Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains
Abstract
Let Ω⊂ℝ2 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 SAB and SCD and study the distribution of the smaller angle Θ∈[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≤θ| SAB∩ SCD≠∅}. 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.
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BibTeX
@misc{manrique2026angle,
title = {{Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains}},
author = {Paulo Manrique},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36517},
eprint = {2609.36517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36517},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains
AU - Paulo Manrique
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36517
UR - https://arxiv.org/abs/2609.36517
AB - Let Ω⊂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∈[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≤θ| S_AB∩ S_CD≠∅}. 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.
ER -