My spherical geometry is a bit rusty but looking at the figure below:
… my intuition tells me that angles $\phi$ and $\theta$ (measured in radians) are connected with the following equation:
$\phi = \theta*cos(\delta)$
… where $\delta$ is the angle corresponding to the green arc.
At least the above holds true for the special cases $\delta=0$ and $\delta=\pi/2$. Does the above equation hold true in general and what is the terminology that describes the various angles and circles I've drawn?
update on terminology
It is now clear that the angles can be more properly described as follows:
- angle $\theta$ is the arc between points $A'$ and $B'$ on a latitude circle at latitude $\delta$ (see the drawing on this answer which makes this point clear)
- angle $\phi$ is the arc between points $A'$ and $B'$ on a great circle
Best Answer
The distance between the endpoints is scaled down like the circle radius by a factor of $\cos\delta$. However, that rather makes $$ \sin \frac\phi2=\cos\delta\sin \frac\theta2.$$