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

Abstract

Let $\Omega \subset {\mathbb{R}}^2$ 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 $\Omega$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $\Theta \in [0,\pi /2]$ formed by them, conditional on the event that they intersect. Using the radial function of $\Omega$, 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 $\Omega$ 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.