The spatial data in GIS's from ARC/INFO [50] down to the Digital Chart of the World [51] is organized into layers, each of which is a subdivision of the plane with explicitly stored neighbor relations. Custom maps are produced by selecting layers of interest and overlaying them to produce new maps. This ``overlay'' process is the primary tool for map analysis.
As a theoretical geometric problem, computing the regions defined by a set of segments or pixels is perhaps not so difficult. But as an engineering problem, tuning algorithms is a challenge. When one considers the errors and different levels of generalization in the input, the task becomes more interesting. If the same feature, say, a stream bed, appears in more than one map, an overlay will likely have many spurious polygons in the vicinity of the stream bed. It would then be desirable to remove one copy from the overlay. Even better would be to average the two maps in some principled manner to improve overall accuracy.
A special form of the general polygon matching is to match two maps and try to determine what has changed from one to the other. This is important for observing historical change, checking map completeness, rectifying common boundaries, or using a higher resolution map to improve a lower resolution map. Again, generalization and error make this an intriguing problem.