On computing connected components of line segments

It is shown that given a set of n line segments, their connected components can be computed in time O(n^4/3log³n). A bound of o(n^4/3) for this problem would imply a similar bound for detecting, for a given set of n points and n lines, whether some point lies on some of the lines. This problem, known as Hopcroft´s problem, is believed to have a lower bound of Ω(n^4/3). For the special case when for each segment both endpoints fall inside the same face of the arrangement induced by the set of segments, we give an algorithm that runs in O(nlog³n) time