Catalan surface

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
A Catalan surface.

In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose rulings 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.[1]

The parametric equations of the Catalan surface are [2]

Special cases[edit]

If all the rulings 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.

See also[edit]