Angle distribution of two random chords in the disc: A sine law

TL;DR

This paper derives a closed-form probability density function for the angle between two random chords in a disc, revealing a sine law when their intersection is inside the disc, with applications in engineering and biology.

Contribution

It provides a novel closed-form expression for the angle distribution of two random chords, highlighting a sine law in the intersection event.

Findings

Probability density function derived in closed form

Sine law established for the intersection event

Applicable to models in engineering and biology

Abstract

Motivated by models in engineering and also biology we determine in closed form the probability density function of the angle shaped by two random chords in a fixed disc. Our main result focus on the event in which the intersection locates inside the fixed disc and establishes a sine law.