三维空间关系的描述及其定性推理
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摘要
空间关系主要包括拓扑关系、方向关系、距离关系等3类，是GIS学科中的重要理论问题之一。它们的研究内容又可细分为空间关系描述和空间关系推理。以往的研究集中在二维空间关系的描述和推理上，而对三维空间关系方面的研究甚少。本文针对三维空间关系理论开展研究。
(1)拓扑关系的描述和推理。采用单纯形数据模型，用九交矩阵工具详细地研究了三维空间中点(0-单纯形)、线(1-单纯形)、面(2-单纯形)和体(3-单纯形)间的拓扑关系，获得了九交矩阵所对应的几何图形。用集合论研究了拓扑关系的定性推理，得到了组合推理表。(2)方向关系的描述和推理。建立了三维空间方向关系的描述模型，并对单方向关系的推理进行了详细研究，导出了组合推理表，给出了单方向关系的推理规律。(3)距离关系的描述和推理。建立了三维空间距离关系的定性描述系统，引入了区间数工具，根据区间数的运算以及定性距离满足的三个约束条件，导出了在不同情况下定性距离推理结果的通项公式。给出了二元关系“>>”的定义，并分析了“>>”的含义。(4)混合空间关系定性推理。在二维空间中，分析了利用Allen区间关系对方法分别描述方向区域和常用的8大拓扑关系(disjoint，meet，overlap，cover，coveredby，contain，inside，equal)的不足，并改进了现有的方法。此外，探讨了如何由方向关系推导拓扑关系问题，得到了相应的组合推理表。在三维空间中，用Allen区间关系对方法分别描述了方向区域和常用的8大拓扑关系，导出了方向关系的推理规律，并给出了方向关系和拓扑关系之间的组合推理表。(5)综合拓扑方向关系描述和推理。针对二维空间中现有拓扑方向关系描述模型的不足，改进了现有模型。在三维空间中，建立了拓扑方向关系的描述模型。根据拓扑关系的不同，分别在二维、三维空间中研究了拓扑方向关系的定性推理，给出了拓扑关系为disjoint和meet时拓扑方向关系的组合推理表。(6)位置关系定性描述和推理。分别在二维、三维空间中，采用基于投影的方向关系建立了位置关系定性描述模型。如，根据目标对象B与参照物A的位置关系和目标对象C与参照物B的位置关系，导出了C与参照物A的位置关系公式。根据位置关系公式，进一步研究了位置关系的定性推理，并用MATLAB语言模拟了位置关系的定性推理。
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.
(1)拓扑关系的描述和推理。采用单纯形数据模型，用九交矩阵工具详细地研究了三维空间中点(0-单纯形)、线(1-单纯形)、面(2-单纯形)和体(3-单纯形)间的拓扑关系，获得了九交矩阵所对应的几何图形。用集合论研究了拓扑关系的定性推理，得到了组合推理表。(2)方向关系的描述和推理。建立了三维空间方向关系的描述模型，并对单方向关系的推理进行了详细研究，导出了组合推理表，给出了单方向关系的推理规律。(3)距离关系的描述和推理。建立了三维空间距离关系的定性描述系统，引入了区间数工具，根据区间数的运算以及定性距离满足的三个约束条件，导出了在不同情况下定性距离推理结果的通项公式。给出了二元关系“>>”的定义，并分析了“>>”的含义。(4)混合空间关系定性推理。在二维空间中，分析了利用Allen区间关系对方法分别描述方向区域和常用的8大拓扑关系(disjoint，meet，overlap，cover，coveredby，contain，inside，equal)的不足，并改进了现有的方法。此外，探讨了如何由方向关系推导拓扑关系问题，得到了相应的组合推理表。在三维空间中，用Allen区间关系对方法分别描述了方向区域和常用的8大拓扑关系，导出了方向关系的推理规律，并给出了方向关系和拓扑关系之间的组合推理表。(5)综合拓扑方向关系描述和推理。针对二维空间中现有拓扑方向关系描述模型的不足，改进了现有模型。在三维空间中，建立了拓扑方向关系的描述模型。根据拓扑关系的不同，分别在二维、三维空间中研究了拓扑方向关系的定性推理，给出了拓扑关系为disjoint和meet时拓扑方向关系的组合推理表。(6)位置关系定性描述和推理。分别在二维、三维空间中，采用基于投影的方向关系建立了位置关系定性描述模型。如，根据目标对象B与参照物A的位置关系和目标对象C与参照物B的位置关系，导出了C与参照物A的位置关系公式。根据位置关系公式，进一步研究了位置关系的定性推理，并用MATLAB语言模拟了位置关系的定性推理。
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.
引文
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