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.