Point-in-Polygon Analysis Under Certainty and Uncertainty

Abstract

The point-in-polygon query in geographical information systems under certainty and uncertainty is formally analyzed in this paper. It is argued that points and polygons can be precise, fuzzy (imprecise), and random (with error) with different schemes of representations. Under certainty, points and polygons can generally be represented by their characteristic functions. Under imprecision induced uncertainty, they can be represented by fuzzy sets characterized by membership functions. If uncertainty is induced by randomness, points and polygons can be described by locational error models in which probability arguments are employed. Points and polygons under certainty turn out to be a special case of that under imprecision and randomness induced uncertainty. Since points and polygons may be precisely, imprecisely or randomly captured or recognized within a spatial information system, the point-in-polygon query is then rather complicated, and its entertainment is not straight forward. In general, the point-in-polygon query can be entertained under nine basic situations. It consists of the queries of whether a precise or fuzzy or random point is in a precise or fuzzy or random polygon. As a consequence, the answer to the query may take on various forms with certain types of uncertainty arguments attached. It involves the integrative utilization of fuzzy set and probability theories to derive the results. The present analysis clarifies some unresolved issues of the point-in-polygon query and provides a generalization to its entertainment. Furthermore, it sheds light on the way certainty and uncertainty can be addressed and implemented in spatial information systems.