MonotoneChains are a way of partitioning the segments of an edge to allow for fast searching of intersections. They have the following properties:
the segments within a monotone chain will never intersect each other
the envelope of any contiguous subset of the segments in a monotone chain is simply the envelope of the endpoints of the subset.
Property 1 means that there is no need to test pairs of segments from within the same monotone chain for intersection. Property 2 allows binary search to be used to find the intersection points of two monotone chains. For many types of real-world data, these properties eliminate a large number of segment comparisons, producing substantial speed gains.