Angle between two random segments

Abstract

The study of \textit{random segments} is a classical problem in geometrical probability whose answer depends on the mechanism used to generate the segments. We consider four independent random points uniformly distributed in the unit disk and form the two labeled segments $S_{AB}$ and $S_{CD}$. The random variable of interest is the usual angle in $[0,\pi ]$ between the vectors $B-A$ and $D-C$, conditional on the event that the two segments intersect. By introducing normal and tangential coordinates for each supporting line, we obtain an integral expression for the conditional density. The signed tangential coordinate of the intersection point is retained throughout the argument, which prevents the reflected-root overcounting that arises when only its squared norm is used. The same change of variables also recovers known distributions for the distance of a random chord from the center and for the length of a random segment. The resulting intersection probability is $\frac{1}{3}\left(1-\frac{35}{12{\pi }^2}\right)$.