how to solve a set of sphere equations?

Hi everyone,

I need to find out the point where 3 spheres intersect but I am having a hard time using MATLAB to solve this. Here are the equations:

(X0-X1)^2 + (Y0-Y1)^2 + (Z0-Z1)^2 = a^2

(X0-X2)^2 + (Y0-Y2)^2 + (Z0-Z2)^2 = a^2

(X0-X3)^2 + (Y0-Y3)^2 + (Z0-Z3)^2 = a^2

Where X0, Y0 and Z0 are the variables (the intersecting point coordinates) and "X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 a" are constants which I'd like to leave in this symbolic form so I can decide their value later.

Replacing the constants for actual values would make the solution very easy to find, I know, but it would ruin the academic value of the solution. I'm looking for a way to solve this that would return a new set of equations of the type X0 = ... Y0 = ... Z0 = ...

Answer 1

[Daniel]

A quick google search indicate that this problem given a set of sphere equations is very common, but no one ever posts a solution to the problem. I couldn't find a single reference on a general solution to this problem, come on guys, take this challenge and help me finish my robot here :)