A quantitative description model for direction relations based on direction groups

The description models for spatial relations, especially those for direction relations, have gained increasing attention in GIS and Cartography community in recent decades. In this paper, such a quantitative model for spatial direction relations is discussed. It has been suggested that people often describe directions between two objects using multiple directions but not a single one; therefore a description model for direction relations should use multiple directions, i.e. direction group. A direction group consists of two components: the azimuths of the normals of direction Voronoi edges between two objects and the corresponding weights of the azimuths. The former can be calculated by means of Delaunay triangulation of the vertices and the points of intersection of the two objects; the latter can be calculated using the common areas of the two objects or the lengths of their direction Voronoi diagram (DVD) edges.The advantages of this model exist in four aspects: (1) direction computations are converted into a 1-dimension space problem and use lines (DVDs) to solve it, therefore direction computation process is simplified; (2) once Dir(A,B), the directions from A to B, is obtained, Dir(B,A) can be got without complex computations; (3) A quantitative direction group can be transformed into a qualitative one easily; (4) quantitative direction relations between objects can be recorded in 2-dimension tables, which is very useful in spatial reasoning.