CATALAN SURFACE

 Surface studied by Catalan in 1855. Eugene Charles Catalan (1814-1894): Franco-Belgian mathematician. Other name: ruled surface with directrix plane.

 Cartesian parametrization: , reunion of the lines Du passing through   and directed by  with . Reduced Cartesian equation of Catalan surfaces with directrix plane xOy with only one line in each plane parallel to xOy:  (reunion of the lines ), giving a conoid for .

A Catalan surface is a ruled surface the generatrices of which remain parallel to a fixed plane, called the directrix plane, in other words, a ruled surface the directrix cone of which is planar.

Examples: the cylinders, the conoids, the ruled helicoids with directrix plane.

The family of lines based on two given curves and parallel to a  given plane generates a surface of Catalan. Taking by
example xOy as plane, we get:

 Cartesian parametrization : , meeting of lines going through  and . Example of a Catalan surface based on two sinusoids.

Do not mistake these for Catalan's minimal surface.