0-Pierced Triangles within a Poisson Overlay

TL;DR

This paper studies triangles formed by Poisson point processes in a plane with respect to their intersection properties with Poisson lines, building on Ambartzumian's 1982 angle density results.

Contribution

It introduces the concept of 0-pierced triangles within a Poisson overlay and explores their properties based on elementary methods and open questions.

Findings

Derived properties of 0-pierced triangles

Connected triangle angles to Poisson processes

Raised open questions for further research

Abstract

Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is 0-pierced if no member of the line process intersects any side of T. Our starting point is Ambartzumian's 1982 joint density for angles of T; our exposition is elementary and raises several unanswered questions.