  We consider an affine Euclidean lattice and record the directions of all
lattice vectors of length at most $T$. Str\"ombergsson and the second author
proved in [Annals of Math.~173 (2010), 1949--2033] that the distribution of
gaps between the lattice directions has a limit as $T$ tends to infinity. For a
typical affine lattice, the limiting gap distribution is universal and has a
heavy tail; it differs distinctly from the gap distribution observed in a
Poisson process, which is exponential. The present study shows that the
limiting two-point correlation function of the projected lattice points exists
and is Poissonian. This answers a recent question by Boca, Popa and Zaharescu
[arXiv:1302.5067]. The existence of the limit is subject to a certain
Diophantine condition. We also establish the convergence of more general mixed
moments.
