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INVOLUTE OF A PLANE CURVE

Notion studied by L'Hôpital in 1647, Fontenelle in 1712 (who gave it its name), then by Cesaro in 1888. |

If M_{0} is the current
point on ,
the current point M of an involute is the locus of the points ,
which yields:
Cartesian parametrization: . Complex parametrization: . Radius of curvature: . |

The *involutes* of a plane curve
are the curves traced by the end of a wire tightened along
and winding itself along .
In other words, they are the traces on the plane of a point on a line pivoting
without slipping on
(therefore, they are special cases of roulettes).

Thus, they also are the orthogonal trajectories of the family of the tangents to the curve, or also the curves for which the initial curve is the evolute.

The involutes of a curve are parallel
to one another.

It will be noticed (cf. figures) that an inflexion point on the initial curve yields a cuspidal point of the second kind on the involute, as the Marquis de l'Hospital had already understood in 1696 (see this text). |

Examples: (see also at evolute)

initial curve |
origin |
involute, or one of them |

circle | any point | involute of a circle! |

point | any point | circle |

cycloid | vertex | the same cycloid, translated |

cardioid | vertex | similar cardioid with ratio 3 |

nephroid | vertex | similar nephroid with ratio 2 |

nephroid | cuspidal point | Cayley sextic |

epicycloid
with parameter q |
vertex | similar epicycloid with ratio
(q + 2)/q |

deltoid | vertex | similar deltoid with ratio 1/3 |

astroid | vertex | similar astroid with ratio 1/2, Maltese cross |

hypocycloid
with parameter q |
vertex | similar hypocycloid with ratio
(q-2)/q |

logarithmic spiral | centre | logarithmic spiral |

catenary | vertex | tractrix |

logarithmic | any point | involute of an exponential |

involute of a circle | cuspidal point | Norwich spiral |

See the generalisation
to 3D curves.

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© Robert FERRÉOL 2017