3维空间线与体对象间的拓扑关系完备性研究

Completeness Research on Topological Relations Between 3D Spatial Objects of Lines and Bodies

线与体是3维空间中两类重要的空间对象,对其之间的拓扑关系进行研究在地矿开采、交通运输等很多领域都有非常重要的应用价值。为了准确高效地对3维空间线与体,特别是复杂对象间的拓扑关系进行分析,首先定义了一类基本空间对象,同时用9交模型来表达空间对象之间的拓扑关系;然后利用线与体之间的9条否定规则推导出基本线与体之间可能存在的11种拓扑关系,再利用复合推导的方法来得到复合线对象与体之间的19种拓扑关系;最后得出了用于推导分析复杂线与体之间拓扑关系的算法。经过详细分析与讨论,证明了这些拓扑关系具有完备性。实践表明,利用完备拓扑关系,不仅可以保证推导分析算法的正确性,也能极大简化复杂线对象与体对象间拓扑关系的分析。

Lines and bodies are two important spatial objects in 3D space. Research on topological relations between lines and bodies have great value in many application fields such as mining, military and transportation. In order to analyze the topological relations between 3D lines and bodies especially complicated lines and bodies correctly and efficiently, a group of sample spatial objects are~ defined. Then nine-intersection model is used to describe the topological relations between spatial objects. Using nine negative conditions between lines and bodies, eleven types of topological relations between sample lines and bodies are deduced. Then the composed reasoning method is used to deduce the topological relations between compound lines and bodies and there are altogether nineteen types. Finally the algorithm to analyze the topological relations between complicated lines and bodies is deduced. After detailed analysis and discussion, the completeness of topological relations between lines and bodies is proved. Therefore, the algorithm can guaranteed the correctness and greatly simplify the analysis of the topological relations between complicated lines and bodies.