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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