Topologically Consistent Embedding of Manifold Cells into a Polyhedral Space Partition: A General Algorithm for Spatial Modeling of the Built Environment

Spatial models used in architecture, civil engineering, and geodesy require topological consistency to support downstream applications such as energy simulation, indoor navigation, and urban planning. Existing Boolean operation engines process individual mesh objects but do not guarantee the consistency of a complete space partition, particularly when unbounded space must be represented explicitly. This paper presents a general embedding algorithm that inserts an arbitrary polyhedral boundary representation B into an existing space partition A, which constitutes a manifold, gapless partition of Euclidean space including unbounded cells, through a sequence of topologically controlled split operations. Unbounded edges are treated as rays and intersected directly by the geometric kernel, so no separate virtualization of the geometry at infinity is required. The algorithm proceeds in three phases: face-by-face comparison with tolerance-based classification to detect all intersections, incremental collection of split data for edges, faces, and cells through a multi-pass traversal, and execution of split work steps on the topological kernel in ascending dimension order. By operating exclusively through split operations, validity is checked incrementally at each split rather than by a separate global repair pass, so A is kept a valid space partition throughout without any smoothing or repair of an invalid result. A systematic test case matrix covering nine categories of volumetric relationships and boundary contact types was developed. Selected representative scenarios from this matrix were implemented and checked for topological correctness on the resulting partition, including preservation of manifold connectedness and gap-free space coverage without post hoc repair. The algorithm generalizes the parametric design operators introduced in a preceding conference paper to the full three-dimensional case and provides a theoretical foundation for consistent spatial modeling of the built environment.