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
is given by

where
is the unit vector characterizing the ruling, with
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.

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.