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FOLIOID

Curve studied by P. van Geer in 1918.
Loria 3D p. 240 |

Polar equation: with n real > 0, i.e.
with .
Algebraic curve iff n is rational:
if p and q are the numerator and denominator of n, the degree is 2(p +q) if p and q are odd, and 4(p + q) otherwise.
For n =1, it is the circle with centre (b,
0) and radius a.
For a = b (e = 1), it is the rose . |

Case *e* < 1: the curve is rather close to a double conchoid of a rose: ):

In blue, the two conchoids of a rose, and in red the folioid, here with *n* = 4.

Case *e* > 1: the curve is composed of *p* closed curves if *p* and *q* are odd, and 2*p* curves otherwise.

n = 1: circle |
n = 2 |
n = 3 |
n = 4 |
n = 5 |

n = 1/2 |
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 |

If it is wrapped around the plane of the circle into a cone with vertex *O* and half-angle , with axis *Oz*, the projection on *xOy* of this wrapped around circle is the folioid . This provides a construction of folioids based on a simple circle in the case *n* < 1.

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© Robert FERRÉOL
2017