Angle between two random segments

TL;DR

This paper derives the probability density function of the angle between two intersecting random segments inside a unit disk, based on four independent uniformly distributed points, revealing insights into their geometric probability behavior.

Contribution

It provides the first explicit integral expression for the distribution of the angle between two random segments formed by four uniform points in a disk.

Findings

Derived the density function of the angle between segments

Analyzed the distribution behavior of the random angle

Enhanced understanding of geometric probability in random segments

Abstract

The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following setting is considered. Let four independent random points that follow a uniform distribution on the unit disk. Two random segments are built with them, which always are inside of the disk. We compute the density function of the angle between these two random segments when they intersect each other. This type of problem tends to be complex due to the high stochastic dependency that exists between the elements that form them. The expression obtained is in terms of integrals, however it allows us to understand the behavior of the distribution of the random angle between the two random segments.