Steiner Surface
A projection of the Veronese Surface into 3-D (which must contain singularities) is called a Steiner surface. Aclassification of Steiner surfaces allowing complex parameters and projective transformations was accomplished in the 19thcentury. The surfaces obtained by restricting to real parameters and transformations were classified into 10 types by Coffmanet al. (1996). Examples of Steiner surfaces include the Roman Surface (Coffman type 1) and Cross-Cap (type 3).
The Steiner surface of type 2 is given by the implicit equation
and can be transformed into the Roman Surface or Cross-Cap by a complex projective change of coordinates (but not bya real transformation). It has two pinch points and three double lines and, unlike the Roman Surface or Cross-Cap,is not compact in any affine neighborhood.
The Steiner surface of type 4 has the implicit equation
and two of the three double lines of surface 2 coincide along a line where the two noncompact ``components'' aretangent.See also Cross-Cap, Roman Surface, Veronese Variety
References
Coffman, A. ``Steiner Surfaces.'' http://www.ipfw.edu/math/Coffman/steinersurface.html. Coffman, A.; Schwartz, A.; and Stanton, C. ``The Algebra and Geometry of Steiner and Other Quadratically Parametrizable Surfaces.'' Computer Aided Geom. Design 13, 257-286, 1996. Nordstrand, T. ``Steiner Relative.'' http://www.uib.no/people/nfytn/stmtxt.htm. Nordstrand, T. ``Steiner Relative [2].'' http://www.uib.no/people/nfytn/stm2txt.htm.
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