Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning counterclockwise). Now, I introduce a new origin $O'$ with cartesic coordinates in the original frame $(-d,0)$, both $d<R$ as $d>R$ are considered. Now, I define from $O'$, $\phi$ as the angle between the positive x-axis and a point on the circle.
if the probability density as a function of $\theta$ is equal to a constant $C$ everywhere(and hence normalized to $\pi C$), we should be able to write $$C\, d\theta=f(\phi)d\phi$$.
Now my question is: What is the expression for $f$?
I initially thought this would be an easy problem, but it turns out less trivial than we were expecting. We encountered this problem by studying galilei transformations of circularly symmetric collisions.
I don't know if this helps, but you can connect $\theta$ and $\phi$ with the following relationship obtained using trigonometry: $$\tan\phi=\frac {R\sin\theta}{d+R\cos\theta}$$