On 2020-05-09 8:48 AM (-4), William F Pokorny wrote:
> On 5/8/20 9:29 PM, Cousin Ricky wrote:

>> shape appears to be exact, but the max_gradient is a nightmare, and I 

>> > https://news.povray.org/4409cd2a%40news.povray.org
> 
> Ah... Thanks. I would probably not have jumped in knowing all that 
> ahead. I was only aware of Bruno's work and his only because I was 
> looking for sphere_sweep and blob test cases for my solver work.
> 
> I've captured all the information in my f_torus working directory.
> 
> My gut when I first read your list of requirements was that there was 
> probably no exact solution. That the requirements force some 
> discontinuity / fuzz. Is this thinking proven wrong somewhere?

Based on my work in 2008, I'm pretty sure there is an exact solution. 
The key seemed to be in solving a quartic equation, although the 
equation itself would not be the whole solution.

> Sebastian's images mention compares to a mesh. Of what mesh was he 
> speaking? Something created from an exact solution? Something in your 
> package somewhere?

I created the mesh using RoundEdge::RE_Elliptorus_mesh(), then 
co-located it with Sebastian's isosurface.  The mesh2 is built up in a 
parametric fashion using sines and cosines.

> Did you see my recent post to povray.general on the parametric 
> performance being better than we've thought all these years? Default 
> gradient of 1.0 should be more like 0.001.

No, I haven't see it yet.

> Supposing there is an exact solution for the elliptical tori you want. 
> Any 8th order equation will be somewhat sloppy with respect to solutions 
> (165 terms were mentioned in the one post. If not sparse...).

I'm aware of that.  I even wrote a JavaScript that shows me the POV-Ray 
order of coefficients for the poly object, although it has since been 
rendered redundant by 3.7's new polynomial syntax.

Every poly I've had occasion to use has been sparse.  While I don't have 
an intuitive feel for these things, I don't see why an elliptical toroid 
would break this pattern.

> My personal interests lean toward something with good gradients in a 
> range which will play well with all the other stuff I'm doing. Solutions 
> a little sloppy, but fast, and good enough are OK with me. :-)

I guess I could take "good enough" if it can sneak by our eyes' 
simultaneous contrast.  But simultaneous contrast is surprisingly 
sensitive.  The attached renders of the lever cap cam of a hand plane 
use deformations of an isosurface oblong toroid that was "good enough," 
but it is easy to see the mismatch between the toroid and the scaled 
cylinder.  This was a few months before my initial upload of RoundEdge; 
I was hopeful of finding a solution, but decided to post the module 
without the toroid rather than include a poor solution.

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