How to find the center of a sphere given the radius and 3 points on the surface?

Question

I need to find the center coordinates of a sphere. I have 3 points on the surface and the radius.

Shall I use the equation of sphere: $(x_1-x)^2+(y_1-y)^2+(z_1-z)^2=r^2$ , plug in 3 points and solve the 3 equations? How can I solve such non linear equations?

Or Shall I follow the geometric approach as described here: sphere center ?

Or something else?

Thanks

Answer 1

If you subtract one of the equations that you’ve derived from the other two, you’ll get a system of two linear equations. The solution to that system is a line along which the center of the sphere must lie. You can then find the one or two points on this line that are at a distance $r$ from any of the three given surface points, which involves solving a simple quadratic equation.

However, this is pretty much what the linked solution, in particular the second one that wasn’t accepted, does. The two linear equations you get above represent two of the plane bisectors of pairs of the three given points. The line p(t) in that solution is their intersection, and he even serves the solution to the final quadratic equation up on a plate.