# A distance-based topological relation model between spatial regions

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## Abstract

Although the definitions of formal models used to represent spatial relations have gained increasing attention over the past 30 years, the linkage between topology and distance has not yet been effectively established. A topological relation model called the distance-based topological relation model (D-TRM) that considers both the topology and distance of spatial regions is proposed. The D-TRM is divided into two subtypes: the actual DTRM (AD-TRM) and the signed DTRM (SD-TRM). The actual distance is based on the distance in a two-dimensional space. The signed distance is based on the sign of the actual distance. Eight topological relations, namely, *disjoint*, *meet*, *overlap*, *cover*, *contain*, *equal*, *coveredBy* and *inside*, represented by the AD-TRM and SD-TRM are shown. The mutual exclusiveness among these eight topological relations represented by the SD-TRM is proven. The topological relation representations from the 9-intersection model (9IM), the splitting measures of the 9IM (SP-9IM), the SD-TRM and the AD-TRM are discussed, and the interoperability of each of the above models is summarised. The topological relation representation between the AD-TRM and the comprehensive set of 11 metric refinements is compared. The results show the following: (1) as the generalisation of the AD-TRM, the SD-TRM can concisely represent the topological relations; (2) the topology and distance between two spatial regions can be represented by the AD-TRM in a unified framework; (3) the AD-TRM provides a greater level of detail than the 9IM and (4) the D-TRM can express more distance information than the comprehensive set of 11 metric refinements.

## Keywords

Topological relation Metric refinements Distance-based topological relation model (D-TRM) Spatial region## Notes

### Acknowledgments

We appreciate the detailed suggestions and comments from the editor and the anonymous reviewers.

### Funding

The work described in this article was supported by the following research programs: The National Basic Research Program of China (973 Program) [grant number 2015CB954103], the National Natural Science Foundation of China [grant number 41622108, grant number 41301417, grant number 41701441], and the Priority Academic Program Development of Jiangsu Higher Education Institutions [grant number 164320H116].

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