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CATALAN'S MINIMAL SURFACE
Surface studied in 1855 by Catalan.
Eugene Charles Catalan (18141894): FrancoBelgian mathematician. 
Cartesian parametrization: .
Simply periodic minimal surface. 
Catalan's minimal surface is the surface obtained by taking
(and then )
in the Weierstrass parametrization of a minimal
surface: .
With
in the parametrization above, we get ;
the projection of the coordinate lines at constant r on a horizontal
plane are trochoids,
and the coordinate lines at constant v are parabolas.
Moreover, the section of Catalan's surface by xOy is a cycloid, which is a geodesic of the surface. 

Opposite, an animated view of the surface "associated" to Catalan's surface, i.e. the surfaces obtained by taking in the Weierstrass parametrization. Their parametrization is . 
Here is the original text by Catalan in which he publishes his surface as an example of application of a general formula for minimal surfaces:
The surface represented by these three equations can be generated in
the following way:
Define OSA the cycloid described by the point S belonging
to the circumference CI, and the cycloidOPB, envelope of
the moving radius CS, P being the contact point. If we create,
in a plane perpendicular to that of the figure, a parabola the projection
of the directrix of which is P, and such that S is the vertex,
this curve (with variable size) generates the surface.
Do not mistaken this surface for the Catalan surfaces.
Catalan's minimal surface, by Alain Esculier
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© Robert FERRÉOL 2017