# Refinement and Computation Method for Line/Body Topological Relationships
## 1 Introduction  
Three-dimensional (3D) geographic information has become important in our everyday life [1]. A three-dimensional realistic geospatial scene is an important new infrastructure in our daily life, and its construction includes the establishment of the ground, underground, indoor, and outdoor geospatial framework of the three-dimensional structure and spatiotemporal relationships, etc. Three-dimensional topological relationships, as the most important and fundamental spatial relationships in a 3D GIS, serve as the theoretical foundation for quality control, updating processing, and spatial analysis of three-dimensional spatial data in realistic 3D construction and 3D GIS [2, 3, 4, 5]. As topological relations are the most important spatial relations [6, 7], many researchers have proposed several topological relationship models to distinguish the relations between two-dimensional objects, e.g., the 4-intersection (4I) model [8], 9-intersection (9I) model [9], the region connection calculus (RCC) model [10], the Voronoi-based 9-intersection (V9I) model [11], the Voronoi-based spatial algebraic model [12], and the Euler-number-based whole-object intersection and differences (E-WID) model [13]. [PERSON] et al. proposed the dimension-extended 9-intersection model (DE+9 IM) [14]. Among the above models, 4I and 9I are widely applied in the GIS society, e.g., OGC lists the 9I model as a reference model for 2D topological relationships. Proposed by [PERSON] and [PERSON] in 1991, 4I model is based on the point set topology theory [8], which decomposes spatial objects into interiors and boundaries where \(\partial\mathbf{A}\), \(\mathbf{A}^{\circ}\), \(\partial\mathbf{B}\), and \(\mathbf{B}^{\circ}\), respectively, represent the boundary and interior of \(\mathbf{A}\) and the boundary and interior of \(\mathbf{B}\). The 4I model uses a 2 \(\times\) 2 matrix to represent the topological relationships between objects, but its discriminatory ability is limitedand, theoretically, capable of distinguishing only 16 types of topological relationships. [PERSON] et al. further proposed a dimension-extension-based 4I model, which, however, also has limited discriminatory ability [15]. To overcome the limited discriminatory ability of the 4I model, [PERSON] and [PERSON] introduced the exterior of spatial objects into the calculation of topological relationships where \(A^{-}\) and \(B^{-}\), respectively, represent the exterior \(A\) and \(B\). The 9I model uses a \(3\times 3\) matrix to represent the topological relationships between objects, theoretically capable of distinguishing \(2^{9}=512\) types of topological relationships. However, in practice, it can only distinguish 8 types of region/region, 19 types of line/region, 33 types of line/line, 3 types of point/region, 3 types of point/line, and 2 types of point/point topological relationships. Due to the widespread influence of the 9I model in two-dimensional space and its relatively simple model structure compared with models created in the same period, it has been extended to three-dimensional space by multiple scholars. This extension has led to the development of methods for describing topological relationships between three-dimensional vector spatial entities, providing initial descriptions of basic topological relationship types between points, lines, regions, and bodies. For example, [PERSON] summarized the topological relationships between three-dimensional spatial entities that can be distinguished using the 9I model [5]. [PERSON] et al. proposed a 3D vector model consisting of basic vector types, predicates, and operations to implement and test the 9I relationships between three-dimensional vector data [2]. [PERSON] and [PERSON] extended RCC*-9 to adopt simplified 3D features and achieved 3D RCC*-9 relations [16]. However, the aforementioned three-dimensional topological relationship models are basic relationship models that can only distinguish a very limited number of basic topological relationship types.