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.

Equations

The vector equation of a Catalan surface is given by
where r = s is the space curve and L is the unit vector of the ruling at u = u. All the vectors L are parallel to the same plane, called the directrix plane of the surface. This can be characterized by the condition: the mixed product = 0.
The parametric equations of the Catalan surface are

Special cases

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.