical direction, supported by .

Line | level | slope | helix | crest, or thalweg |

Definition | Constant altitude | Maximal slope | Constant slope | |

Differential condition |

- the gravitational lines, depending on a given gravitational field .

Line | flow | catenary | brachistochrone |

Definition | line followed by a massive point. | - shape assumed by a homogeneous wire placed on the surface
- homogeneous line the center of gravity of which has minimal altitude. |
- minimal travel time |

Differential condition | where |

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© Robert FERRÉOL
2018