I am working on a web site describing "Steiner surfaces," which are
geometric shapes
defined by specific types of polynomial equations.  They are of some
interest in the theory of
computer graphics, since Steiner surface patches contain many conic
curves (ellipses, etc.) and
have nice ray-tracing properties.

In a recent paper, my co-authors and I proved there are only finitely
many equivalence classes
of Steiner surfaces, up to linear transformations of the parametric
domain and the ambient
three-dimensional projective space.  The results of the classification
include some well-known
shapes, for example, Steiner's Roman surface, Steiner's Cross-cap
surface,
Whitney's Umbrella, and ordinary spheres and ellipsoids, but also some
less
well-known quartic and cubic shapes.

The web site lists some equations for representatives of each
equivalence class, and I've posted
some very simple POV-ray scenes and source code.  I've also written a
very brief background
discussion of projective geometry, but this is probably only useful for
experts who already know it,
and the site has links to other web pages related to Steiner surfaces.

Finally, you'll find an "include" file for several Steiner quartics and
cubics,
modeled after A. Enzmann's shapesq.inc file.

Here's the URL:
 http://www.ipfw.edu/math/Coffman/steinersurface.html


Adam Coffman
Assistant Professor of Mathematics
Indiana Purdue Univ. Fort Wayne

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