A Formalization of Topological Relations Between Simple Spatial Objects

  • Gutemberg Guerra- Filho
  • Claudia Bauzer Medeiros
  • Pedro J. de Rezende
Part of the Advances in Geographic Information Science book series (AGIS)


This paper presents a new framework for modeling topological relations among objects of type point, line, and region. The main contributions are in two directions: First, the formalism proposed allows specifying all possible relations by means of symmetric matrices (whereas the usual formulation of such relations does not have this property). Symmetric matrices enable the efficient and automatic verification of valid matrices associated only with the possible topological relations. Second, it allows the specification of cases where two objects are spatially related in more than one way (e.g., a line that crosses a given region in one part and is adjacent to the same region region in another part). This increases the flexibility that users are offered to model queries on spatial databases using topological relations.


Topological relations 3-axis-Intersection Conceptual neighborhood diagram 


  1. Alboody A, Inglada J, Sedes F (2009) Enriching the spatial reasoning system RCC8. SIGSPATIAL Spec 1(1):14–20Google Scholar
  2. Alexandroff P (1961) Elementary concepts of topology. Dover Publications, Inc., New YorkGoogle Scholar
  3. Clementini E, Felice PD (1995) A comparison of methods for representing topological relationships. Inf Sci Appl 3(3):149–178Google Scholar
  4. Clementini E, Felice PD, van Oosterom P (1992) A small set of formal topological relationships suitable for end-user interaction. Technical Report, University of L’Aquilla, ItalyGoogle Scholar
  5. Clementini E, Sharma J, Egenhofer M (1994) Modelling topological spatial relations: strategies for query processing. Comput Graph 18(6):815–822Google Scholar
  6. Deng M, Cheng T, Chen X, Li Z (2007) Multi-level topological relations between spatial regions based upon topological invariants. GeoInformatica 11(2):239–267Google Scholar
  7. Egenhofer M (1991) Extending SQL for cartographic display. Cartogr Geogr Inf Syst 18(4):230–245Google Scholar
  8. Egenhofer M, Al-Taha K (1992) Reasoning about gradual changes of topological relationships. In: Frank ICA, Formentini U (eds) Theories and methods of spatio-temporal reasoning in geographic space. Springer, Pisa, pp 196–219Google Scholar
  9. Egenhofer M, Franzosa R (1991) Point-set topological spatial relations. Int J Geogr Inf Syst 5(2):161–174Google Scholar
  10. Egenhofer M, Herring J (1990) A mathematical framework for the definition of topological relationships. In: Brassel K, Kishimoto H (eds) Proceedings fourth international symposium on spatial data handling, vol 2. International geographical union, Zurich, Switzerland, pp 803–813Google Scholar
  11. Egenhofer M, Herring J (1991) Categorizing binary topological relationships between regions, lines, and points in geographic databases. Technical report, Department of Surveying Engineering, University of Maine, OronoGoogle Scholar
  12. Egenhofer M, Herring J (1991) Categorizing topological spatial relations between point, line, and area objects. Technical Report, University of MaineGoogle Scholar
  13. Egenhofer M, Mark D (1995) Modelling conceptual neighbourhoods of topological line-region relations. Int J Geogr Inf Syst 9(5):555–565Google Scholar
  14. Frank A (1982) Mapquery–database query language for retrieval of geometric data and its graphical representation. ACM Comput Graph 16(3):199–207Google Scholar
  15. Freksa C (1992) Temporal reasoning based on semi-intervals. Artif Intell 54:199–227Google Scholar
  16. Herring J (1991) The mathematical modeling of spatial and non-spatial information in geographic information systems. In: Mark D, Frank A (eds) Cognitive and linguistic aspects of geographic space. Kluwer Academic Publishers, Dordrecht, pp 313–350Google Scholar
  17. Herring J, Larsen R, Shivakumar J (1988) Extensions to the SQL language to support spatial analysis in a topological database. In: Proceedings GIS/LIS’88, San Antonio, TX, pp 741–750Google Scholar
  18. Kurata Y (2009) Semi-automated derivation of conceptual neighborhood graphs of topological relations. In: Proceedings of the international conference on spatial information theoryGoogle Scholar
  19. Kurata Y, Egenhofer M (2007) The \(9^{+}\)-intersection for topological relations between a directed line segment and a region. In: Proceedings of the workshop on behavioral monitoring and interpretation, vol 42, pp 62–76Google Scholar
  20. McKenney M, Pauly A, Praing R, Schneider M (2005) Dimension-refined topological predicates. In: Proceedings GIS’05, Bremen, GermanyGoogle Scholar
  21. Papadias D, Egenhofer M, Sharma J (1996) Hierarchical reasoning about direction relations. In: Proccedings 4th ACM workshop on advances in, GIS, pp 107–114Google Scholar
  22. Pigot S (1991) Topological models for 3d spatial information systems. In: Mark D, White D (eds) Proceedings of autocarto 10 (Falls church: american society for photogrammetry and remote sensing, pp 368–392Google Scholar
  23. Raper J, Bundock M (1991) UGIX: A layer-based model for a GIS user interface. In: Mark D, Frank A (eds) Cognitive and linguistic aspects of geographic space. Kluwer Academic Publishers, Dordrecht, pp 449–476Google Scholar
  24. Roussopoulos N, Faloutsos C, Sellis T (1988) An efficient pictorial database system for PSQL. IEEE Trans Software Eng 14(5):630–638Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gutemberg Guerra- Filho
    • 1
  • Claudia Bauzer Medeiros
    • 2
  • Pedro J. de Rezende
    • 2
  1. 1.Department of Computer Science and EngineeringUniversity of Texas at ArlingtonArlingtonUSA
  2. 2.Institute of ComputingUNICAMPBrazil

Personalised recommendations