Intersection of Two Spheres

For quite sometime I have been intrigued by the curve created by the intersection of two spheres with the following properties/conditions:

Let both spheres have the same radius. Let's say a radius of 2.

Let one sphere be centered at (0,0,0) and the other sphere be centered at

(2,2,2).

I think the equations for the two spheres are:

x^2+y^2+z^2=4

and

(x-2)^2+(y-2)^2+(z-2)^2=4

I have seen a method finding the intersection when both spheres have the same coordinates on the y and z axis and differing coordinates on the x axis. This involves combining the two equations and solving for x.

This would yield a circle parallel to the yz plane.

So, here is what I did:

(x-2)^2+(y-2)^2+(z-2)^2=x^2+y^2+z^2

Multiplying through and rearranging I got

x+y+z=3.

I think this is the equation of the plane where the two spheres intersect.

I am not sure what to do next.

I have read that it will be a circle and that there will be two equations describing the curve.(Now that i think about it i guess the intersection of 2 spheres will always be a circle)

I had built a model about 10 years ago out of balsa wood and the curve really did not look circular. But maybe that was due to the inaccuracies of the model.

How do I find an equation that describes this curve.

I looked in my old calculus book but could find nothing. :(

Any help would be greatly appreciated.