Topological relations between directed line segments (DLs) may contribute to queries and analyses related to noninstantaneous phenomena whose position changes over time. Although considerable research has been conducted to study topological relation models and the specification of the topological relations that exist in reality, further work is required to consider what types of topological relations between DLs in a cyclic space can be realized. This research is a contribution to the clarification of the topological relations between DLs in a cyclic space that can occur in reality. A DL is divided into four primitives: a starting point, an ending point, an interior, and an exterior. A topological relation model between two DLs in a cyclic space with a 4 × 4 matrix is proposed in this article. A total of 38 topological relations between DLs in the cyclic space are distinguished, and the matrix patterns and the corresponding geometric interpretations of the 38 topological relations are shown to prove the existence of the topological relations. Eleven negative conditions are summarized to prove the completeness of the 38 topological relations. The cyclic interval relations, spherical topological relations, and topological relations presented in this research are compared. The results show the following: (1) the proposed topological relation model can well represent the topological relations between DLs, and (2) the proposed 11 negative conditions can be used to prove the completeness of the 38 topological relations.

中文翻译：

---

循环空间中有向线段之间的拓扑关系

有向线段（DL）之间的拓扑关系可能有助于查询和分析与位置随时间变化的非瞬时现象有关的问题。尽管已经进行了大量的研究来研究拓扑关系模型和现实中存在的拓扑关系的规范，但是需要做进一步的工作来考虑可以实现循环空间中DL之间的什么类型的拓扑关系。这项研究为澄清现实中可能发生的循环空间中DL之间的拓扑关系做出了贡献。DL分为四个基元：起点，终点，内部和外部。本文提出了具有4×4矩阵的循环空间中两个DL之间的拓扑关系模型。区分了循环空间中DL之间的38个拓扑关系，并显示了38个拓扑关系的矩阵模式和相应的几何解释，证明了该拓扑关系的存在。总结了11个负条件，以证明38个拓扑关系的完整性。比较了本研究提出的周期间隔关系，球面拓扑关系和拓扑关系。结果表明：（1）提出的拓扑关系模型可以很好地表示DL之间的拓扑关系；（2）提出的11个负条件可以证明38个拓扑关系的完整性。并给出了38种拓扑关系的矩阵模式和相应的几何解释，证明了该拓扑关系的存在。总结了11个负条件，以证明38个拓扑关系的完整性。比较了本研究提出的周期间隔关系，球面拓扑关系和拓扑关系。结果表明：（1）提出的拓扑关系模型可以很好地表示DL之间的拓扑关系；（2）提出的11个负条件可以证明38个拓扑关系的完整性。并用38个拓扑关系的矩阵模式和相应的几何解释证明了拓扑关系的存在。总结了11个负条件，以证明38个拓扑关系的完整性。比较了本研究提出的周期间隔关系，球面拓扑关系和拓扑关系。结果表明：（1）提出的拓扑关系模型可以很好地表示DL之间的拓扑关系；（2）提出的11个负条件可以证明38个拓扑关系的完整性。比较了球形拓扑关系和本研究提出的拓扑关系。结果表明：（1）提出的拓扑关系模型可以很好地表示DL之间的拓扑关系；（2）提出的11个负条件可以证明38个拓扑关系的完整性。比较了球形拓扑关系和本研究提出的拓扑关系。结果表明：（1）提出的拓扑关系模型可以很好地表示DL之间的拓扑关系；（2）提出的11个负条件可以证明38个拓扑关系的完整性。