{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T09:03:43Z","timestamp":1768554223927,"version":"3.49.0"},"reference-count":46,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T00:00:00Z","timestamp":1756080000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation (US)","award":["2019470"],"award-info":[{"award-number":["2019470"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>There is voluminous literature concerning the scope of topological relations that span various embedding spaces from R1 to R2, Z2 , S1 and S2 , and T2. In the case of the *1 spaces, those relations have been considered as conceptualizations of both spatial relations and temporal relations. Missing from that list are the set of digital relations that exist within Z1 , representing discretized time, discretized ordered line segments, or discretized linear features as embedding spaces. Discretized time plays an essential role in timeseries data, 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\u2013time cubes or spatio-temporal stacks. This paper explores the digital relations that exist in Z1 interpreted as a regular topological space under the digital Jordan curve model as well as a folded-over temporal interpretation of that space for use in spatio-temporal information systems and geo-foundation models. The digital Jordan curve model represents the maximum expressive power between discretized objects, making it the ideal paradigm for a decision support system model. It identifies 34 9-intersection relations in Z1 , 42 9-intersection + margin relations in Z1 , and 74 temporal relations in Z1 , utilizing the 9+-intersection, the commercial standard for spatial information systems for querying topological relations. This work creates opportunities for better spatio-temporal reasoning capacity within spatio-temporal stacks and a more direct interface with intuitive language concepts, instrumental for effective utilization of spatial tools. Three use cases are demonstrated in the discussion, representing each of the utilities of Z1 within the spatial data science community.<\/jats:p>","DOI":"10.3390\/ijgi14090327","type":"journal-article","created":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T08:44:43Z","timestamp":1756111483000},"page":"327","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Digital Relations in Z1: Discretized Time and Rasterized Lines"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7104-9982","authenticated-orcid":false,"given":"Matthew P.","family":"Dube","sequence":"first","affiliation":[{"name":"Department of Computer Information Systems, University of Maine at Augusta, Augusta, ME 04330, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Goertzel, B., Geisweiller, N., Coelho, L., Jani\u010di\u0107, P., and Pennachin, C. (2011). Real-World Reasoning: Toward Scalable, Uncertain Spatiotemporal, Contextual and Causal Inference, Atlantic Press.","DOI":"10.2991\/978-94-91216-11-4"},{"key":"ref_2","unstructured":"Frank, A.U., and Kuhn, W. (1995). Na\u00efve geography. International Conference on Spatial Information Theory, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Xiao, N., Kwan, M., Goodchild, M.F., and Shekhar, S. (2012). An ordering of convex topological relations. International Conference on Geographic Information Science, Springer.","DOI":"10.1007\/978-3-642-33024-7"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Raubal, M., Mark, D.M., and Frank, A.U. (2013). Cognitive and Linguistic Aspects of Geographic Space, Springer.","DOI":"10.1007\/978-3-642-34359-9"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"207","DOI":"10.2307\/142726","article-title":"The choreography of existence: Comments on Hagerstrand\u2019s time-geography and its usefulness","volume":"53","author":"Pred","year":"1978","journal-title":"Econ. Geogr."},{"key":"ref_6","first-page":"137","article-title":"A brief review of the evolution of GIScience since the NCGIA Research Agenda initiatives","volume":"26","author":"Claramunt","year":"2023","journal-title":"J. Spat. Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1145\/182.358434","article-title":"Maintaining knowledge about temporal intervals","volume":"26","author":"Allen","year":"1983","journal-title":"Commun. ACM"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1080\/02693799108927841","article-title":"Point-set topological spatial relations","volume":"5","author":"Egenhofer","year":"1991","journal-title":"Int. J. Geogr. Inf. Syst."},{"key":"ref_9","unstructured":"Egenhofer, M.J., and Herring, J.R. (NCGIA Technical Report, 1990). Categorizing binary topological relations between regions, lines, and points in geographic databases, NCGIA Technical Report."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0020-0255(95)00289-8","article-title":"A model for representing topological relationships between complex geometric features in spatial databases","volume":"90","author":"Clementini","year":"1996","journal-title":"Inf. Sci."},{"key":"ref_11","unstructured":"Nebel, B., Rich, C., and Swartout, W. (1992). A spatial logic based on regions and connection. Principles of Knowledge Representation and Reasoning, Morgan Kauffman."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1005632801858","article-title":"Temporal prepositions and temporal generalized quantifiers","volume":"24","author":"Pratt","year":"2001","journal-title":"Linguist. Philos."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dube, M.P., and Hall, B.P. (2025). Conceptual neighborhood graphs: Event detectors, data relevancy, and language translation. Geogr. Accord. ChatGPT, in press.","DOI":"10.20944\/preprints202508.1602.v1"},{"key":"ref_14","first-page":"40","article-title":"Definitions of line-line relations for geographic databases","volume":"16","author":"Egenhofer","year":"1993","journal-title":"IEEE Data Eng. Bull."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ruas, A., and Gold, C. (2008). Conceptual neighborhoods of topological relations between lines. Headway in Spatial Data Handling, Springer.","DOI":"10.1007\/978-3-540-68566-1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1080\/02693799508902056","article-title":"Modelling conceptual neighbourhoods of topological line-region relations","volume":"9","author":"Egenhofer","year":"1995","journal-title":"Int. J. Geogr. Inf. Syst."},{"key":"ref_17","unstructured":"Cohn, A.G., Giunchiglia, F., and Selman, B. (2000). A model for reasoning about topologic relations between cyclic intervals. Principles of Knowledge Representation and Reasoning, Morgan Kaufman."},{"key":"ref_18","first-page":"25","article-title":"Spherical topological relations","volume":"1","author":"Egenhofer","year":"2005","journal-title":"J. Data Semant. III"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.jlap.2005.04.003","article-title":"On the complemented disk algebra","volume":"66","author":"Li","year":"2006","journal-title":"J. Log. Algebr. Program."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Abel, D.J., and Ooi, B.C. (1993). Topological relations between in regions in R2and Z2. International Symposium on Spatial Databases, Springer.","DOI":"10.1007\/3-540-56869-7"},{"key":"ref_21","unstructured":"Egenhofer, M.J., and Herring, J.R. (1995). Topological relations between discrete regions. International Symposium on Spatial Databases, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Fabrikant, S., Raubal, M., Bertolotto, M., Davies, C., and Freundschuh, S. (2015). From metric to topology: Determining relations in discrete space. International Conference on Spatial Information Theory, Springer.","DOI":"10.1007\/978-3-319-23374-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"100906","DOI":"10.1016\/j.cola.2019.100906","article-title":"Beyond the digital Jordan curve: Unconstrained simple pixel-based raster relations","volume":"54","author":"Dube","year":"2019","journal-title":"J. Comput. Lang."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.ijar.2019.10.008","article-title":"Binary topological relations on the digital sphere","volume":"116","author":"Dube","year":"2020","journal-title":"Int. J. Approx. Reason."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1016\/0097-8493(94)90007-8","article-title":"Modelling topological spatial relations: Strategies for query processing","volume":"18","author":"Clementini","year":"1994","journal-title":"Comput. Graph."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Schwabish, J. (2021). Better Data Visualizations, Columbia University Press.","DOI":"10.7312\/schw19310"},{"key":"ref_27","unstructured":"Damiani, M.L., Renz, M., Eldawy, A., Kroger, P., and Nascimento, M.A. (2023, January 13\u201316). Geo-foundation models: Reality, gaps, and opportunities. Proceedings of the 31st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, Hamburg, Germany."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1111\/1467-9671.00012","article-title":"Use of a three-domain representation to enhance GIS support for complex spatiotemporal queries","volume":"3","author":"Yuan","year":"1999","journal-title":"Trans. GIS"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1111\/tgis.12328","article-title":"Classification of topological relations between spatial objects in two-dimensional space within the dimensionally extended 9-intersection model","volume":"22","author":"Shen","year":"2018","journal-title":"Trans. GIS"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s101090100066","article-title":"An integrated representation of spatial and temporal relationships between evolving regions","volume":"3","author":"Claramunt","year":"2001","journal-title":"J. Geogr. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Cova, T.J., Miller, H.J., Beard, M.K., Frank, A.U., and Goodchild, M.F. (2008). The 9+-intersection: A universal framework for modeling topological relations. International Conference on Geographic Information Science, Springer.","DOI":"10.1007\/978-3-540-87473-7"},{"key":"ref_32","unstructured":"Adams, C.A., and Franzosa, R.D. (2008). Introduction to Topology: Pure and Applied, Pearson Prentice-Hall."},{"key":"ref_33","unstructured":"Agrawal, D., Zhang, P., El Abbadi, A., and Mokbel, M. (2010, January 2\u20135). 9+-intersection calculi for spatial reasoning on the topological relations between heterogeneous objects. Proceedings of the 18th SIGSPATIAL International Conference on Advances in Geographic Information Systems, San Jose, CA, USA."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"150","DOI":"10.3390\/ijgi14040150","article-title":"Conceptual neighborhood graphs of topological relations in Z2","volume":"14","author":"Hall","year":"2025","journal-title":"Int. J. Geo-Inf."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Fabrikant, S., Raubal, M., Bertolotto, M., Davies, C., and Freundschuh, S. (2015). Swiss canton regions: A model for complex objects in geographic partitions. International Conference on Spatial Information Theory, Springer.","DOI":"10.1007\/978-3-319-23374-1"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Tenbrink, T., Stell, J., Galton, A., and Wood, Z. (2013). The topology of spatial scenes in Z2. International Conference on Spatial Information Theory, Springer.","DOI":"10.1007\/978-3-319-01790-7"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Duckham, M., Stewart, K., and Pebesma, E. (2014). Oriented regions for linearly conceptualized features. International Conference on Geographic Information Science, Springer.","DOI":"10.1007\/978-3-319-11593-1"},{"key":"ref_38","unstructured":"Lewis, J.A. (2019). A Qualitative Representation of Spatial Scenes in R2with Regions and Lines. [Ph.D. Thesis, University of Maine]."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1080\/00029890.1979.11994873","article-title":"Digital topology","volume":"86","author":"Rosenfeld","year":"1979","journal-title":"Am. Math. Mon."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0095-8956(89)90027-0","article-title":"Discrete Jordan curve theorems","volume":"47","author":"Vince","year":"1989","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1006\/jvlc.1997.0071","article-title":"Modes of overlap","volume":"9","author":"Galton","year":"1998","journal-title":"J. Vis. Lang. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Mate, S., Burkle, T., Kapsner, L.A., Toddenroth, D., Kampf, M.O., Sedlmayr, M., Castellanos, I., Prokosch, H.U., and Kraus, S. (2019). A method for the graphical modeling of relative temporal constraints. J. Biomed. Inform., 100.","DOI":"10.1016\/j.jbi.2019.103314"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"35","DOI":"10.3233\/FI-2013-894","article-title":"Discrete duality for rough relation algebras","volume":"127","author":"Duntsch","year":"2013","journal-title":"Fundam. Informa."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s10707-009-0097-0","article-title":"Qualitative change detection using sensor networks based on connectivity information","volume":"15","author":"Jiang","year":"2011","journal-title":"Geoinformatica"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0004-3702(92)90090-K","article-title":"Temporal reasoning based on semi-intervals","volume":"54","author":"Freksa","year":"1992","journal-title":"Artif. Intell."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Fabrikant, S.I., Reichenbacher, T., van Kreveld, M.J., and Schlieder, C. (2010). The family of conceptual neighborhood graphs for region-region relations. International Conference on Geographic Information Science, Springer.","DOI":"10.1007\/978-3-642-15300-6"}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/14\/9\/327\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:35:36Z","timestamp":1760034936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/14\/9\/327"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,25]]},"references-count":46,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["ijgi14090327"],"URL":"https:\/\/doi.org\/10.3390\/ijgi14090327","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,25]]}}}