Topological relations between directed line segments in the cyclic space
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Topological relations between directed line segments in the cyclic space Jingwei Shen1 · Kaifang Shi1 · Min Chen2,3 Received: 11 December 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Topological relation · Directed line segments · Cyclic space · Negative conditions JEL Classification C88
* Min Chen [email protected] Extended author information available on the last page of the article
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1 Introduction A line segment is defined as a simple line connected with two endpoints. As a type of line segment, a directed line segment (DL) consists of an ordered pair of points, which are the starting point and ending point (ISO 19107 2003). Many geographic objects in the real world, such as one-way streets and bus routes, exhibit directed linear distributions. Figure 1 illustrates a bus route that is a common example of direct line segments. DLs are fundamental geometric elements used in geographic databases to represent directed linear entities (Kurata and Egenhofer 2006). A DL may be embedded in a one-dimensional space (Allen 1983), a cyclic space (Hornsby et al. 1999), two-dimensional space (Gao et al. 2008; Formica et al. 2017), and three-dimensional space (Kurata 2008). A cyclic space appears as a ring. Many object locations may vary in a cyclic pattern over time. The following two examples present cyclic patterns in terms of location changes. First, driven by northeast trade winds and westerlies, the
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