Polyhedral Partitioning of Space for Digital Modelling of Topology and Geometry
Final Report Abstract
The topological and geometrical description of objects plays a fundamental role in various fields, including architecture, civil engineering, geoinformatics, and city modeling. Objects in these domains can exhibit a wide range of characteristics, including convex and concave shapes, simply and multiply connectivity, bounded and unbounded nature, and the presence of holes and cavities. While existing literature has addressed individual requirements and combinations thereof, there is currently no software tool that comprehensively supports modeling objects with all these properties. The research project aimed to develop both a theoretical framework and a pilot implementation for objects with the specified properties. The approach involved partitioning the 3-dimensional unbounded Euclidean space, where the user interacts with unoriented domains stored in the user model, while the core model focuses on oriented objects and explicitly stores neighboring relations. Two key operations, domain splitting and domain merging, can modify a valid partition, provided that the result remains valid. The implementation ensures consistency between the user model and core model, and it exhibits robustness without contradictions between the explicitly stored topology and location test algorithms. The project's findings underscored the need to critically examine existing concepts in geometric modeling, extend them, and introduce novel ones. All algorithms in the framework adhere to a rank concept, where the rank of a domain corresponds to its dimension. For instance, splitting a domain of rank r requires prior splits of domains with lower ranks. This strict application of the rank concept enables a clear and comprehensible description of the theory as well as an understandable implementation. Key topological concepts in the framework include twins, polygons, polyhedrons, and dihedral cycles. Dihedral cycles are employed to describe the 3-dimensional properties of the partition along their edges. An edge defined in the user model is mapped to at least two arrows in the core model, while a face in the user model corresponds to two facets in the core model. For traversing from one facet to the next around an edge, two dihedral cycles are internally stored, one for each direction. A novel concept, termed "anchor and clint zones", was introduced for location tests, enabling efficient testing for both convex and concave domains. The research also involved creating various types of models for validation, including traditional building models with 3- dimensional components, skeleton models where building components are represented in reduced dimensions (e.g., walls as 2-dimensional objects), and models with partitioned unbounded exteriors. In conclusion, our research has not only addressed the fundamental requirements of topological and geometrical descriptions of objects in various fields but has also provided a comprehensive theoretical framework and a robust implementation that paves the way for modeling complex 3-dimensional objects in architecture, civil engineering, geoinformatics, and city modeling.
Publications
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Modeling bounded and unbounded space with polyhedra: Topology and operators for manifold cell complexes. Advanced Engineering Informatics, 54, 101790.
Huhnt, Wolfgang; Sternal, Maximilian & Pahl, Peter Jan
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Topological queries in a space partitioning model: Definition,visualization and exports of results, in: Proceedings of 33. Forum Bauinformatik, mediaTUM-Portal
Städtler, A. & Sternal, M.
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Robust Modeling ofPolyhedral Space Partitions. Lecture Notes in Civil Engineering, 427-442. Springer Nature Switzerland.
Sternal, Maximilian & Huhnt, Wolfgang
