# Refinement and Computation Method for Line/Body Topological Relationships
## 2 Method for Refinement of Topological Relationships between Line and Body
### Definition of 3D Objects and Line/Body Unit Intersections with Manifold Topology
The situation of line/body intersection points is relatively simple, as they can only intersect on the surface of the body, as shown in the two cases in Figure 3. However, the situation of line/body intersection lines can be much more complex. Obviously, in natural language, people often communicate whether the line and its endpoints are on  
Figure 3: Example of multiple unit intersections. the boundary of the body. It is relatively easy to identify whether the endpoint is on the boundary, but whether the line as a whole is on the boundary or inside the body cannot be determined solely by the endpoints. For example, the two endpoints of the intersecting lines _cd_, _de_, _ef_, and _fg_ are all on the boundary of body \(A\), but the intersecting lines _ef_ and _cd_ are on the boundary of body \(A\), while _de_ and _fg_ are inside body \(A\). It is noted that \(f_{E}(\textbf{A}/\textit{ef})\) = \(1,f_{E}(\textbf{A}/\textit{cd})\) = \(1,f_{E}(\textbf{A}/\textit{de})\) = \(0\), and \(f_{E}(\textbf{A}/\textit{fg})\) = \(0\). Therefore, the Euler number can better distinguish these two cases. Upon closer analysis of _ed_, _fg_, _ef_, and _cd_, it is evident that their connectivity status still differs, and this difference remains important in three-dimensional data conflict detection and processing. For instance, assuming line \(B\) represents a pedestrian walkway and \(A\) represents a building structure, although _ef_ and _cd_ are both on the boundary of \(A\), the two segments of the pedestrian walkway connected by _ef_, namely, _ed_ and _fg_, are both inside \(A\), suggesting that _ef_ is likely an indoor pedestrian walkway (such as an elevator or staircase) within the building. On the other hand, one end of the pedestrian walkway connected by _cd_ is inside \(A\), while the other end is outside \(A\), indicating that _cd_ is likely an outdoor pedestrian walkway. From Figure 3, it can be observed that each intersection line consists of two endpoints, and each endpoint may have different connectivity states. There may be no connecting line segment at the endpoint (denoted as _null_), as seen in endpoint \(a\) of intersection line 1 in Figure 3. The connecting line segment may be outside the body an object (denoted as _out_), as seen in endpoint \(b\) of intersection line 1 and endpoint \(c\) of intersection line 2 in Figure 3. The connecting line segment may lie on the boundary of the body object (denoted as _on_), as seen in both endpoints, \(e\) and \(d\), of intersection line 3 in Figure 3. The connecting line segment may be inside the body object (denoted as _in_), as seen in both endpoints, \(e\) and \(f\), of intersection line 4 in Figure 3. Therefore, this study distinguishes the types of line/body unit intersections based on the dimension of the unit intersection (_f_\({}_{D}\)), the number of intersections between the endpoints of the intersection line and the body surface (_fg_), the Euler number of the difference of the body with the unit intersection (_f_\({}_{E}\)), and the connectivity state predicate of the endpoints of the unit intersection (_fg_) (Equation (2)). \[T_{i}(A,B)=(f_{D},f_{S},f_{E},f_{R}) \tag{2}\]  
According to Equation (2), the basic types of line/body unit intersections can be derived based on the values of \(f_{D},f_{S},f_{E},f_{R}\).