Catalan surface

In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose generators are parallel to a fixed plane.

This can be characterized by the condition: the mixed product [L(u), L' (u), L" (u)] = 0.

[1] The parametric equations of the Catalan surface are [2]

If all the generators of a Catalan surface intersect a fixed line, then the surface is called a conoid.

Catalan proved that the helicoid and the plane were the only ruled minimal surfaces.