Bézier representation for cubic surface patches (Q1208633)

It would be nice to have an easy method to model cubic surfaces in computer aided design. The authors offer a method of parametrization that yields a biquadratic Bézier representation of a surface patch given by 4 vertices of the surface \(F_ 3\) that lie pairwise on a straight line contained in the surface. The parametrization is based on a nice idea and deserves some study. \{Unfortunately, the method does not work in general. Of the 27 straight lines on a nondegenerate real cubic surface over \(C^ 3\), only 3 have to be real and these may form a triangle. In addition, there are 18 classes of singular cubics and two classes of ruled cubic surfaces. Also, if one already has two straight lines on the surface, one has all the tools to write down its equation in Cayley-Salmon normal form and the Clebsch birational representation of the surface onto the projective plane and with it a parametric representation of the implicitly given surface. It would be worthwhile to find an easy algorithmic implementation of the Clebsch representation or other reductions to normal forms. All mathematics necessary to make cubic surfaces useful in CAD is assembled in article IIIC10a by \textit{W. Fr. Meyer} of the old Encyklopädie der mathematischen Wissenschaften, 3. Band, 2. Teil, 2. Hälfte, p. 1437- 1531 (1928)\}.