I tried to solve this analytically but gave up and ran a simulation. I would still like to know if it can be done explicitly.
Consider a circle radially divided into 'zones', so say zone 1 is the annulus that goes from R1 to R2, and zone 2 is the annulus that goes from R3 to R4. How can I calculate the average distance between two points from zone 1 to zone 2?
I ended up with an equation that looked like:$$d(z1, z2) = \frac{\int_{R1}^{R2}\int_{R3}^{R4}\int_{0}^{2\pi} \sqrt{r + r' -2\sqrt{r r'}cos(\theta)} dr dr' d\theta}{\int_{R1}^{R2}\int_{R3}^{R4}\int_{0}^{2\pi} dr dr' d\theta}$$
The normal way of solving this is to transform to different polar coordinates, but I don't see how that can work here as the annuli limits become sooooo complicated.
A huge thanks if anyone knows how to do this integral.
The reason that it is $r$ and not $r^2$ is because you have to transform $r$ to $\sqrt r$ to get an even distribution over the annuli. Otherwise the smaller the annulus the higher the density of points becomes.
I thought it would resolve to an elliptical function problem, but even in the infinitely thin case, the answer still requires the $\theta$ integral and the equation reduces to $\sqrt{ 1 + k^2 sin \theta}$ rather than the usual $\sqrt {1 - k^2 sin \theta}$, soil I wasn't sure that it was an elliptical function anyway.
Thanks for your thoughts.