# Refinement and Computation Method for Line/Body Topological Relationships
## 1 Introduction
They cannot yet distinguish the refinement types of topological relationships between 3D entities that contain multiple intersections. In the context of the refinement of topological relationships involving multiple intersections, [PERSON] and [PERSON] established the BBIS (boundary-boundary intersection set) model to describe refined relationships between regions [17]. [PERSON] et al. introduced a hierarchical model for line/line relationships [18]. [PERSON] et al. introduced a multi-level relationship model for region/region relationships based on topological invariants [19]. [PERSON] et al. proposed the topological chain model to describe refined relationships between lines, which they applied in conflict detection for rivers and contour lines [20]. [PERSON] et al. developed the WWIS (whole-whole intersection set) model based on the intersection set operation results of spatial objects as a whole. They provided a comprehensive and distinct description of refined relationships between regions, lines, and lines/regions by using node degrees and Euler numbers [13; 21]. In addition to the refined relationships mentioned above, [PERSON] et al. developed a model of conceptual neighborhoods' topological line-region relationships [6]. [PERSON] et al. proposed the 9+-intersection model for directed line segments and regions [22]. [PERSON] et al. distinguished 152 kinds of topological relationships for simple fuzzy regions [23]. [PERSON] et al. proposed a model and computational method for describing relationships between lines and regions [24]. [PERSON] et al. proposed the directed polyline-polygon model [25]. [PERSON] et al. proposed the 25I model by decomposing spatial objects into vertex, edge, face, interior, and exterior [26]. [PERSON] et al. introduced a topological relationship model that distinguishes between directed lines and directed regions [27] and the 27-intersection model in two-dimensional space [28]. Based on the 9I model, [PERSON] et al. proposed the generalized 9-intersection model for regions with multiple holes [29]. [PERSON] et al. proposed a three-dimensional electronic navigation charts (3D ENC) feature spatial relation model (3 DSRM) based on point-set topology theory, which combines 3D topological relations, distance relations, and directional relations [30]. In summary, the refined topological relationships between lines and bodies are still an open issue to date. Therefore, we propose a model to represent the refined topological relationships between lines and bodies using dimension information, Euler numbers, the number of intersections between the line and the body's boundary, predicates, and topological sequences in this study. As [PERSON] et al. 's WWIS + BBIS refined models, based on the intersection components of the spatial objects as a whole, along with their E-WID coarse model, have formed a complete framework for describing and computing two-dimensional topological relationships using manifold without theoretical deficiency, this framework has the potential to be extended to the 3D space. This study adopts the intersection component concept of this framework and, by defining the 3D line/body intersection component (unit intersection), develops a method to describe 3D line/body unit intersections and derive their types. It establishes an expression model for the refined relationships of 3D lines/bodies. Furthermore, it proposes a calculation algorithm for line/body topological relationships and implements the automatic calculation of line/body refined topological relationships by using an open-source geometric algorithm library.