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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.
Contents
Equations
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
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.