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NODAL CURVE

Curve studied by La Gournerie in 1851. |

Polar equation:
(or )
with
n > 0. |

The *nodal curves* are the Brocard transforms of the Kappa, when the pole is at the centre of the Kappa.

Each curve is composed of an infinite branch, the base, obtained for :

If *n* is rational and its numerator is *p* and its denominator *q*, then the curve is composed of 2*p* branches, images of the base branch by rotation when *q* is odd, and *p* branches when it is even.

Examples:

n = 1 : Kappa |
n = 2: windmill |
n = 3 |
n = 4 |
n = 5 |

n = 1/2:
right strophoid |
n = 3/2 |
n = 5/2 |
n = 7/2 |
n = 9/2 |

n = 1/3 |
n = 2/3 |
n = 4/3 |
n = 5/3 |
n = 7/3 |

n = 1/4 |
n = 3/4 |
n = 5/4 |
n = 7/4 |
n = 9/4 |

n = 1/5 |
n = 2/5 |
n = 3/5 |
n = 4/5 |
n = 6/5 |

All nodal curves are stereographic projections of the clelia.

The inverse (with centre O and radius *a*^{2}) of a nodal curve is the same curve turned by .

Asymptotic line of a 3D helicoid????

Compare to the epispiral.

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© Robert FERRÉOL 2017