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On intersection probabilities of four lines inside a planar convex domain

Part of: General convexity Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  07 December 2022

Davit Martirosyan  and
Victor Ohanyan
Davit Martirosyan*
Affiliation:
Yerevan State University
Victor Ohanyan*
Affiliation:
Yerevan State University
*
*Postal address: Faculty of Mathematics and Mechanics, 1 Alek Manukyan St, Yerevan 0025, Armenia.
*Postal address: Faculty of Mathematics and Mechanics, 1 Alek Manukyan St, Yerevan 0025, Armenia.
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Abstract

Let $n\geq 2$ random lines intersect a planar convex domain D. Consider the probabilities $p_{nk}$, $k=0,1, \ldots, {n(n-1)}/{2}$ that the lines produce exactly k intersection points inside D. The objective is finding $p_{nk}$ through geometric invariants of D. Using Ambartzumian’s combinatorial algorithm, the known results are instantly reestablished for $n=2, 3$. When $n=4$, these probabilities are expressed by new invariants of D. When D is a disc of radius r, the simplest forms of all invariants are found. The exact values of $p_{3k}$ and $p_{4k}$ are established.

Keywords

Stochastic geometryconvex domain invariantscombinatorial algorithmcounting intersection pointsrandom chord length moments

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 53C65: Integral geometry
Secondary: 52A22: Random convex sets and integral geometry 52A10: Convex sets in $2$ dimensions (including convex curves)

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 504 - 527
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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