Outer interval of circle intersection

Is there a consistent way to calculate the outer interval $\left(~\mbox{element of}\ \left[0, 2\pi\right]~\right)$ of a circle created by an intersection ?.

I calculated the intersection points and the angels with ${\rm atan2}$, but beyond that I'm clueless.

The radii of the circles and the positions of the centers are given. The blue circle would be at the center $\left(0,0\right)$.

For example, in this image I need the green interval:

• What exactly is given in the problem? I would expect it would be the radii of the circles and the distance between the centers. Is that it? – bob.sacamento Aug 2 '14 at 16:51
• Note that the problem is a bit simpler if the radii are identical. Is that the case here? – Semiclassical Aug 2 '14 at 16:57
• The radii are not identical. The radii of the circles and the distance between the centers is given though. – Dayrush Aug 2 '14 at 17:09

The measure of the red arc is $$\alpha=2\arccos\left(\frac{d^2+r_2^2-r_1^2}{2r_2d}\right)$$ where $r_1$ is the radius of the blue circle and $d$ is the distance between the centers.
Note: $\arccos$ is usually written acos in programming languages.
• The green arc is $\beta=2\pi-\alpha$ – Paracosmiste Aug 2 '14 at 18:13