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MANNHEIM CURVE

Notion studied by Mannheim
in 1859, name given by Wölffing in 1899.
Amédée Mannheim (1831-1906): French mathematician and artillery captain. |

If the intrinsic
equation 1 of the rolling curve is ,
then the Cartesian equation of the Mannheim curve is: ,
the axis Ox being the rolling axis.
If the intrinsic equation 2 of the rolling curve is , then the Cartesian parametrization of the Mannheim curve is: . |

The *Mannheim curve* associated to a curve is the
locus of the centre of curvature at the contact point of this curve rolling
without slipping on a line.

Examples:

initial curve | Mannheim curve |

circle | line |

alysoid (including the catenary) | parabola |

cycloid | circle |

cycloidal curve | ellipse |

pseudo-cycloidal curve | hyperbola |

logarithmic spiral | line |

Cornu spiral | rectangular hyperbola |

curve with sinusoidal radius | sinusoid |

involute of a circle | parabola |

catenary of equal resistance | catenary |

Ribaucour
curve of index k |
Ribaucour curve of index k – 1 |

pseudo-spiral
of index n |
curve |

See an application of the Mannheim curves for the wheel-road couples.

A possible generalisation is for the curve to roll on
any curve instead of a straight line, for example on a circle. In this
latter case, the locus of the centre of curvature can be called polar Mannheim
curve. If the intrinsic equation
1 of the rolling curve is ,
then the polar equation of the polar Mannheim curve is: ,
where *a* is the radius of the circle.

NOTE: in
the literature, another notion can be referred to as "Mannheim curve".
It consists in the curves the radius of curvature of which is proportional
to the signed distance to a fixed point.

This notion is tackled on the page dedicated to the
Norwich
spiral.

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© Robert FERRÉOL 2017