Integrating Space, Time, Version and Scale Using Alexandrov Topologies (1303.2595v1)

Abstract: This article introduces a novel approach to spatial database design. Instead of extending the canonical Solid-Face-Edge-Vertex schema by, say, "hypersolids" these classes are generalised to a common type SpatialEntity, and the individual BoundedBy relations between two consecutive classes are generalised to one BoundedBy relation on SpatialEntity instances. Then the pair (SpatialEntity, BoundedBy) is a so-called incidence graph. The novelty about this approach uses the observation that an incidence graph represents a topological space of SpatialEntity instances because the BoundedBy-relation defines a so-called Alexandrov topology for them turning them into a topological space. So spatial data becomes part of mathematical topology and topology can be immediately applied to spatial data. For example, continuous functions between two instances of spatial data allow the consistent modelling of generalisation. Further, it is also possible to establish a formal topological definition of spatial data dimension, and every topological data model of arbitrary dimension gets a simple uniform data model. This model covers space-time, and the version history of a spatial model can be represented by an Alexandrov topology, too. By integrating space, time, version, and scale into one single schema, topological queries across those aspects are enabled through topological constructions. In fact, the topological constructions cover a relationally complete query language for spaces and can be redefined to operate accordingly on their graph representations. With these observations a relational database schema for a spatial data model of dimension 6 and more is developed. The schema seamlessly integrates 4D space-time, levels of detail and version history, and it can be easily expanded to also contain non-spatial information or be linked to other data sources.