An Algebraic Model for Spatial Objects with Indeterminate Boundaries

Conventional modelling in geographic information systems (G IS) made us fam iliar with geographic objects whose spatial extents are clear and universally recognized. When we think of geographic objects with indeterminate boundaries, we immediately face the difficulty of drawing them and representing them in a computer format. The difficuily comes from a well established approach in GIS modelling which tries to simplify each geographic entity and represent its shape by using simple objects of Eucl idean geometry, like points, lines and regions. The boundaries of such objects are fixed by geometric definitions. This simpl ification process can make it difficult to represent the real nature of geographic entities when the application requires more detail. Burrough (1992) thinks that it is necessary to use basic geographic enti ties that take into account the complexity and uncertainty in posi. tion, topology and attributes. Objects with indeterminate boundaries can be partitioned in two broad categories: (a) objects which have sharp boundaries whose position and shape are unknown or cannot be measured exactly; and (b) objects which do not have wei1defined boundaries or for which it is useless to fix a boundary. The first kind of indeterminate boundary is what is usually called positional uncertainty and involves all geographic data. due to errors in measurements and the finite representation of computer formats. Positio nal uncertainty affects GIS reliability especially in certain thematic maps, such as soil maps, land use maps, and geological maps. Different models have been developed for the representation of positional uncertainties, such as fuzzy models (Dutta, 1991; Leung et 01., 1992; Banai, 1993 ; Heuvelink and Burrough, 1993; Leung and Leung, 1993) and probabilistic models (Blakemore, 1984; Goodchild and Dubuc, 1987; Finn. 1993; Shibasaki, 1993). The second ki nd of indeterminate boundary is very different since the uncertainty is not due to the limitations of current technology, but intrinsically belongs to the