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RADIAL CURVE OF A GIVEN CURVE

Notion studied by Tucker in 1864. |

The radial curve of a curve ,
associated to a fixed point O, is the locus of the points P
defined by
where
is the vector joining the current point on
to its centre of curvature; in other words, it is the locus of the end
of the radius-of-curvature vector, attached to a fixed point. |

For an initial curve
with current point ,
the radial curve is the set of points .
Cartesian parametrization of the radial curve: . Complex parametrization: . If the intrinsic equation 2 of the initial curve is , then the polar equation of its radial curve is . |

The radial curve of an algebraic curve is an algebraic curve of the same degree as its evolute.

Examples:

initial curve |
associated radial curve |

circle | circle |

ellipse | sextic
with equation i.e. (Loria p. 308) |

parabola | duplicatrix cubic |

cycloid
(with rolling circle with radius R) |
circle with radius 2R |

deltoid | regular trifolium |

astroid | quatrefoil |

epicycloid with parameter
q |
rose |

hypocycloid with parameter
q |
rose |

catenary of equal strength | line |

catenary | kampyle of Eudoxus |

tractrix | kappa |

involute of a circle | Archimedean spiral |

clothoid | lituus |

logarithmic spiral | logarithmic spiral |

Ribaucourt curve
of index k
the cases k = -2, -1, and 2 amount to the above cases of the
parabola, the catenary and the cycloid. |
Clairaut's curve
of index k-1 |

pseudo-spiral
of index n
the cases n = 0 , 1, and -1/2 amount to the above cases of the
circle, the clothoid and the involute of a circle. |
Archimedean spiral of index |

See an application of radial curves for
wheel-road
couples.

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© Robert FERRÉOL 2017