find equation for plane touching 3 spheres
I am stuck with a the following problem: I am searching the equation for a plane that touches 3 spheres.
Assuming that these spheres are defined by the midpoints P,Q,R (I'll use capital letters for vectors now), and their radii p,q,r. The plane that touches the spheres will have a normalvector N that connects P,Q,R with three different point on the plane U,V,W, where the distance |P-U| = p; |Q-V| = q;|R-W| = r. The normalvector N can be defined through the three points on the plane as the crossproduct: (V-W)x(U-W)
So I come up with the equations:
U = P + p N/|N|
V = Q + q N/|N|
W = R + r N/|N|
N = (V-W)x(U-W)
My known variables are P,Q,R and p,q,r. Now I am searching for U,V,W in order to describe the plane.
But to get U,V,W seems impossible to me, I have no idea how to solve those equations for U,V,W - any suggestions or ideas would be more than welcome!