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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