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.
The vector equation of a Catalan surface is given by
r = s(u) + v L(u),
where r = s(u) is the space curve and L(u) is the unit vector of the ruling at u = u. All the vectors L(u) are parallel to the same plane, called the directrix plane of the surface. This can be characterized by the condition: the mixed product ['''''L'''''(''u''), '''''L' '''''(''u''), '''''L" '''''(''u'')] = 0.https://books.google.com/books?id=K31Nzi_xhoQC&dq=catalan+surface&pg=PA279
The parametric equations of the Catalan surface are http://www.mathcurve.com/surfaces/catalan/catalan.shtml
x=f(u)+vi(u), y=g(u)+vj(u), z=h(u)+vk(u)
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.