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CATENOID



Surface studied by Euler in 1740.
The name comes from catena: chain, which is also the Latin name of the catenary. Other name: alysseid, from the Greek alusion "small chain" (given by Bour in 1862). 
Cylindrical equation: .
Cartesian parametrization: (). Parametrization for which the coordinate lines are the rhumb lines forming an angle with the parallels, which are also the asymptotic lines for = 45° [Fedenko ex 807] : : . First fundamental quadratic form: where . Surface element: . Second fundamental quadratic form: . Gaussian curvature: . Area of the portion : . Volume: . 
The catenoid is the surface of revolution generated by the rotation of a catenary around its base.
We get the parametrization by taking and in the Weierstrass parametrization of a minimal surface.
Consider two parallel circular rings with diameters D
at distance d;
It can be proved that if d/D < 0.66, there exist 3 minimal surfaces supported by these two rings: 2 catenoids and the socalled Goldschmidt surface, composed of the two disks delimited by the rings. It can be proved that if d/D < 0.53, the surface with minimal area between these 3 is one of the 2 catenoids (in red opposite); but starting from 0.53, it is weirdly the Goldschmidt surface which has minimal area. And starting from 0.66, it is the only surface anyway: there no longer are catenoids supported by the rings. Remarks:

Animation showing the profile of the 2 catenoids for d/D ranging from 0.1 to 0.66; 
On this picture, the theoretical limit 0.53 of blowup seems to be exceeded
by far...

A catenoid can be continuously and isometrically transformed
into a right helicoid,
the surface remaining constantly minimal.
Equations of this transform: The intermediate surfaces are the minimal helicoids. 
See also the skew catenoid,
the trinoid, and the
axial revolution of the catenary.
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© Robert FERRÉOL 2019