> In the long-run, wouldn't it make
> more sense to actually create function based, spline-type surfaces?  Is
that
> what NURBS are?  (If so, then this has probably been discussed before.)


Yes, that's what NURBS are AFAIK. However, even NURBS and some other smooth
objects (like bezier patches) which don't have a simple, predefined equation
governing them are approximated by triangles. The reason is that triangles
are a lot faster to render. Try rendering a parametric object in POV-Ray; I
don't believe it approximates the surface with triangles and it severely
impacts the speed.

Another difficulty is that triangle meshes, NURBS, etc all define a surface;
they don't define the "inside" of the object as well as x^2+y^2+z^2 - r^2
defines the inside of a sphere.

Function defined surfaces would certainly be more useful if these problems
didn't exist, but the speed of polygon-based surfaces is worth the
inconveniences.

> Oh, also I've apparently been saying "smooth triangle problem" when I
should
> have really been saying "shadow line problem".  I assumed they were both
the
> same thing.

They're different but since they both appear with smooth triangles they're
often grouped together.

The smooth triangle "normal flipping" bug (or not bug) occurs around the
edges of an object no matter where you look at the object from, creating a
black halo. The "shadow line" issue occurs at the points of an object with
smoothed triangles or perturbed normals which are along the line where the
light stops hitting the object. This creates a sharp transition from light
into shadow where one would want it to be smooth.

 - Slime
 [ http://www.slimeland.com/ ]

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