Counting pairs in the circular analysis region. Left: two-dimensional representation of the function ϕ_s(θ) for various values of the separation s and the position θ in a circular field of view of radius R. Right: scaling of N_p(s), the number of pairs of points separated by a distance s within a circular field of view of radius R. “Inner” concerns those pairs integrally contained within the circular domain, while “Ext” concerns those pairs with only one end within the domain. “All” is the sum of the two numbers. The curves actually show δ(sN_p(s))/δs, which is the differential number of pairs per interval of s expressed in units of the radius R. Here ℓ ≪ R is the side length of an elementary pixel, so that the total number of pixels in the circle is πR²/ℓ².