On Finding Ordinary or Monochromatic Intersection Points

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 $O(n^2)$.Comment: 21 pages, 4 figure