On intersection probabilities of four lines inside a planar convex domain

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 al...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of applied probability Jg. 60; H. 2; S. 504 - 527
Hauptverfasser: Martirosyan, Davit, Ohanyan, Victor
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge, UK Cambridge University Press 01.06.2023
Schlagworte:
Algorithms
Combinatorial analysis
Domains
Integrals
Invariants
Original Article
Stochastic geometry
60D05
52A10
combinatorial algorithm
counting intersection points
52A22
53C65
random chord length moments
convex domain invariants
ISSN:0021-9002, 1475-6072
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2022.60