Surface interpolation, spatial adjacency and GIS

Spatial information may be viewed in two ways: as coordinates measured with a ruler along imaginary axes; or as the adjacency relationships between objects. The former has been the most common historically, but the latter appears more relevant to the handling of spatially distributed objects in a computer. The interpolation problem, as implemented using weighted-average techniques, depends on the selection of appropriate neighbours to the sample location being estimated. Traditionally this was done using “metric” methods, with sometimes unsatisfactory results. An alternative “adjacency relationship” approach can be used, based on the construction of the Voronoi regions about each point, that removes many of the inconsistencies of the earlier methods and, in addition, provides an integrated weighting method for each adjacent point. Interpolants can then be modeled to provide appropriate levels of continuity. The technique may be extended to data objects besides points , in part icular to l ine segments, permit t ing consistent integrat ion of elevat ion information at break-l ines and contour segments as well as at point locations. While the concepts discussed here are applied to two spatial (independent) dimensions plus “elevation”, the Voronoi method may readily be applied to higher dimensions.