Distribution of lattice points visible from the origin

Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C¹ boundary. For each integer Q ≥ 1, we consider the integer lattice points from Ω_Q = {(Qx, Qy); (x, y) ∈ Ω} which are visible from the origin and prove that the 1^st consecutive spacing distribution of the angles formed with the origin exists. This is a probability measure supported on an interval [m_Ω, ∞), with m_Ω > 0. Its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel.