Angular Distance Between Two Points on a Sphere

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Angular Distance Between Points on a Sphere

To find the angular distance between two points on a sphere, suppose that the two points have a right ascension (RA) of $\phi_{1}$ and $\phi_{2}$, and a declination (DEC) of $\theta_{1}$ and $\theta_{2}$. Alternatively, the angles $\phi$ and $\theta$ could correspond to geographical longitude and latitude respectively. The angle at the center of the sphere separating the two points is:
\[
\Psi = \arccos{(\sin{\theta_{1}} \sin{\theta_{2}} + \cos{\theta_{1}} \cos{\theta_{2}} \cos{(\phi_{1} -\phi_{2})})}.
\]
The arc length on the spherical surface is equal to the radius of the sphere times $\Psi$ (in radians). When the two points are extremely close together (or otherwise when $|\cos{\Psi}|$ is close to unity), different formulas must be used to ensure sufficient precision (the haversine formula, or the Vincenty formula).