Finding the Circumcenter of a Sphere

Date: 09/24/2003 at 07:01:08
From: Chich-Hsiung Yeh
Subject: How to find the circumcenter (Xc,Yc,Zc) of a sphere

I am trying to write a computer program to find the circumcenter
(Xc,Yc,Zc) of a sphere given 3 points (x1,y1,z1), (x2,y2,z2) and
(x3,y3,z3) on the sphere, along with the radius.  But I don't know how
to do this.  Can you help me? 

Date: 09/24/2003 at 08:17:00
From: Doctor George
Subject: Re: How to find the circumcenter (Xc,Yc,Zc) of a sphere

Hi,

Thanks for writing to Doctor Math.

Here is the system of equations.

    (x1-xc)^2 + (y1-yc)^2 + (z1-zc)^2 = r^2
    (x2-xc)^2 + (y2-yc)^2 + (z2-zc)^2 = r^2
    (x3-xc)^2 + (y3-yc)^2 + (z3-zc)^2 = r^2

Here are the steps.

1. Subtract the second equation from the first.

2. Subtract the third equation from the first.

3. Solve the two new equations for xc in terms of zc and
   yc in terms of zc.

4. Substitute xc and yc into one of the sphere equations
   to get an equation in only zc.

5. Solve for zc.

6. Use zc and the solutions from step 3 to get xc and yc.

For another insight, if we rewrite the equations like this...

    (xc-x1)^2 + (yc-y1)^2 + (z1-z1)^2 = r^2
    (xc-x2)^2 + (yc-y2)^2 + (z2-z2)^2 = r^2
    (xc-x3)^2 + (yc-y3)^2 + (z3-z3)^2 = r^2

we can reframe the problem as solving for the intersection of three 
spheres.

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/