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Re: Sphere formula

• To: mathgroup at smc.vnet.net
• Subject: [mg109488] Re: Sphere formula
• From: "S. B. Gray" <stevebg at ROADRUNNER.COM>
• Date: Thu, 29 Apr 2010 02:54:22 -0400 (EDT)
• References: <hr65rj$jq6$1@smc.vnet.net>
• Reply-to: stevebg at ROADRUNNER.COM

On 4/27/2010 1:05 AM, S. B. Gray wrote:
> 1. The center of a sphere through 4 points has a very nice determinant
> form. (http://mathworld.wolfram.com/Sphere.html) What I want is a nice
> formula for the center of a sphere through 3 points, where the center is
> in the plane of the three points. I have a formula but it's a horrible
> mess of hundreds of lines, even after FullSimplify.
>
> 2. (Unlikely) Is there a way to get Mathematica to put a long formula into a
> matrix/determinant form if there is a nice one?
>
> Any tips will be appreciated.
>
> Steve Gray
>
Thanks to everyone who answered my question, but there is a simpler 
answer. I forgot the simple fact that any linear combination of two 
vectors lies in the plane of the two vectors.

Let the three points be p1,p2,p3. Consider the linear function
p=b(p2-p1)+c(p3-p1) where b,c are to be determined and p is the desired 
center. Now do

Solve[{Norm(p-p1)==Norm(p-p2),Norm(p-p1)==Norm(p-p3)},{b,c}].

This gives b,c and therefore p, which will be equidistant from p1,p2, 
and p3 and lie in their plane. Very simple. (I used (p-p1).(p-p1) etc. 
instead of Norm.)

Steve Gray