Catalan surface

Equations

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.https://books.google.com/books?id=K31Nzi_xhoQC&dq=catalan+surface&pg=PA279

The parametric equations of the Catalan surface are http://www.mathcurve.com/surfaces/catalan/catalan.shtml

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

Special cases

If all the generators 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, Florida:CRC Press, 2006. https://www.crcpress.com/product/isbn/9781584884484 ()
• V. Y. Rovenskii, Geometry of curves and surfaces with MAPLE https://books.google.com/books?id=K31Nzi_xhoQC&dq=conoid+maple&pg=PA277 ()