{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:11:24Z","timestamp":1773655884620,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T00:00:00Z","timestamp":1609891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41971360"],"award-info":[{"award-number":["41971360"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>Refined topological relations play an important role in spatial database quality control. Currently, there is no unified and reasonable method to represent refined line\/region and line\/line topological relations in two-dimensional (2D) space. In addition, the existing independent line\/region and line\/line models have some drawbacks such as incomplete type discrimination and too many topological invariants. In this paper, a refined line\/region and line\/line topological relations are represented uniformly by the sequence, dimension, and topological type of the intersection components. To make the relevant definitions conform to the traditional cognitions in 2D Euclidean space, the (simple) spatial object is defined based on manifold topology, and the spatial intersection components are defined based on the whole-whole object intersection set. Then the topological invariant of node degree is introduced, and the adjacent point kinds (e.g., \u201cNull\u201d, \u201cOn\u201d, \u201cIn\u201d, and \u201cOut\u201d) are defined to distinguish the intersection component types. Excluding impossible and symmetrical types, 29 types of intersection-lines (including 21 between lines\/regions and 8 between lines\/lines), and 6 types of intersection-points (including 2 between lines\/regions and 4 between lines\/lines) are classified. On this basis, a node degree-based whole-whole object intersection sets (N-WWIS) model for refined line\/region and line\/line topological relations is presented, and it can be combined with the Euler number-based whole object intersection and difference (E-WID) model (coarse level) to form a hierarchical representation method of topological relations. Furthermore, a prototype system based on the N-WWIS model for automatic topological integrity checking is developed and some evaluation experiments are conducted with OpenStreetMap (OSM) data is presented based on the classification of intersection components. The experimental results show that the N-WWIS model will enable the geographic information systems (GIS) community to develop automated topological conflict checking and dealing tools for spatial data updates and quality control.<\/jats:p>","DOI":"10.3390\/ijgi10010015","type":"journal-article","created":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T09:15:16Z","timestamp":1609924516000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Refined Lines\/Regions and Lines\/Lines Topological Relations Model Based on Whole-Whole Objects Intersection Components"],"prefix":"10.3390","volume":"10","author":[{"given":"Xiaoguang","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry, Central South University, Ministry of Education, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongyuan","family":"He","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry, Central South University, Ministry of Education, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dongyang","family":"Hou","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry, Central South University, Ministry of Education, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rui","family":"Li","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry, Central South University, Ministry of Education, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heng","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry, Central South University, Ministry of Education, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,6]]},"reference":[{"key":"ref_1","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. Sci."},{"key":"ref_2","unstructured":"Egenhofer, M., and Herring, J. (1991). Categorizing Binary Topological Relationships between Regions, Lines, Points in Geographic Databases. A Framework for the Definitions of Topological Relationships and An Algebraic Approach to Spatial Reasoning within This Framework, NCGIA Technical Reports; National Center for Geographic Information and Analysis."},{"key":"ref_3","unstructured":"McMaster, R., and Armstrong, M. (1993). A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis. Autocarto 11, American Society for Photogrammetry and Remote Sensing."},{"key":"ref_4","first-page":"149","article-title":"A Comparison of Methods for Representing Topological relationships","volume":"3","author":"Clementini","year":"1995","journal-title":"Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/13658810151072831","article-title":"A Voronoi-based 9-intersection model for spatial relations","volume":"15","author":"Chen","year":"2001","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Shen, J., Zhou, T., and Chen, M. (2017). A 27-Intersection Model for Representing Detailed Topological Relations between Spatial Objects in Two-Dimensional Space. ISPRS Int. Geo-Inf., 6.","DOI":"10.3390\/ijgi6020037"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1080\/02693799508902056","article-title":"Modeling Conceptual Neighborhoods of Topological Line-Region Relations","volume":"9","author":"Egenhofer","year":"1995","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_8","unstructured":"Gottfried, B. (2007). The 9+ intersection for topological relationships between a directed line segment and a region. Workshop on Behavior and Monitoring Interpretation, CEUR-WS.org."},{"key":"ref_9","first-page":"13","article-title":"Enriching the semantics of the directed polyline\u2013polygon topological relationships: The DLP-intersection matrix","volume":"192","author":"Formica","year":"2017","journal-title":"J. Geogr. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1111\/tgis.12615","article-title":"Topological relations between a directed line and a directed region","volume":"24","author":"Shen","year":"2020","journal-title":"Trans. GIS"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s10707-006-0004-x","article-title":"Multi-level Topological Relations between Spatial Regions Based Upon Topological Invariants","volume":"11","author":"Deng","year":"2007","journal-title":"Geoinformatica"},{"key":"ref_12","unstructured":"Kaufmann, M., and San, M. (1992). A spatial logical based on regions and connection. Proceedings of the 3rd International Conference on Knowledge Representation and Reasoning, Cambridge, MA, USA, 25\u201329 October 1992, Springer."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Abel, D., and Ooi, B.C. (1993). Qualitative and Topological Relationships in Spatial Databases. Proceedings of the Third International Symposium on Advances in Spatial Databases, Singapore, 23\u201325 June 1993, Springer.","DOI":"10.1007\/3-540-56869-7"},{"key":"ref_14","first-page":"50","article-title":"A Voronoi-based spatial algebra for spatial relations","volume":"12","author":"Li","year":"2002","journal-title":"Prog. Nat. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1983","DOI":"10.1080\/13658816.2013.780607","article-title":"A Euler-number-based Topological Computation Model for Land Parcel Database Updating","volume":"27","author":"Zhou","year":"2013","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_16","first-page":"25","article-title":"A Hierarchical Representation of Line-Region Topological Relations","volume":"XXXVII","author":"Deng","year":"2008","journal-title":"Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/69.667085","article-title":"Topological invariants for lines","volume":"10","author":"Clementini","year":"1998","journal-title":"IEEE Trans. Knowl. Data. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/s12517-019-4273-8","article-title":"Detailed model of topological and metric relationships between a line and region","volume":"12","author":"Wu","year":"2019","journal-title":"Arab. J. Geosci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Li, Z.L., and Deng, M. (2006). A hierarchical approach to the line-line topological relations. Progress in Spatial Data Handling, Springer.","DOI":"10.1007\/3-540-35589-8_24"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1080\/02693799508902030","article-title":"On the Equivalence of Topological Relations","volume":"9","author":"Egenhofer","year":"1995","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1023\/A:1026570130172","article-title":"Basic Topological Models for Spatial Entities in 3-dimensional Space","volume":"4","author":"Li","year":"2000","journal-title":"GeoInformatica"},{"key":"ref_22","first-page":"264","article-title":"Extended model of topological relationships between spatial objects in geographic information systems","volume":"9","author":"Liu","year":"2007","journal-title":"Int. J. Appl. Earth Obs. Geoinf."},{"key":"ref_23","unstructured":"(2003). ISO. 19107, Geographic Information\u2013Spatial Schema, Technical Report."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Lee, J.M. (2011). Introduction to Topological Manifolds, Springer.","DOI":"10.1007\/978-1-4419-7940-7"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Tu, L.W. (2011). An Introduction to Manifolds, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-7400-6"},{"key":"ref_26","first-page":"1","article-title":"Connective Spaces","volume":"39","author":"Muscat","year":"2006","journal-title":"Mem. Fac. Sci. Eng. Shimane Univ."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1145\/1132863.1132865","article-title":"Topological relationships between complex spatial objects","volume":"31","author":"Schneider","year":"2006","journal-title":"ACM Trans. Database Sys."},{"key":"ref_29","unstructured":"Armstrong, M.A. (1979). Basic Topology, McGraw-Hill Book Company."},{"key":"ref_30","first-page":"128","article-title":"Solutio problematis ad geometriam situs pertinentis","volume":"8","author":"Euler","year":"1741","journal-title":"Comment. Acad. Sci. Petropolitanae"},{"key":"ref_31","unstructured":"Frank, H. (1969). Graph Theory, Addison-Wesley."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Graph Theory, Springer.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zhao, Y.J., Zhou, X.G., Li, G.Q., and Xing, H.F. (2016). A Spatio-Temporal VGI Model Considering Trust-Related Information. ISPRS Int. Geo-Inf., 5.","DOI":"10.3390\/ijgi5020010"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1707","DOI":"10.3390\/ijgi4031707","article-title":"Dynamic Integrating OSM data to Borderland Database","volume":"4","author":"Zhou","year":"2015","journal-title":"ISPRS Int. Geo-Inf."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Girindran, R., Boyd, D., Rosser, J., Vijayan, D., Long, G., and Robinson, D. (2020). On the Reliable Generation of 3D City Models from Open Data. Urban Sci., 4.","DOI":"10.3390\/urbansci4040047"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Giovanella, A., Bradley, P.E., and Wursthorn, S. (2019). Evaluation of Topological Consistency in CityGML. ISPRS Int. J. Geo-Inf., 8.","DOI":"10.3390\/ijgi8060278"}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/10\/1\/15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:07:29Z","timestamp":1760159249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/10\/1\/15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,6]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["ijgi10010015"],"URL":"https:\/\/doi.org\/10.3390\/ijgi10010015","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,6]]}}}