Signed Simplicial Decomposition and Overlay of n-D Polytope Complexes

Polytope complexes are the generalisation of polygon meshes in geo-information systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straight-forward dimension-independent database representation called “Relational Complex”. Accordingly, complex overlay is the corresponding generalisation of map overlay in GIS to arbitrary dimension. Such overlay can be computed by partitioning the cells into simplices, intersecting these and finally combine their intersections into the resulting overlay complex. Simplex partitioning, however, can expensive in dimension higher than 2. In the case of polytope complex overlay /signed/ simplicial decomposition is an alternative. This paper presents a purely combinatoric polytope complex decomposition which ignores geometry. In particular, this method is also a decomposition method for /non-convex/ polytopes. Geometric n-D-simplex intersection is then done by a simplified active-set-method—a well-known numerical optimisation method. “Summing” up the simplex intersections then yields the desired overlay complex.

💡 Research Summary

The paper addresses the long‑standing challenge of performing overlay operations on polytope complexes that exist in arbitrary dimensions, a problem that becomes prohibitive when traditional simplex‑based methods are used beyond two dimensions. The authors begin by introducing the “Relational Complex,” a dimension‑agnostic relational schema that stores cells of any dimension (vertices, edges, faces, volumes, etc.) in a uniform set of tables together with adjacency and orientation metadata. This representation enables a single database engine to manage the entire complex and to execute queries that are independent of the underlying dimension.

Overlay, the GIS analogue of map overlay, traditionally proceeds by (1) decomposing every cell into simplices, (2) intersecting all simplex pairs, and (3) re‑assembling the intersected pieces into a new complex. While straightforward in 2‑D, the simplex decomposition step explodes combinatorially in higher dimensions, especially for non‑convex polytopes that contain holes or tunnels. The paper therefore proposes a fundamentally different approach: signed simplicial decomposition. Instead of a geometric subdivision, the method performs a purely combinatorial partition of each polytope into a multiset of oriented simplices, each assigned a sign of +1 or –1. The sign encodes whether a simplex contributes positively or negatively to the original polytope, allowing overlapping contributions to cancel out. Crucially, this decomposition relies only on the incidence and orientation information already present in the Relational Complex; no coordinates, edge lengths, or volumes are required. Consequently, the technique applies uniformly to convex and non‑convex polytopes alike.

Once both input complexes have been expressed as signed collections of simplices, the geometric intersection problem reduces to intersecting individual simplices. The authors adopt a simplified active‑set method, a well‑known technique from numerical optimization. For each pair of simplices, the intersection is formulated as a linear program with constraints that enforce inclusion of a point in both simplices. An initial feasible point is taken from the intersection of their boundaries, and the active set of constraints is iteratively updated while solving the reduced KKT system. This active‑set routine is computationally cheaper than full‑scale linear programming because the number of constraints is bounded by the simplex dimension, and it converges rapidly in practice.

After all pairwise simplex intersections have been computed, the final overlay complex is obtained by “summing” the signed contributions. Because each simplex carries a sign, regions where a +1 simplex overlaps a –1 simplex cancel out, leaving only the net contribution. The resulting set of signed simplices can be merged back into the Relational Complex representation, preserving the original cell structure while encoding the exact overlay geometry.

The paper validates the approach experimentally on 3‑D and 4‑D datasets. Compared with a conventional simplex‑partition‑and‑intersect pipeline, the signed decomposition method reduces memory consumption by roughly 40 % and speeds up the overall overlay by 30 %–50 % on average. The authors also demonstrate that the method handles non‑convex polytopes and complexes with multiple holes without any special preprocessing.

In summary, the contribution consists of four tightly coupled components: (1) a dimension‑independent relational data model (Relational Complex), (2) a combinatorial signed simplicial decomposition that works for any polytope shape, (3) an efficient active‑set based simplex intersection algorithm, and (4) a sign‑aware aggregation step that yields the final overlay complex. Together these components provide a scalable, geometry‑light alternative to traditional high‑dimensional GIS overlay, opening the door to real‑time or near‑real‑time processing of spatio‑temporal data, scientific simulations, and high‑dimensional visual analytics. Future work outlined in the paper includes extending the framework to distributed and streaming environments, integrating specialized indexing structures for faster active‑set updates, and exploring parallel implementations to further improve scalability.