This seems like it should be fairly simple, but it has me completely stumped. Imagine I have a latitude line at angle $\theta_1$ on the surface of the unit sphere in 3D. This is a "small circle", meaning it's a circle that's not a great circle.
Now suppose I have, in addition, a second coordinate system where the axis is at an angle $\phi$ to the original axis, and a second latitude line at an angle $\theta_2$ in the new coordinate system:
If these two curves cross, they do so at two points. I want to know the angle they make on the surface of the sphere. That is, the angle between the vectors tangent to the two circles at a point where they cross:
What is this angle $\psi$, as a function of $\phi$, $\theta_1$ and $\theta_2$?