The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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


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 

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

- Doctor George, The Math Forum 
Associated Topics:
College Conic Sections/Circles
College Higher-Dimensional Geometry
High School Conic Sections/Circles
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.