How to find the distance between two points on a sphere in polar coordinates with fixed $\phi$

Question

I'm trying to calculate the distance between to points on a sphere with fixed $\phi$. For clarity, the sphere has radius $R$ and is centered at the origin. So, if we let $\phi = \frac{\pi}{2}$, then clearly the distance between those two points is just $2R$, and if we let $\phi = 0$, then the distance is just $0$. Apparently, the distance is given by $2R\sin\phi$ in polar coordinates. This makes sense to me intuitively, as in the examples above, $\sin\phi$ would be $1$ and $0$ respectively, but I can't seem to make sense of the $\phi$ values in between. I'm guessing some sort of trig is required. Any help would be greatly appreciated. Thanks in advance!

Answer 1

It seems the questioner wants to ask this question:

If a plane perpendicular to the $z$ axis cuts a sphere of radius $R$ so as to subtend elevation angle $\phi$, what is the diameter of the circle of intersection?

This diameter is obviously $2 R \sin \phi$ (blue).