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
| r(u,v)=s(u)+vL(u), |
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
| [L(u),L^'(u),L^('')(u)]=0. |
If the generators intersect in a fixed line, the surface becomes a conoid.
See also
Catalan Minimal Surface, ConoidExplore with Wolfram|Alpha
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References
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.Referenced on Wolfram|Alpha
Catalan Ruled SurfaceCite this as:
Weisstein, Eric W. "Catalan Ruled Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalanRuledSurface.html