An approach is provided for converting a polygon described as an ordered list of perimeter points into a set of connected quadrilaterals suitable for use in an advanced integrated circuit router. Edges are constructed between the points of the polygon. Then, one or more edges are selected as starting locations. Pairs of edges or portions of pairs of edges are matched to form sequences of quadrilaterals. Methods are provided to determine when edges should be split or skipped to ensure that all quadrilaterals are convex or meet other criteria. Other methods are provided to determine when the matching process should be terminated and restarted at another location. Finally, the sequences of quadrilaterals are joined together to form a data structure suitable for use within an integrated circuit router.