# Refinement and Computation Method for Line/Body Topological Relationships
## 2 Method for Refinement of Topological Relationships between Line and Body  
As mentioned above, the existing 3D topological relationship model cannot distinguish the refinement relations between 3D objects with multiple intersections. However, in real application scenarios, there are many cases with multiple intersections, and different types of intersections often require different processing methods. In this section, we provide an application scenario first and then define the 3D object using the manifold and model the framework for refinement of topological relationships between the line and the body. Application Scenario for Topological Relationships between the Line and the Body with Multiple Intersections  
In the 3D real world, there are many real scenes where a line and a body have multiple intersections, e.g., an electric-power line, a water pipeline, a gas pipeline and a building. Furthermore, with the rapid development of indoor positioning technologies today, integrated indoor-outdoor navigation services have become a hot research topic in the GIS field. Constructing high-quality integrated indoor-outdoor pedestrian networks is a primary task for integrated indoor-outdoor navigation [31]. While in the indoor-outdoor navigation system, there may be several intersections between a pedestrian pathway and a building (Figure 1). Figure 1 illustrates the integrations between indoor and outdoor pedestrian pathways and buildings. Currently, people generally utilize multi-source (including crowdsourced) data to generate integrated indoor-outdoor pedestrian pathways. However, due to the influence of data quality from multiple sources and indoor positioning accuracy, the generated integrated indoor-outdoor pedestrian pathway data often suffer from various quality issues. Figure 1 presents an example from the open-source dataset depicting the relationship between integrated indoor-outdoor pedestrian pathways and buildings in Hong Kong. In this dataset, buildings are represented as three-dimensional bodies while pedestrian pathways are abstracted as 3D lines. The core attributes of pedestrian pathways include features such as \"Indoor\" and \"Outdoor\". Figure 1a illustrates a two-dimensional  
Figure 1: Example of multiple intersections between a 3D building and pedestrian pathway: (**a**) example of multiple intersections between the line and the body (2D); (**b**) example of multiple intersections between the line and the body (3D). projection of a building and a pedestrian pathway dataset. In this projection, **p1** to **p5** represent five nodes of the pedestrian pathway, while **e1** to **e4** represent the four segments formed by these five nodes. It is evident from Figure 1a that **p1** and **p5** are located outside the building, while **p2**, **p3**, and **p4** are situated on the surface of the building. Additionally, **e1** and **e4** are positioned outside the building, while **e2** and **e3** are within. However, the attribute of this pedestrian pathway is labeled as \"Indoor\". Clearly, there are multiple intersections between the 3D building and the pedestrian pathway, with some of these intersections involving conflicts between topological and semantic attributes. During data quality inspection and processing, it is necessary to address each intersection separately based on its specific characteristics. For example, if **p1** and **p2** belong to the same point within the tolerance range, **e1** can be deleted; otherwise, the attributes of **e1** will be changed to \"Outdoor\". Similarly, if **p4** and **p5** are within the same point tolerance range, **e4** can be removed; otherwise, its attribute should be changed to \"Outdoor\". Existing 3D basic topological relationship models for lines/bodies have not yet expressed the refinement of each intersection, making it challenging to support applications such as 3D conflict detection and processing involving multiple intersections.