The Dual is the Context: Spatial Structures for GIS

Abstract

GIS (Geographic Information Systems) are concerned with the manipulation and analysis of spatial data at a "Geographic" scale. Apart from issues of storage, database query and visualization, they must deal with several significantly different types of spatial information. These may be roughly classified as: discrete objects; networks; polygonal maps; and surfaces. Each of these has a specific set of assumptions associated with it, a specific data structure, and a specific set of algorithms. This produces a high level of complexity in the construction, manipulation, analysis and comparison of these datasets. We demonstrate that when "here " is replaced by "closest to here " the resulting proximal query (a Voronoi diagram) may be used to manipulate the four categories described above, with a resulting simplification of the system. All discrete objects become "fields ", with a value at any location. These fields provide the 'context' of the object, and many applications require both the primal and dual for analysis: the Quad-Edge is therefore the data structure of choice. Thus in 2D the 'context' the dual set of adjacent neighbours and neighbourhoods, as identified by the Voronoi diagram greatly assists in the interpretation and analysis of the data. In 3D the same thing holds true: the dual graph of the 3D object set in 3D GIS perhaps a model of the rooms of a building interior gives a clear structure for expressing their adjacency relations. A direct modification of the Quad-Edge - splitting the four parts into two pairs, each containing one primal and one dual half-edge -provides a structure appropriate for CAD-type nonmanifold modelling, which we demonstrate by the construction of a large building complex, using Euler-type operators. Thus, again, the dual graph provides the context for the (primal) geometric data.