# 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  
Given that manifold topology better aligns with the cognitive habits of individuals in the GIS field regarding spatial objects, [PERSON] et al. have adopted it to define two-dimensional objects and differentiate two-dimensional topological relationships. Due to the complexity of the geometric shapes and topological properties of three-dimensional spatial objects, the following sections initially utilize manifold topology to define 3D simple points, lines, bodies, and their interiors and boundaries [32; 33] (see Figure 2). Figure 2: Diagram of the interior and boundary of \(n\) (\(0\leq n\leq 3\))-dimensional manifolds, with the yellow portions representing their neighborhood: (**a**) neighborhood of point interior and point boundary; (**b**) neighborhood of body boundary and body interior; (**c**) neighborhood of line boundary and line interior. **Definition 1:**_A simple point is a zero-dimensional manifold with only an interior and no boundary (Figure 2a)._  
**Definition 2:**_A simple body is a three-dimensional manifold with a connected interior and a connected boundary, and is homeomorphic to a solid cube (Figure 2b)._  
**Definition 3:**_A simple line is a one-dimensional manifold with a connected interior and two distinct boundary points (Figure 2c)._  
It should be noted that three-dimensional manifolds (bodies) can have tunnels or voids. Such three-dimensional manifolds with voids or tunnels are referred to as complex manifolds. For simplicity, we do not discuss complex manifolds in this article. Therefore, the lines and bodies studied here are topologically equivalent to connected, orientable, compact one-dimensional and three-dimensional manifolds. Thus, the unit intersections of lines/bodies can be defined as follows:  
**Definition 4:**_Let **A** and **B** be the 3D body and line objects, respectively, and **I** be the largest connected subset of **A**\(\cap\)**B**. Let \(P\) = {\(P_{1}\), \(P_{2}\),..., \(P_{m}\)} be a subset of **I** where \(P\in\textbf{I}\). If \(P_{j}\) (\(0\leq j\leq m\)) is a zero-dimensional manifold (point), then \(P_{j}\) is a unit intersection (point) of **A**\(\cap\)**B**. Let \(\text{L}=\text{I}_{1}\), \(l_{2}\),..., \(l_{n}\)} where \(L\in\textbf{I}\). If \(l_{i}\) (\(0\leq i\leq n\)) is a one-dimensional manifold (line) and the interior of \(l_{i}\) equals the largest connected subset of **A**\(\cap\)**I**\({}_{i}\) or **A**\({}^{\circ}\cap\)**I**\({}_{i}\), then \(l_{i}\) is a unit intersection (line) of **A**\(\cap\)**B**. Moreover, \(\textbf{I}=P\cup\text{L}=\text{U}_{i=1}^{n}l_{i}\) (\(i\geq 0\)) \(\cup\text{U}_{j=1}^{m}p_{j}\) (\(j\geq 0\))  
As shown in Figure 3, the intersection of body **A** and line **B** has four connected subsets, namely, the polyline (_ab_) (_cdefg_) and isolated points \(h\) and \(i\). Among them, \(h\) and \(i\) are isolated points, and two unit intersections (points) of **A**\(\cap\)**B** are marked as 6 and 7. According to the definition of unit intersections, the polyline (_cdefg_) needs to be subdivided into four unit intersections: _cd_, _de_, _ef_, and _fg_. Therefore, the four connected subsets of the intersection between body **A** and line **B** include seven unit intersections, namely, intersection lines _ab_, _cd_, _de_, _ef_, and _fg_, and intersection points \(h\) and \(i\). The intersection lines can be preliminarily classified into three types: 1 the interior of **ab** and endpoint \(a\) are inside body **A**, while endpoint \(b\) is on the boundary of **A**; 2 the endpoints and interior of _cd_ and _ef_ are all on the boundary of **A**; 3 the interior of _de_ and _fg_ is inside body **A**, and both endpoints are on the boundary of **A**. The connectivity of intersection points \(i\) and \(h\) is also different.