Voronoi Diagrams and Spatial Adjacency

One of the basic concerns of Geographic Information Systems and their application is the spatial relationships between objects on the map. This has been attacked from two directions: by the construction/linking of polygon boundary segments; and by including spatial operators within data-base retrieval requests. In both cases only an incomplete picture has been obtained due to the lack of any consistent definition of spatial adjacency. Previous work on interpolation problems showed that a basic definition of spatial adjacency was required even for point objects. This could be supplied by examining the contacts between the Voronoi polygons formed by "blowing bubbles" around each data point until the bubbles met. The resulting space-covering tiling gives a consistent adjacency description of objects on a map, as adjacent objects have Voronoi polygons that touch. This concept may be extended from point objects to line segments, and the appropriate Voronoi polygons formed. In this case the polygons around each object may have parabolic as well as linear boundary segments. The resulting network of adjacency relations forms a triangulation linking all map objects to their neighbours irrespective of actual contact, providing a general definition of adjacency. A variety of graph-traversal operations may be performed on this adjacency network, permitting solutions of various spatial mapping problems. Examples include interpolation, polygon skeletonization and rastervector integration. It is hoped that the Voronoi concept may provide a common framework for several classes of spatial problems.