On Finding Ordinary or Monochromatic Intersection Points

TL;DR

This paper presents efficient algorithms for finding ordinary or monochromatic intersection points in arrangements of lines, hyperplanes, and pseudolines, with optimal or near-optimal time complexities.

Contribution

It introduces algorithms that efficiently find ordinary or monochromatic intersections in arrangements of lines, hyperplanes, and pseudolines, extending previous methods to higher dimensions and different geometric objects.

Findings

Algorithm finds an ordinary intersection in $O(n \, \log n)$ time for lines.

Extension of the algorithm to hyperplanes in $\mathbb{R}^d$ also runs in $O(n \log n)$ time.

Additional algorithms find intersections in pseudoline arrangements in $O(n^2)$ time.

Abstract

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of $n$ lines in $\mathbb{R}^2$, not all parallel and not all passing through a common point, in time $O(n \log{n})$. The algorithm is then extended to find an ordinary intersection among an arrangement of hyperplanes in $\mathbb{R}^d$, no $d$ passing through a line and not all passing through the same point, again, in time $O(n \log{n})$. Two additional algorithms are provided that find an ordinary or monochromatic intersection, respectively, in an arrangement of pseudolines in time $O(n^2)$.