Catalan Ruled Surface

A Catalan ruled surface is a ruled surface whose generators are all parallel to the same plane. The general form of such a surface generated by a space curve parametrized by s(u) is given by


where L(u) is the unit vector characterizing the ruling, with L(u) always parallel to the same plane (called the directrix plane). The latter condition can be characterized by the scalar triple product identity


If the generators intersect in a fixed line, the surface becomes a conoid.

See also

Catalan Minimal Surface, Conoid

Explore with Wolfram|Alpha


Catalan, E. Mémoire sur les surfaces gauches à plan directeur. Paris, 1843.Klingenberg, W. A Course in Differential Geometry. Springer, 1978.Millman, R. S. and Parker, G. D. Elements of Differential Geometry.0132641437 Prentice-Hall, pp. 31-35, 1977.

Cite this as:

Weisstein, Eric W. "Catalan Ruled Surface." From MathWorld--A Wolfram Web Resource.

Subject classifications