Spherical Topological Relations

  • Max J. Egenhofer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3534)


Analysis of global geographic phenomena requires non-planar models. In the past, models for topological relations have focused either on a two-dimensional or a three-dimensional space. When applied to the surface of a sphere, however, neither of the two models suffices. For the two-dimensional planar case, the eight binary topological relations between spatial regions are well known from the 9-intersection model. This paper systematically develops the binary topological relations that can be realized on the surface of a sphere. Between two regions on the sphere there are three binary relations that cannot be realized in the plane. These relations complete the conceptual neighborhood graph of the eight planar topological relations in a regular fashion, providing evidence for a regularity of the underlying mathematical model. The analysis of the algebraic compositions of spherical topological relations indicates that spherical topological reasoning often provides fewer ambiguities than planar topological reasoning. Finally, a comparison with the relations that can be realized for one-dimensional, ordered cycles draws parallels to the spherical topological relations.


Relation Algebra Topological Relation Universal Relation Topological Relationship Minimum Bounding Rectangle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allen, J.: Maintaining Knowledge about Temporal Intervals. Communications of the ACM 26(11), 832–843 (1983)zbMATHCrossRefGoogle Scholar
  2. 2.
    Balbiani, P., Osmani, A.: A Model for Reasoning about Topological Relations between Cyclic Intervals. In: Cohn, A., Giunchiglia, F., Selman, B. (eds.) Seventh International Conference on Principles of Knowledge Representation and Reasoning, KR2000, Breckenridge, CO, pp. 675–687. Morgan Kaufmann Publishers, San Mateo (2000)Google Scholar
  3. 3.
    Billen, R., Zlatanova, S., Mathonet, P., Bouvier, F.: The Dimensional Model: A Framework to Distinguish Spatial Relationships. In: Richardson, D., van Oosterom, P. (eds.) Advances in Spatial Data Handling: Tenth International Symposium on Spatial Data Handling, pp. 285–298. Springer, Berlin (2002)Google Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1967)zbMATHGoogle Scholar
  5. 5.
    Bruns, T., Egenhofer, M.: Similarity of Spatial Scenes. In: Kraak, M.-J., Molenaar, M. (eds.) Seventh International Symposium on Spatial Data Handling, Delft, The Netherlands, pp. 173–184. Taylor & Francis, London (1996)Google Scholar
  6. 6.
    Clementini, E., di Felice, P.: Approximate Topological Relations. International Journal of Approximate Reasoning 16(2), 173–204 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Clementini, E., di Felice, P., Califano, G.: Composite Regions in Topological Queries. Information Systems 20(7), 579–594 (1995)CrossRefGoogle Scholar
  8. 8.
    Clementini, E., di Felice, P., van Oosterom, P.: A Small Set of Formal Topological Relationships Suitable for End-User Interaction. In: Abel, D.J., Ooi, B.-C. (eds.) SSD 1993. LNCS, vol. 692, pp. 277–295. Springer, Heidelberg (1993)Google Scholar
  9. 9.
    Clementini, E., Sharma, J., Egenhofer, M.: Modelling Topological Spatial Relations: Strategies for Query Processing. Computers and Graphics 18(6), 815–822 (1994)CrossRefGoogle Scholar
  10. 10.
    Cohn, A., Gotts, N.: The "Egg-Yolk" Representation of Regions with Indeterminate Boundaries. In: Burrough, P., Frank, A. (eds.) Geographic Objects with Indeterminate Boundaries, pp. 171–187. Taylor & Francis, London (1996)Google Scholar
  11. 11.
    Cui, Z., Cohn, A., Randell, D.: Qualitative and Topological Relationships in Spatial Databases. In: Abel, D.J., Ooi, B.-C. (eds.) SSD 1993. LNCS, vol. 692, pp. 296–315. Springer, New York (1993)Google Scholar
  12. 12.
    Egenhofer, M.: Deriving the Composition of Binary Topological Relations. Journal of Visual Languages and Computing 5(2), 133–149 (1994a)Google Scholar
  13. 13.
    Egenhofer, M.: Pre-Processing Queries with Spatial Constraints. Photogrammetric Engineering & Remote Sensing 60(6), 783–790 (1994b)Google Scholar
  14. 14.
    Egenhofer, M., Al-Taha, K.: Reasoning About Gradual Changes of Topological Relationships. In: Frank, A.U., Formentini, U., Campari, I. (eds.) GIS 1992. LNCS, vol. 639, pp. 196–219. Springer, Berlin (1992)Google Scholar
  15. 15.
    Egenhofer, M., Franzosa, R.: Point-Set Topological Spatial Relations. International Journal of Geographical Information Systems 5(2), 161–174 (1991)CrossRefGoogle Scholar
  16. 16.
    Egenhofer, M., Franzosa, R.: On the Equivalence of Topological Relations. International Journal of Geographical Information Systems 9(2), 133–152 (1995)CrossRefGoogle Scholar
  17. 17.
    Egenhofer, M., Herring, J.: Categorizing Binary Topological Relationships Between Regions, Lines, and Points in Geographic Databases. In: Egenhofer, M., Herring, J., Smith, T., Park, K. (eds.) A Framework for the Definition of Topological Relationships and an Algebraic Approach to Spatial Reasoning within this Framework, NCGIA Technical Report 91-7, Santa Barbara, CA, National Center for Geographic Information and Analysis (1991)Google Scholar
  18. 18.
    Egenhofer, M., Mark, D.: Modeling Conceptual Neighbourhoods of Topological Line-Region Relations. International Journal of Geographical Information Systems 9(5), 555–565 (1995)CrossRefGoogle Scholar
  19. 19.
    Egenhofer, M., Di Felice, P., Clementini, E.: Topological Relations between Regions with Holes. International Journal of Geographical Information Systems 8(2), 129–144 (1994)CrossRefGoogle Scholar
  20. 20.
    Egenhofer, M., Sharma, J.: Assessing the Consistency of Complete and Incomplete Topological Information. Geographical Systems 1(1), 47–68 (1993)Google Scholar
  21. 21.
    Egenhofer, M., Sharma, J., Mark, D.: A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis. In: McMaster, R., Armstrong, M. (eds.) Autocarto 11, Minneapolis, MN, pp. 1–11 (1993)Google Scholar
  22. 22.
    Freksa, C.: Temporal Reasoning Based on Semi-Intervals. Artificial Intelligence 54, 199–227 (1992)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Gott, N.: Using the ’RCC’ Formalism to Describe the Topology of Spherical Regions. Technical Report 96.24, University of Leeds, Leeds (1996)Google Scholar
  24. 24.
    Hadzilacos, T., Tryfona, N.: A Model for Expressing Topological Integrity Constraints in Geographic Databases. In: Frank, A.U., Formentini, U., Campari, I. (eds.) GIS 1992. LNCS, vol. 639, pp. 252–268. Springer, Heidelberg (1992)Google Scholar
  25. 25.
    Hazelton, N.W., Bennett, L., Masel, J.: Topological Structures for 4-Dimensional Geographic Information Systems. Computers, Environment, and Urban Systems 16(3), 227–237 (1992)CrossRefGoogle Scholar
  26. 26.
    Hernández, D.: Qualitative Representation of Spatial Knowledge. Springer, New York (1994)zbMATHCrossRefGoogle Scholar
  27. 27.
    Hornsby, K., Egenhofer, M., Hayes, P.: Modeling Cyclic Change. In: Chen, P., Embley, D., Kouloumdjian, J., Liddle, S., Roddick, J. (eds.) Advances in Conceptual Modeling, Versailles, France. LNCS, vol. 1227, pp. 98–109. Springer, Berlin (1999)CrossRefGoogle Scholar
  28. 28.
    IBM, IBM Informix Geodetic DataBlade Module. User’s Guide, Version 3.11 White Planes, NY, IBM Corporation (2002), available from,
  29. 29.
    Mark, D., Egenhofer, M.: Modeling Spatial Relations Between Lines and Regions: Combining Formal Mathematical Models and Human Subjects Testing. Cartography and Geographic Information Systems 21(3), 195–212 (1994)Google Scholar
  30. 30.
    Papadias, D., Theodoridis, Y., Sellis, T., Egenhofer, M.: Topological Relations in the World of Minimum Bounding Rectangles: A Study with R-Trees. ACM SIGMOD 4(2), 92–103 (1995)CrossRefGoogle Scholar
  31. 31.
    Papadimitriou, C., Suciu, D., Vianu, V.: Topological Queries in Spatial Databases. In: Fifteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), Montreal, Canada, pp. 81–92. ACM Press, New York (1996)CrossRefGoogle Scholar
  32. 32.
    Pigot, S.: Topological Models for 3D Spatial Information Systems. In: Mark, D., White, D. (eds.) Autocarto 10, Baltimore, MD, pp. 368–392 (1991)Google Scholar
  33. 33.
    Pullar, D., Egenhofer, M.: Toward Formal Definitions of Topological relations Among Spatial Objects. In: Third International Symposium on Spatial Data Handling, Sydney, Australia, pp. 225–241 (1988)Google Scholar
  34. 34.
    Randell, D., Cui, Z., Cohn, A.: A Spatial Logic Based on Regions and Connection. In: Nebel, B., Rich, C., Swartout, W. (eds.) Principles of Knowledge Representation and Reasoning, KR 1992, Cambridge, MA, pp. 165–176 (1992)Google Scholar
  35. 35.
    Sharma, J.: Integrated Topological and Directional Reasoning in Geographic Information Systems. In: Craglia, M., Onsrud, H. (eds.) Geographic Information Research: Trans-Atlantic Perspectives, pp. 435–447. Taylor & Francis, London (1999)Google Scholar
  36. 36.
    Smith, T., Park, K.: Algebraic Approach to Spatial Reasoning. International Journal of Geographical Information Systems 6(3), 177–192 (1992)CrossRefGoogle Scholar
  37. 37.
    Tarski, A.: On the Calculus of Relations. The Journal of Symbolic Logic 6(3), 73–89 (1941)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Usery, L.: University Consortium for Geographic Information Science Research Priorities: Global Representation and Modeling (2002),
  39. 39.
    Winter, S.: Topological Relations between Discrete Regions. In: Egenhofer, M.J., Herring, J.R. (eds.) SSD 1995. LNCS, vol. 951, pp. 310–327. Springer, Berlin (1995)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Max J. Egenhofer
    • 1
  1. 1.National Center for Geographic Information and Analysis, Department of Spatial Information Science and Engineering, Department of Computer ScienceUniversity of MaineOronoUSA

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