Digital Relations in Z1: Discretized Time and Rasterized Lines
Matthew Paul Dube · Preprints.org · 2025
There is a voluminous literature concerning the scope of topological relations that span spaces from R^1 to R^2, Z^2, S^1 and S^2, and T^2. In the case of the *^1 spaces, those relations have been considered both as conceptualizations of both spatial relations and also temporal relations. Missing from that list are the set of digital relations that exist within Z^1, representing discretized time or discretized ordered line segments. Discretized time plays an essential role in timeseries data and in spatio-temporal information systems and geo-foundation models where time is represented in layers of consecutive spatial rasters and/or spatial vector objects colloquially referred to as space-time cubes or spatio-temporal stacks. This paper explores the digital relations that exist in Z^1 interpreted as a regular topological space under the digital Jordan curve as well as a folded-over temporal interpretation of that space for use in spatio-temporal information systems and geo-foundation models. It identifies 34 9-intersection relations in Z^1, 42 9-intersection + margin relations in Z^1, and 74 temporal relations in Z^1, utilizing the 9+-intersection. This workcreates opportunities for better spatio-temporal reasoning capacity within spatio-temporal stacks and more direct interface with intuitive language concepts instrumental for effective utilization of spatial tools.