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