A New Construction of Forests with Low Visibility

TL;DR

This paper introduces a novel method for constructing forests in Euclidean space that ensures any sufficiently long line segment is closely intersected by the forest, with the density of points controlled by a parameter.

Contribution

The paper presents a new construction of point sets in $ ^d$ that guarantees low visibility along line segments, improving upon previous methods by controlling the density and intersection properties.

Findings

Ensures line segments of specified length intersect the forest within epsilon distance.

Constructs forests with density depending only on dimension and epsilon.

Provides explicit bounds on the length of line segments for guaranteed intersection.

Abstract

A set of points with finite density is constructed in $\mathbb{R}^d$, with $d\geq2$, by adding points to a Poisson process such that any line segment of length $O\left(\varepsilon^{-(d-1)}\ln\varepsilon^{-1}\right)$ in $\mathbb{R}^d$ will contain one of the points of the set within distance $\varepsilon$ of it. The constant implied by the big-$O$ notation depends on the dimension only.