Alexander Enzmann wrote:

> Cool pics.  Ever seen the book, "A Topological Picturebook" by Francis?
> There's a bunch of stuff in there that I've always wanted to see
> rendered/animated (particularly the tobacco pouch).  I've seen several
> sphere eversion animations and they are always fun (haven't seen a
> POV-Ray script for generating one yet).
>
> One comment - you might point out on the web site that due to the way
> ray tracing works, you end up missing the 2 dimensional solutions (for
> the Roman surface anyway) that lie on the axes.
>
> As far as the classification of Steiner surfaces - wasn't there a book
> published around 10 years ago that had them all laid out?  Forget the
> title - I happened across it at the Harvard COOP one day and didn't have
> enough cash to buy it.
>
> Xander

....
Thanks for the comments!

The book you (AE) mentioned sounds like it is Francois Apery's

"Models of the Real Projective Plane:
Computer Graphics of Steiner and Boy Surfaces."
(Vieweg, 1987)

I recommend it to anyone interested in math and computer graphics, and
a large part of the book deals with the geometry, topology, and
algebra of Steiner surfaces.  It explains some of the ideas on my web
page, and many more, and lists a few different types of Steiner
surfaces, as well as Boy's surface.

To answer your question about how much of the classification appears
in this book, here's a brief history of the research on Steiner
surfaces.  It's been known (for some time, at least 100 years) that
quartic Steiner surfaces usually have three double lines (Types 1-3),
but that sometimes two of these lines coincide (Types 4 and 5), or all
three coincide (Type 6).  It was also known that there were Steiner
surfaces containing infinitely many lines, and these have cubic or
quadratic equations.

Even before computer graphics, in the days of wire and plaster models,
it was known that Types 1, 2, and 3 looked different in real space
despite being equivalent under re-scaling the coordinates using
complex numbers, for example, starting with the Roman surface:

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

and changing x to i*x (where i^2=-1) and z to i*z

gives the equation:

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

and then multiplying both sides by -1 gives the Type 2 equation:

x^2y^2-x^2z^2+y^2z^2-xyz=0.

The Type 3 (Cross-cap) is also related to these.  It has three double
lines, but two of them are complex, and can be seen in real space only
after a complex change of coordinates (and then, the surface is Type 1
or 2).

Apery's book gives several nice ways to describe Steiner surfaces, and
an elementary proof that the quartics fall into three classes (Types
1-3, 4-5, and 6) when complex transformations are allowed.  The book
also has nice graphics of Types 1 and 3, but not the other Steiner
surfaces, and I haven't seen these other types rendered anywhere else,
either.

What's new (1980's-90's, after Apery's book appeared) is the
development of an algebraic procedure for detecting whether parametric
equations are equivalent under _real_ transformations.
(Unfortunately, this procedure is rather tedious and I must only refer
to the CAGD article for the details.)  It explains, for example, why
there isn't any way to transform the Roman surface equation into the
Type 2 or 3 equation using only real numbers (no "i").  The conclusion
we derived from the classification of real Steiner quartics, using
only real transformations, is that there are exactly three types with
three double lines, two types with two double lines, and only one type
with one double line.  Further, there are exactly three different
types (7,8,9) of ruled cubics that can arise.

I think I'll update the page to clarify some of these points and also
mention the "whiskers" sticking out of some of the surfaces.  Maybe
later this year I'll have some animations ready.  Thanks again for the
feedback.

Adam C.

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