Signed Simplicial Decomposition and Overlay of n-D Polytope Complexes

Polytope complexes are the generalisation of polygon meshes in geoinformation systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straightforward 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.