Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains

Source abstract

Let $Ω\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $Ω$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $Θ\in[0,π/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $Ω$, together with a parametrization of each segment in terms of its supporting line and the positions of its endpoints along that line, we derive an integral representation for the conditional distribution $$\Pr\{Θ\leqθ| S_{AB}\cap S_{CD}\neq\varnothing\}.$$ The resulting expression makes explicit how the geometry of the boundary of $Ω$ determines the angular distribution. The probability of intersection appears naturally as the normalizing constant and is related to the probabilistic version of Sylvester's four-point problem.