Ken wrote:

> ... have nice ray-tracing properties.  ???
>
>   I would be interested to hear an elaboration on that particular statement.
>
>

To address your request for a few more remarks on Steiner surfaces andray
tracing, I'll only be able to make a few comments about geometry.
I'll refer to the literature to let the computer scientists explain
for themselves why they're interested in Steiner surfaces.

1.  For quartics in general, a line (such as a light ray) meets the
surface in at most four points, or else it is completely contained in
the surface.  Lines intersecting one of the Steiner surface's double
lines meet the surface in at most two other points, and lines passing
through the triple point can meet the surface in at most one other
point.

2.  Let P be a point on a Steiner surface, and let L be a line tangent
to the surface at P.  Then there's a conic curve (ellipse, hyperbola,
parabola, or union of lines) lying on the surface, so that L is
tangent to the conic at P.  (I am skipping a few details, for example,
what about the singular points...) This property is called having a
"double infinity" of conics, and the Steiner surfaces are known to be
the only surfaces with this property.  The idea is that every line in
the parametric domain is mapped to a conic curve by the quadratic
rational parametrization.

3.  The intersection of a parametric Steiner surface with one of its
tangent planes is, in general, a union of two conic curves.  (possibly
complex or singular)

4.  The simple form of the parametric and implicit equations means the
normal vector at a point P is easy to calculate, and there are
formulas for the normal vector in terms of the coefficients of the
equations.  Presumably these formulas are computationally faster than
calculating derivatives.

...

If you happen to be near a math/CS library, here are a few more
references to Steiner surfaces, in addition to the Apery book I
mentioned in a previous post.

F. Aries and R. Senoussi, "Approximation de surfaces parametriques par
des carreaux rationnels du second degre en lancer de rayons," Revue de
CFAO et d'informatique graphique (6) 12 (1997), 627--645.

W. L. F. Degen, "The types of triangular Bezier surfaces," in The
Mathematics of Surfaces, VI (Brunel Univ., 1994), 153--170,
Inst. Math. Appl. Conf. Ser. New Ser., 58, G. Mullineux, ed., Oxford
Univ. Press, New York, 1996.

W. Boehm and H. Prautzsch, "Geometric Concepts for Geometric Design,"
published by A. K. Peters, 1994.

T. Sederberg and D. Anderson, "Steiner surface patches," IEEE
Comput. Graph. Appl. 5 (1985) 23--36.

T. Sederberg, "Ray tracing of Steiner patches," Computer Graphics 18
(1984) 159--164.

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