From: Jim Ferry <"jferry"@[delete_this]uiuc.edu>
Subject: Re: Kiss Precise
Date: Wed, 15 Dec 1999 10:10:09 -0600
Newsgroups: sci.math

Jason B wrote:
> 
> Does anyone have any information about this in 3d?
> 
> Are there any catches, or does the
> 
> 2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 formula hold,
> where a,b,c,d are the curvature of the spheres?

This is the extension to n dimensions:

Let s_j be signed reciprocals of the radii of n+2 mutually
tangent (n-1)-dimensional spheres in R^n.  Choose the signs
of the s_j such that externally tangent spheres have the
same sign, and internally tangent spheres have opposite signs.

Then:   n SUM (s_j)^2 = (SUM s_j)^2

Example:  You have a unit sphere and three spheres of equal
size, with centers on the midplane of the unit sphere, all
mutually tangent.  What is the radius of a sphere tangent
to all of these?

Solution:  Let the reciprocal radii be:  -1 for the unit
sphere, s for the 3 midplane spheres, and t for the other
one.  s and t are positive.

We solve for s by slicing in the midplane and doing the
2-d problem: 2 (3s^2 + (-1)^2) = (3s-1)^2.  The positive
solution is s = 1 + 2/sqrt(3).

Now we solve for t:  3(t^2 + 3s^2 + (-1)^2) = (t+3s-1)^2.
This reduces to (t - (1+sqrt(3)))^2 = 0.  (A double root?
Of course!  Why?)  So the radius of the final sphere is
1/t = (sqrt(3) - 1)/2.

|             Jim Ferry              | Center for Simulation  |
+------------------------------------+  of Advanced Rockets   |
| http://www.uiuc.edu/ph/www/jferry/ +------------------------+
|    jferry@[delete_this]uiuc.edu    | University of Illinois |
==============================================================================

From: dmoews@xraysgi.ims.uconn.edu (David Moews)
Subject: Re: Kiss Precise
Date: 15 Dec 1999 13:23:24 -0800
Newsgroups: sci.math

In article <3856B8F4.A070A6D2@uunet.com>, Jason B  <JAB@uunet.com> wrote:
|Does anyone have any information about this in 3d?
|
|Are there any catches, or does the
|
|2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 formula hold, where a,b,c,d
|are the curvature of the spheres?

In general, in n dimensions, you need n+2 pairwise tangent hyperspheres, and 
the coefficient is n instead of 2:

n(a_1^2 + ... + a_{n+2}^2) = (a_1 + ... + a_{n+2})^2,

where the a_is are appropriately signed reciprocal radii.


-- 
David Moews                              dmoews@xraysgi.ims.uconn.edu