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

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(t2n) 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.