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Re: Sphere formula

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  • Subject: [mg109460] Re: Sphere formula
  • From: DrMajorBob <btreat1 at>
  • Date: Wed, 28 Apr 2010 01:59:01 -0400 (EDT)

If you meant 3 points in 2 space (you don't), I think the circle's  
equation can be found as follows:

zero = Det@({
     {x^2 + y^2, x, y, 1},
     {x1^2 + y1^2, x1, y1, 1},
     {x2^2 + y2^2, x2, y2, 1},
     {x3^2 + y3^2, x3, y3, 1}

eqns = Thread[
      Flatten@CoefficientList[(x - a)^2 + (y - b)^2 - rsq - c zero, {x,
          y}], 0] == 0];
Solve[eqns, {a, b, c, rsq}] // Together

(output omitted)

The center is {a, b}, and the radius is Sqrt@rsq.

If you mean 3 points in 3 space (I think you do), you'll need some more  
work... but the above might give you ideas.


On Tue, 27 Apr 2010 03:05:16 -0500, S. B. Gray <stevebg at>  

> 1. The center of a sphere through 4 points has a very nice determinant
> form. ( 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

DrMajorBob at

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