Catalan surface

A ruled surface whose rectilinear generators are all parallel to the same plane. Its line of restriction (cf. Ruled surface) is planar. The position vector of a Catalan surface is $r=\rho(u)+vl(u)$, where $l''(u)\neq0$, $(l,l',l'')=0$. If all the generators of a Catalan surface intersect the same straight line, then the surface is a conoid.

References

• [1] E. Catalan, "Mémoire sur les surfaces gauches à plan directeur" , Paris (1843)
• [a1] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German) Zbl 0366.53001
• [a2] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 31–35 Zbl 0425.53001