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In GIS and spatial databases, cardinal directions are frequently used as selection and join criteria in query languages. However, most cardinal direction models are only able to handle two-dimensional spatial objects. But, e.g., geoscientists and engineers in fields like geography, cartography, soil engineering, and landscape modeling have shown an increasing demand for modeling cardinal directions between objects in the three-dimensional space. Unfortunately, the few available 3D cardinal direction models suffer from several problems like the coarse approximation of the two operand 3D objects in terms of single points and minimum bounding boxes, the lacking property of converseness of the cardinal direction relations computed, and the incomplete coverage of all possible direction relations. All problems mentioned can lead to incorrect results. This paper proposes a new model that solves these problems and in a first stage introduces an objects interaction cube and, as its representation, an objects interaction cube matrix to capture all possible interactions between any two 3D spatial objects. In a second stage, an interpretation technique is applied to the objects interaction cube matrix to determine the cardinal directions.

Modeling cardinal directions in the 3D space with the objects interaction cube matrix