  Let $h$ be a fixed non-zero integer. For every $t\in \mathbb{R}_+$ and every
prime $p$, consider the angles between rays from an observer located at the
point $(-tJ_p^2,0)$ on the real axis towards the set of all integral solutions
$(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h \pmod{p}$ in the square
$[-J_p,J_p]^2$, where $J_p=(p-1)/2$. We prove the existence of the limiting gap
distribution for this set of angles as $p\rightarrow \infty$, providing
explicit formulas for the corresponding density function, which turns out to be
independent of $h$.
