MichaelJF <fri### [at] t-onlinede> wrote:

> > Now, in order to massively cut down on the number of actual spheres that you
> > need to use, I was wondering if you could somehow test for tangency with the
> > outer enclosing sphere and only render those spheres. (maybe 1 additional
> > layer...?)
>
> That was the riddle. To generate all spheres but only once. Without
> testing. If they fulfill the Descartes-rule tangency test are
> superflous. This is what the rule-set of Claudio Esperanca does. There
> is an article by David W. Boyd from 1973, who developed a similar (?)
> rule-set.
>
>
https://www.ams.org/journals/mcom/1973-27-122/S0025-5718-1973-0338937-6/S0025-5718-1973-0338937-6.pdf
>
> But I did and do not understand his arguments.

Ah, but that's not what I'm suggesting.
Yes, it would be best to only _calculate_ the parameters for every unique sphere
only once - however; since we can only see the outer layer of spheres in the
render, then we only need to _save_ those in the include file.
So, you could take the radius of the outer enclosing sphere, subtract the radius
of the current sphere, and see if that's equal to the distance of the sphere's
center from the origin.  If it is, save it, if the distance to the center is
less, then it's inside of the outer shell of spheres and therefore not visible.

I think that would also be a much better object to illuminate from the inside.
:)

I would also think that, given symmetry considerations, one could further only
calculate a subset of the whole spherical collection, and simply rotate a lot of
spheres around the origin in order to further minimize calculations.
(granted, this is fussy, and as Boyd says, "It can be shown, with some effort .
.. ." ;) that this is possible)

- BW

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hi,

"Bald Eagle" <cre### [at] netscapenet> wrote:
> MichaelJF <fri### [at] t-onlinede> wrote:
> > ...
> I especially like the version with the missing spheres - you can clearly see the
> inner Pappas chain(s) inside the outer bounding sphere, which is such a cool
> effect!

+1.  I think that "nibbling" deeper and deeper into the arrangement of spheres
would make a real cool animation.


regards, jr.

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Il 16/09/2024 17:40, MichaelJF ha scritto:
> After showing a first version of osculatory sphere packings in 
> BaldEagle's "Pappus Chain" thread, I couldn't let the topic rest.
> 
> https://news.povray.org/povray.advanced-users/message/ 
> %3C66a14899%40news.povray.org%3E/#%3C66a14899%40news.povray.org%3E
> [...]

The third image is fascinating!

Paolo

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