Several months ago I asked the question referenced here .... Eccentered Circles - determine space between circle at a given location
For this same question I now need to figure out a more generic approach where the angle is not $60$ degrees but now could be anything from $0$ to $359$ degrees with $0$ always being the point at which the $2$ circles intersect (contact one another). How could I figure out what $x$ is in this situation?
David
the simpler way is to parametrize the segment of leght $d=\overline {PQ}$ (see the figure) by means of the central angle $\theta=\angle POM$.
From the figure we have:
$\overline{OP}=R \quad \overline{CQ}=r \quad \overline{OC}=R-r$
and using he cosine rule we have: $$ \overline{PC}=d+r=\sqrt{R^2+(R-r)^2-2R(R-r)\cos \theta} $$ from which we can find $d$ as a function of $\theta$.
If you want to use the angle $\phi=\angle PCM$ as independet variable, than you can find $d$ from (cosine rule for $\overline {OP}$): $$ R^2=(R-r)^2+(r+d)^2+2(R-r)(r+d)\cos \phi $$
But it is a bit more intricate.