Topological relations between directed line segments in the cyclic space

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.