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CYCLOTOMIC SURFACE

Surface studied by Eugene Catalan in 1859; name proposed by Barre de Saint Venant.
From the Greek kuklos: "circle" and temnein
"cut". |

Spherical equation for a cyclotomic surface with axis Oz and directrix with polar equation in the plane xOy: .
Cartesian parametrization: . Cartesian equation: eliminate u between and , hence the general form: .
Polar equation of the section by a plane z = h: . |

A cyclotomic surface is a circled surface generated by a circle with *varying radius* in rotation around one of its diameters (its center *O* remaining fixed).

The surface is completely defined by the point *O* and the section of the surface by the plane passing through *O* and perpendicular to the fixed diameter of the generating circles, such section is called the *directrix* of the surface.

Examples:

If the directrix is a circle with center *O*, then the associated cyclotomic surface is the sphere.

If the directrix is a circle passing by *O*:

Spherical equation: .
Cartesian equation: . Quartic surface. Polar equation of the section by a plane z = h:
(lemniscate of Booth, which is a lemniscate
of Bernoulli for h = a/2).
Volume : . Area: . |

If the directrix is a line that does not pass by *O*:

Spherical equation: .
Cartesian equation: . Quartic surface. |

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© Robert FERRÉOL 2017