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∈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≡ h 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→∞, providing explicit formulas for the corresponding density function, which turns out to be independent of h.