> It's news to me that POV supports splines in the first place.

It does bezier patches which are the closest you are going to get to 
describing "any curved surface" mathematically.

> Only if you have the original curve to hand.

What if you designed the curve by seeing how it turned out after 
sub-divison?  Blender works like this, you can edit the vertices of a very 
rough triangle mesh and see how the perfectly smooth surface reacts in real 
time.  You are then defining the "original curve" by the crude mesh.

> If you *insist* on using triangles, you're going to need a hell of a lot 
> of them to fake the appearence of a good curve. That means you either need 
> a triangle mesh of absurd dimensions, or you need to generate the 
> triangles on the fly.

Exactly.  This is what all film-quality renderers do.

> What all known computer games do is use a static, very low resolution 
> triangle mesh and then smother it with lashes of low-level trickery to 
> give a vague semblance of curvature.

Of course, because that method gets the highest quality output in realtime. 
And it's very easy to have two (or more) versions of a mesh at different 
resolutions, eg for stills rendering, close-up real-time rendering, far away 
rendering etc.  When you have a mathematical surface rendered directly it's 
very difficult to speed it up!

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