Celestial coordinates

If the two objects have celestial spherical-polar coordinates (in practice Right Ascension and Declination) $\alpha_{1},\delta_{1}$ and $\alpha_{2},\delta_{2}$ then the criterion is that $D$ should be less than or equal to the great circle distance between the two coordinates:

$$D \leq \arccos ( {\rm abs} ( \sin \delta_{1} \sin \delta_{2}... ...(\alpha_{1} - \alpha_{2} ) \cos \delta_{1} \cos \delta_{2} ) )$$ (8)

Equation  is the natural form for the great circle distance, simply derived by applying spherical trigonometry to the two coordinates. In practice it has the disadvantage that because of numerical errors it is inaccurate when the great circle distance is a small angle. There are algebraically equivalent formulations which retain numerical accuracy for small angles. In catpair the great circle distance is calculated using the appropriate SLA routine¹¹, which uses such a formulation in order to ensure accuracy for small angles.