Neighborhood Reasoning of Rectangular Direction Constraints

In the process of spatial calculation, spatial object is often described as minimum bounding rectangle (MBR), which makes the rectangular constraint a key subset of spatial relationship. On the basis of rectangular algebra, we illustrate the 169 rectangular direction constraints with a 2×2 matrix, which is called F-matrix, basing on the interval relations between projected intervals of rectangles. According to the neighborhood relation between rectangular direction constraints, we build the neighbor grid for rectangular directions using a 4-dimensional coordinate system. In the research, we calculate the distance between rectangular directions via the shortest path between the corresponding vertexes in the grid. The relational distance indicates the neighborhood of two relations, and analyzes how a rectangular direction turns into another one due to the deformation of rectangles, such as sca-ling and translation. During the rectangular deformation, a set of new rectangular relations will be created. According to the initial and final rectangular constraints, we use the Cartesian products of corresponding feature value tuple intervals, to calculate the F-matrixes of newly created rectangular relations. Besides, we also explore and predict the corresponding rectangular directions during the rectangular deformation, for example, if the current constraint is meeting, the next rectangular constraint must be disjointed or overlaid. In the last section of paper, we analyze and conclude the characteristics of corresponding F-matrixes during the deformation of rectangles. According to the current rectangular constraint and impending deformation, more detailed predictions can be made.