# Refinement and Computation Method for Line/Body Topological Relationships
## 5 Conclusions and Discussion  
The existing 3D topological relationship models are all basic relationship models that cannot distinguish the refined topological relationship between the line and the body with multiple intersections. Building upon the WWIS 2D topological refinement model, we have developed a 3D refinement of the topology description model for lines and bodies. This model defines the concept of line/body unit intersections and proposes a method for distinguishing between them based on dimension, Euler numbers, and predicates for connecting intersection endpoints. On this basis, we have derived 2 types of intersection points and 21 types of intersection lines between lines and bodies, illustrating the description method for the refinement of topology relationships involving multiple intersections. Furthermore, we have developed algorithms for determining the types of adjacency endpoints for intersection points and intersection lines, as well as for calculating the Euler numbers and the number of intersection points between endpoints of intersection lines and the surface of bodies. In summary, the main contributions of this paper are listed as follows:  
1. Based on manifold topology theory, a 3D line and body are defined, followed by the definition of unit intersections between the line and the body. 2. Based on the unit intersections, basic intersection types between the line and the body are deduced and distinguished using the dimensions of unit intersections, Euler numbers, the number of intersections between unit intersections and body boundaries, and predicates. Finally, 23 basic types of line/body are distinguished, and through these 23 basic types and the topological sequence of basic types, all refinement relations are represented. 3. This paper describes the computation method of basic types and validates the effectiveness of the model using real three-dimensional data. Finally, we have implemented the computation of 3D line/body refined topological relationships using the Nef polyhedron model for representation, C++ programming language, and CGAL open-source library. We have validated the effectiveness of our approach and demonstrated its feasibility for applications such as conflict detection between building structures and pedestrian pathways using real-world building and pedestrian pathway data from Hong Kong. Since this paper only focuses on the refinement of topological relationships between simple lines and bodies, the body objects cannot have channels or voids. Future research will focus on other spatial entities, such as the refinement of topological relationships \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**C** & **Type** & **No.** & **C** & **Type** & **No.** \\ \hline (a) & (0, \textless{}out, null\textgreater{}) & 7 & (b) & (0, \textless{}out, out\textgreater{}) & 1 \\ \hline (c) & (1, 0, 2, \textless{}null, null\textgreater{}) & 1115 & (d) & (1, 1, 1, \textless{}null, on\textgreater{}) & 4 \\ \hline (e) & (1, 1, 1, \textless{}null, in\textgreater{}) & 5 & (f) & (1, 1, 1, \textless{}null, null\textgreater{}) & 5 \\ \hline (g) & (1, 1, 1, \textless{}null, out\textgreater{}) & 342 & (h) & (1, 2, 1, \textless{}null, null\textgreater{}) & 1 \\ \hline (i) & (1, 2, 1, \textless{}null, out\textgreater{}) & 1 & (j) & (1, 2, 1, \textless{}null, in\textgreater{}) & 4 \\ \hline (k) & (1, 2, 1, \textless{}out, out\textgreater{}) & 3 & (l) & (1, 2, 1, \textless{}out, in\textgreater{}) & 1 \\ \hline (m) & (1, 2, 1, \textless{}in, in\textgreater{}) & 1 & (n) & (1, 2, 0, \textless{}null, null\textgreater{}) & 1 \\ \hline (o) & (1, 2, 0, \textless{}null, out\textgreater{}) & 2 & (r) & (1, 2, 0, \textless{}out, out\textgreater{}) & 21 \\ \hline (s) & (1, 2, 0, \textless{}out, in\textgreater{}) & 2 & (u) & (1, 2, 0, \textless{}in, in\textgreater{}) & 1 \\ \hline (v) & (1, 2, 0, \textless{}on, in\textgreater{}) & 1 & (w) & (1, 2, 0, \textless{}on, on\textgreater{}) & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Types and quantities of unit intersections.