Angular distribution towards the points of the neighbor-flips modular curve seen by a fast moving observer

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 $(-t{J}_p^2,0)$ on the real axis towards the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h(\mathrm{m}\mathrm{o}\mathrm{d}p)$ in the square $[-J_p,J_p{]}^2$, where $J_p=(p-1)\mathrm{/}2$. We prove the existence of the limiting gap distribution for this set of angles as $p\to \mathrm{\infty }$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$.