Catalan surface
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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.
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 [http://www.mathcurve.com/surfaces/catalan/catalan.shtml x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) \,
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 [https://books.google.com/books?id=K31Nzi_xhoQC&dq=conoid+maple&pg=PA277 ( )
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