Spatial Relations And Their Qualitative Reasoning In A Three-dimensional Space

Spatial relations, including topological, directional and distance relations, have been recognized as one of the fundamental research themes in the field of Geographic Information System (GIS). For research convenience, each of the three spatial relations can further be divided into the representation and reasoning of spatial relations. Previous studies focused on the representation and reasoning of spatial relations in the two-dimensional (2D) space. Progress on similar studies in the three-dimensional (3D) space has been very limited, however. This dissertation is an investigation of the theory of spatial relations in the 3D space.First, topological relations among 0-simplex, 1-simplex, 2-simplex and 3-simplex in 3Dspace were studied with the 9-intersection matrix model in the basis of the simplex datamodel. The geometric interpretations toward these relations were also provided. Based on thesets theory, the qualitative reasoning of topological relations was investigated, and thereasoning results were presented in the form of composition tables. Second, the representationmodels of directional relations were built in 3D space. The reasoning of a single directionrelation was specifically studied to derive the composition tables and to explore the laws ofsingle direction relation reasoning. Third, a framework for the representation of qualitativedistance in 3D space was proposed. The concept of interval number was introduced and theoperations of the interval number were defined to derive the universal formulae for the resultsof qualitative distance reasoning under the three constraints of qualitative distance. Thedefinition of binary relation >> was provided and discussed. Fourth, the limitations to usethe Allen's interval relation pair in describing the directional regions and the commonly-used8 topological relations (disjoint, meet, overlap, cover, coveredby, contain, inside, equal) in 2Dspace were analyzed and refined. The topological relations were reasoned from the directionalrelations and the results were shown in composition tables. The directional regions and thecommonly-used 8 topological relations in 3D space were described using the Allen's intervalrelation pair. The qualitative reasoning laws of the directional relations in 3D space wereexplored with the theory of interval relation pair. The mixed reasoning between the directionaland topological relations was studied, and the results were presented in combined tables. The representation models of the topological and directional relations in 2D space were first analyzed, and then an improved model was put forward to describe the mixed relations. The description model of the integrated topological and directional relations in 3D space was also developed and the directional regions described by the interval pair were analyzed. The qualitative reasoning of integrated topological and directional relations was studied according to the different topological relations, and the combined tables of the integrated topological and directional relations were obtained when the topological relations were disjoint or meet in 2D and 3D space. The description model of integrated qualitative distance and directional relations was built with the project-based method in 2D- and 3D space, respectively. Fro example, the formulae of the positional relations between the primary object C and the reference object A was derived from the positional relations between the primary object B and the reference object A as well as the positional relations between the primary object C and the reference object B. The qualitative reasoning of the position relations was studied using positional relation equations and was simulated in 2D- and 3D- space with MATLAB programming language, respectively.