Catalan surface

A Catalan surface.

This article is about the ruled surfaces. For the minimal surface, see Catalan's minimal 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.

The parametric equations of the Catalan surface are

$x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) \,$

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.

References

A. Gray, E. Abbena, S. Salamon,Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, FL:CRC Press, 2006. (ISBN 9781584884484)
Hazewinkel, Michiel, ed. (2001), "Catalan surface", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
V. Y. Rovenskii, Geometry of curves and surfaces with MAPLE (ISBN 978-0-8176-4074-3)

Catalan surface

See also

References