Steiner Surface

A projection of the Veronese Surface into 3-D (which must contain singularities) is called a Steiner surface. A classification of Steiner surfaces allowing complex parameters and projective transformations was accomplished in the 19th century. The surfaces obtained by restricting to real parameters and transformations were classified into 10 types by Coffman et 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

$$x^2y^2-x^2z^2+y^2z^2-xyz=0,$$

and can be transformed into the Roman Surface or Cross-Cap by a complex projective change of coordinates (but not by a 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

$$y^2-2xy^2-xz^2+x^2y^2+x^2z^2-z^4=0,$$

and two of the three double lines of surface 2 coincide along a line where the two noncompact ``components'' are tangent.

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.