Sometimes we need the output polygons themselves, not just their areas. Note that we already determine the vertices, and the direction from each vertex to the next. It is conceptually simple to link the adjacent vertices together and so have a complete map overlay system. Of course, there are practical problems, such as numerical roundoff, to consider, otherwise topological errors, such as the wrong vertices being linked, could cause the output to be meaningless. Producing the overlay polygons themselves would probably double the size and execution time of the program.