On 2020-05-02 9:27 AM (-4), Bald Eagle wrote:
> Hopefully this is to spec.
> 
> The green circles are torus{} objects to show that the cross section is indeed
> exactly circular.
> 
> Not my best work, and it's only a parametric, not an implicit equation, but
> perhaps  I can "implicitize" it.

Converting it to an isosurface with a table gradient will be the hard 
part.  I don't know what to do with parametrics; I might have put in the 
work if they didn't take longer to render than refraction and dispersion 
with media and radiosity.

These are some diagrams that I worked out in 2008.  They led me to a 
quartic equation, which I predicted (but haven't verified) would become 
8th order when converted to 3-D.  I never did solve the equation.  My 
last attempt that I remember was in 2010, and those worksheets are 
currently in storage (along with most of my possessions) while home 
repairs progress(?).

Note that the minor radius does not project to the origin except at the 
elliptical vertices and co-vertices.

Even without direct conversion to 3-D, the solution could be adapted for 
isosurface ellipses with roughly isodirectional gradients, which could 
easily be converted to toroids with RE_fn_Blob2()--although blobbing a 
shape that has already been treated with RE_fn_Blob2() would be rather slow.

Also attached is a test of an equation posted by Nicolas George at the 
time.  It looks remarkably similar to Bill's May 8th fail.  Nicolas 
appears to have taken exactly the approach that I had rejected, namely 
to project the minor radius back to the origin.  In fairness to him, I 
had not posted my diagrams.  His post is at
   https://news.povray.org/48fe0ebe%241%40news.povray.org
in response to my thread.

Post a reply to this message

Attachments:
Download 'elliptor-eq.png' (52 KB) Download 'elliptor-fig.png' (18 KB) Download 'elliptical_torus-iso.jpg' (72 KB)

Preview of image 'elliptor-eq.png'

Preview of image 'elliptor-fig.png'

Preview of image 'elliptical_torus-iso.jpg'