{
  "abstract": "Let $Ω\\subset\\mathbb{R}^2$ 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 $Ω$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $Θ\\in[0,π/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $Ω$, 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 $Ω$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36517",
  "author_total": 1,
  "authors": [
    {
      "name": "Paulo Manrique"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6qm251c777tcp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36517",
  "fulltext_url": null,
  "id": "6qm251c777tcp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36517",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214904926",
  "pdf_url": "https://arxiv.org/pdf/2609.36517",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6qm251c777tcp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36517",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36517"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains",
  "title_text": "Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/6qm251c777tcp",
  "venue": "arXiv",
  "versions": []
}