next surface previous surface 2D curves 3D curves surfaces fractals polyhedra

CATALAN'S MINIMAL SURFACE

Surface studied in 1855 by Catalan.
Eugene Charles Catalan (1814-1894): Franco-Belgian 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


next surface previous surface 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL  2017