I've made some progress on the spreadsheet, and have some questions and comments
on the topic of polynomials and parametrics and isosurfaces.

In this thread:
http://news.povray.org/povray.binaries.images/thread/%3Cweb.591ae564741e6af9c437ac910%40news.povray.org%3E/

I was working on the Dupin cuclide shape and stated:

> ... Dupin cyclide in both isosurface (implicit) and
> parametric form,
.....
> It is, however, SLLLLllllllllllooooooooooowwwwwwww.

Le_Forgeron <jgr### [at] freefr> wrote:
> It is not slow with a polynomial.

But now check this out:
http://www.povray.org/documentation/view/3.7.1/74/

"Note:Since version 3.5, POV-Ray includes the new isosurface object which makes
the polynomial object more or less obsolete. The isosurface is more versatile
(you can specify any mathematical function, not just polynomials), easier to
use. You can write the function as is, without needing to put values in a
gigantic vector.
Isosurfaces often render considerably faster than equivalent polys."

Whaaaaat?


This was interesting too, as I never knew this was the case:

"However, the most mathematically oriented still like polys because isosurfaces
are calculated just by approximating the right value, while the poly is
calculated in a mathematically exact way. Usually isosurfaces are more than good
enough for most applications, though."


I spent a good deal of time looking into how to find out how many terms there
would need to be for a polynomial of order greater than 7, since the polynomial
that kicked off this line of inquiry is 8th order, and the table in the docs
only goes up to 7.

The docs seem to be fragmented, and scattered over many different incarnations,
with bits and pieces of data here, there and everywhere.

One of those bits was:
http://www.povray.org/documentation/view/3.6.1/298/
There are n such terms where n = ((Order+1)*(Order+2)*(Order+3))/6.

I'd love to know where this equation comes from.

Also, considering that quintics are apparently of ridiculous complexity to
solve, and higher order polynomials rapidly become more insane,  I'd like to get
a little insight as to how POV-Ray deals with 15th order polys and, if the
version of the docs I'm reading is right - _35th_ order.

http://www.povray.org/documentation/3.7.0/t2_3.html

"Note: A maximum of 35th degree polynomial can be represented with the poly
object. If a higher degree polynomial or other non-polynomial function has to be
represented, then it is necessary to use the isosurface object."

(When did the limit go from 15 to 35?   WHO recoded that in the source??)

Also, I would certainly be interested in seeing the equation and render for the
highest order poly object anyone has rendered....  :D

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