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Intersections of random chords of a circle

Cynthia Bortolotto, Victor Souza

TL;DR

This work studies the distribution of the intersection point distance $\ell$ from the origin for two random lines $L_1,L_2$ generated by four independent uniform points on the unit circle. It provides two proofs (geometric and analytic) of the main result: for $0\le r$, $\mathbb{P}(\ell\le r)=\frac{16}{\pi^3}\int_0^{\min\{1,r\}} \frac{\arccos(t/r)\,\arcsin(t)}{\sqrt{1-t^2}}\mathrm{d}t$, and in particular, for $0\le r\le1$, $\mathbb{P}(\ell\le r)=\frac{2}{\pi^2}\mathrm{Li}_2(r^2)$. The paper then situates this continuous-line model within the discrete problem of diagonal intersections in regular polygons (Karamata), establishes a general integral-transform framework $\mathcal{I}$ and $\mathcal{J}$ that converts a line-distribution measure $\mu$ into the corresponding intersection distribution, and discusses the Bertrand paradox via several sampling schemes (I–IV) with explicit formulas and infinite mean of $\ell$ under most rotationally invariant measures. It also proves an analytic identity linking the dilogarithm to the geometric integral and outlines rich directions, including higher dimensions and connections to hyperbolic geometry and polylogarithms. Overall, the paper deepens the bridge between planar geometry, random line models, and special functions, with potential applications to extremal questions and geometric probability paradoxes.

Abstract

Where are the intersection points of diagonals of a regular $n$-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

Intersections of random chords of a circle

TL;DR

This work studies the distribution of the intersection point distance from the origin for two random lines generated by four independent uniform points on the unit circle. It provides two proofs (geometric and analytic) of the main result: for , , and in particular, for , . The paper then situates this continuous-line model within the discrete problem of diagonal intersections in regular polygons (Karamata), establishes a general integral-transform framework and that converts a line-distribution measure into the corresponding intersection distribution, and discusses the Bertrand paradox via several sampling schemes (I–IV) with explicit formulas and infinite mean of under most rotationally invariant measures. It also proves an analytic identity linking the dilogarithm to the geometric integral and outlines rich directions, including higher dimensions and connections to hyperbolic geometry and polylogarithms. Overall, the paper deepens the bridge between planar geometry, random line models, and special functions, with potential applications to extremal questions and geometric probability paradoxes.

Abstract

Where are the intersection points of diagonals of a regular -gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

Paper Structure

This paper contains 17 sections, 4 theorems, 64 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Intersection points of diagonals
  3. Extending Karamata's result
  4. A conceptual overview
  5. Special values
  6. The Bertrand paradox
  7. Proofs of the main theorem
  8. Geometric proof
  9. Analytic proof
  10. Bertrand's paradox and beyond
  11. An analytic identity
  12. Further directions
  13. Hyperbolic volumes
  14. Special values
  15. Polylogarithms
  16. ...and 2 more sections

Key Result

Theorem 1

Let $X_1$, $X_1'$, $X_2$ and $X_2'$ be points in drawn independently and uniformly from the unit circle centered at the origin. Let $P$ be the intersection point of the line through $X_1$ and $X_1'$ with the line through $X_2$ and $X_2'$. If we write $\ell$ for the distance of $P$ to the origin, the for any $r > 0$. In particular, for $0 \leq r \leq 1$, it holds that

Figures (10)

  • Figure 1: A regular 15-gon with all diagonals and their intersection points.
  • Figure 2: Multiple of dilogarithm function and it's derivative.
  • Figure 3: Intersection points of diagonals of a regular 12-gon according to their multiplicity.
  • Figure 4: A regular 11-gon with intersection points of diagonal lines, some outside the circumcircle.
  • Figure 5: Two possible extensions of the dilogarithm function to $\mathbb{R}_{\geq 0}$.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • proof : Proof of \ref{['thm:analytic']}
  • proof : Proof of \ref{['cor:integrability']}
  • Corollary 4
  • proof