Catalan surface
Introduction  A 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:'''''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.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.
See also * Generalized helicoid
References * A. Gray, E. Abbena, S. Salamon, ''Modern differential geometry of curves and surfaces with Mathematica'', 3rd ed. Boca Raton, Florida:CRC Press, 2006. ()* * V. Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE''  ()