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

Let $h$ be a fixed non-zero integer. For every $t\in R_{+}$ 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 toward the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h\left(\mathrm{mod},,p\right)$ in the square ${[-J_p,J_p]}^2$, where $J_p=(p-1)/2$. This set of points can be seen as a generic model for any target set with points randomly distributed on the integer coordinates of a square, in which, apart from a small number of exceptions, exactly one point lies above any abscissa.

We prove the existence of the limiting gap distribution for this set of angles as $p\to \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$. The resulted gap distribution function shows the existence of a sequence of threshold points between which the distribution of seen angles has different shapes. This provides a tool of reference in guiding the observer, which allows one to find and control the position relative to the universe of observed points.