Qualitative spatial reasoning about distances and directions in geographic space

Abstract

Most known methods for spatial reasoning translate a spatial problem into an analytical formulation in order to solve it quantitatively. This paper describes a method for formal, qualitative reasoning about distances and cardinal directions in geographic space. The problem addressed is how to deduce the distance and direction from point A to C, given the distance and direction from A to B and B to C. We use an algebraic approach, discussing the manipulation of distance and direction symbols (e.g. ‘N’, ‘E’, ‘S’ and ‘W’, or ‘Far’ and ‘Close’) and define two operations, composition and inverse, applied to them. After a review of other approaches, the desirable properties of deduction rules for distance and direction values are analyzed. This includes an algebraic specification of the ‘path’ image schema, from which most of the properties of distance and direction manipulation follow. Specific systems for composition of distance are explored. For directions, a formalization of the well-known triangular concept of directions (here called cone-shaped directions) and an alternative projection-based concept are explored. The algebraic approach leads to the completion of distance or direction symbols with an identity element, standing for the direction or distance from a point to itself. The so completed axiom system allows deductions, at least ‘Euclidean-approximate’, for any combination of input values.