The Strong Dirac conjecture, open in some form since 1951 [5], is that every set of n points in R includes a member incident to at least n/2 − c lines spanned by the set, for some universal constant c. The less frequently stated dual of this conjecture is that every arrangement of n lines includes a line incident to at least n/2 − c vertices of the arrangement. It is known that an arrangement o...

A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppstein’s notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric sym...

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n). No known input causes our algorithm to use area Ω(n ) for any > 0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puz...

Given a line arrangement A with n lines, we show that there exists a path of length n/3−O(n) in the dual graph of A formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k blue and 2k red lines with no alternating path longer than 14k. Further, we show that any line arrangement with n lines has a coloring ...

The realizability problem for rank 3 oriented matroids (see [1]) is equivalent to the pseudoline stretchability problem (see [4]). This paper uses an example to illustrate a new approach to this problem. The main theorem of this paper, like the trigonometric form of Ceva’s theorem, shows a non-trivial relationship amongst the angles in a specific line arrangement figure (c). We work in polar co...