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Subject: Re: How can I increase this solve syntax ?
Date: Wed, 25 May 2011 00:51:04 +0000 (UTC)
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"Florin Neacsu" wrote in message <irh71j$kqo$1@newscl01ah.mathworks.com>...
> Hi,
> 
> These might be of some interest to OP 
> http://www.diku.dk/hjemmesider/ansatte/rfonseca/implementations/apollonius3d.pdf
> 
> http://home.bway.net/lewis/lewbrid.pdf      (problem 4 /pg 13 deals with his 2nd question)
> 
> Regards,
> Florin
- - - - - - - - -
  Hello Florin.  If Mahdi permits the radius r0 of the moveable sphere to vary suitably, then it could become possible to adjust r0 in such a way that the sphere is externally tangent to each of a set of four given spheres, though not with the method I have presented as it stands.  (It should be noted that not all arrangements of four given spheres would permit such a solution.)

  I believe I see a way in which the method could be modified to solve that problem, though.  It can be shown that the path followed by c0 and c00 as r0 varies would trace out one entire branch of a certain planar 3D hyperbola whose parameters could presumably be expressed explicitly.  Doing the same thing with a different set of three out of the four spheres would produce a second hyperbola.  If the four spheres are so situated that a solution is possible, these hyperbolas must intersect somewhere and do so with the same value of r0, and that should be something that is subject to direct computation using the two hyperbolas' parameters.

  In the report at "http://home.bway.net/lewis/lewbrid.pdf" which you referred to I believe it is problem 3 that pertains to Mahdi's question.  In it they speak of an answer to this problem having 18366 terms.  That suggests to me that they are using a very difficult approach to the problem.  I can't imagine the above two hyperbolas and their intersection involving anything remotely approaching such complexity.  Do you think I am being overconfident?

Roger Stafford