Patterns of lines intersecting on plane

Abstract

In this work, we are interested in patterns of n lines on the plane such that lines produce M intersecting points. One famous related problem is called Fourier's 17 line problem. Although there are only a few known necessary conditions for realization, some arrangements may not even drawable. Consequently, we try to find more necessary conditions for this problem that can lead to possible arrangements and realize the arrangements. We provide a systematic algorithm to draw the possible arrangements. We establish lemmas for the direct drawing and find some certain cases of arrangements which are drawable. We also provide another approach to this problem by determining the existence of solutions that correspond to the systems of equations with conditions for each arrangement.