On the number of touching pairs in a set of planar curves

TL;DR

This paper establishes an upper bound on the number of touching points in a set of planar curves where each pair meets exactly once, using structural properties like quasi-grids to analyze curve arrangements.

Contribution

It introduces a bound of $O(t^2n)$ on touchings in such curve families and develops a method based on identifying quasi-grids for structural analysis.

Findings

Bound of $O(t^2n)$ on touchings in curve families.

Introduction of quasi-grids as a structural tool.

Insight into high-touching curve arrangements.

Abstract

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has $O(t^2n)$ touchings, where $t$ is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.