Intuition for Parametrization of an intersection of two circles

Question

In this question, the parametrization of two circles was given based on their intersection. The idea seems simple and quite correct and intuitive. However, I don't understand the intuition behind the prametrization here i.e for the area UQWD. Specifically, extending the polar coordinates to $\pi- l(r)/2(r) $ and $\pi+ l(r)/2(r) $? It seems basic but I have spent a while trying to figure out its intuition. NB: I have repeated the diagram here to make it easily accessible

Answer 1

For a fixed value of $r$, the angle $\theta$ sweeps from point $A$ to point $B$. The length of the arc from $A$ to $B$ is $l(r)$. But arclength $s$ is just $r\theta$. So the angle swept out from $A$ to $B$ is $l(r)/r$. Half of this angle, $l(r)/(2r)$, is swept out from $A$ to the horizontal; the other half is swept from the horizontal to $B$.

To express the region in polar coordinates centered at the user circle, we measure $\theta$ relative to the horizontal. Point $A$ then corresponds to $\theta=\pi-l(r)/(2r)$, while point $B$ corresponds to $\theta=\pi+l(r)/(2r)$.