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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.

Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution. It is also the only minimal surface with a circle as a geodesic.

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 so-called 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.

 - the constant 0.66 (Laplace's limit constant) is an approximated value of 1/ sinh u, where u is defined by u = coth u, u > 0 
 - the constant 0,53 is an approximated value of the solution of 2ch((x^2+1)/2) = x+1/x.


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 blow-up 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