A Model for Ternary Projective Relations between Regions

  • Roland Billen
  • Eliseo Clementini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2992)


Current spatial database systems offer limited querying capabilities beyond topological relations. This paper introduces a model for projective relations between regions to support other qualitative spatial queries. The relations are ternary because they are based on the collinearity invariant of three points under projective geometry. The model is built on a partition of the plane in five regions that are obtained from projective properties of two reference objects: then, by considering the empty/non empty intersections of a primary object with these five regions, the model is able to distinguish between 31 different projective relations.


Convex Hull Projective Geometry Reference Object Projective Property Spatial Object 
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.
    Clementini, E., Di Felice, P.: Spatial Operators. ACM SIGMOD Record 29(3), 31–38 (2000)CrossRefGoogle Scholar
  2. 2.
    Clementini, E., Di Felice, P., Hernández, D.: Qualitative representation of positional information. Artificial Intelligence 95, 317–356 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    Dugat, V., Gambarotto, P., Larvor, Y.: Qualitative Theory of Shape and Orientation. In: Proc. of the 16th Int. Joint Conference on Artificial Intelligence (IJCAI 1999), pp. 45–53. Morgan Kaufmann Publishers, Stockolm (1999)Google Scholar
  5. 5.
    Egenhofer, M.J.: Deriving the composition of binary topological relations. Journal of Visual Languages and Computing 5(1), 133–149 (1994)CrossRefGoogle Scholar
  6. 6.
    Egenhofer, M.J., Herring, J.R.: Categorizing Binary Topological Relationships Between Regions, Lines, and Points in Geographic Databases. In: Department of Surveying Engineering, University of Maine, Orono (1991)Google Scholar
  7. 7.
    Freksa, C.: Using Orientation Information for Qualitative Spatial Reasoning. In: Frank, A.U., Campari, I., Formentini, U. (eds.) Theories and Models of Spatio-Temporal Reasoning in Geographic Space, pp. 162–178. Springer, Berlin (1992)Google Scholar
  8. 8.
    Gapp, K.-P.: Angle, Distance, Shape, and their Relationship to Projective Relations. In: Proceedings of the 17th Conference of the Cognitive Science Society, Pittsburgh, PA (1995)Google Scholar
  9. 9.
    Gapp, K.-P.: From Vision to Language: A Cognitive Approach to the Computation of Spatial Relations in 3D Space. In: Proc. of the First European Conference on Cognitive Science in Industry, Luxembourg, pp. 339–357 (1994)Google Scholar
  10. 10.
    Goyal, R., Egenhofer, M.J.: Cardinal directions between extended spatial objects. IEEE Transactions on Knowledge and Data Engineering (2003) (in press)Google Scholar
  11. 11.
    Hernández, D.: Qualitative Representation of Spatial Knowledge. LNCS (LNAI), vol. 804. Springer, Heidelberg (1994)zbMATHCrossRefGoogle Scholar
  12. 12.
    Isli, A.: Combining Cardinal Direction Relations and other Orientation Relations in QSR. In: AI&M 14-2004, Eighth International Symposium on Artificial Intelligence and Mathematics, Fort Lauderdale, Florida, January 4-6 (2004)Google Scholar
  13. 13.
    Kray, C., Blocher, A.: Modeling the Basic Meanings of Path Relations. In: Proc. of the 16th International Joint Conference on Artificial Intelligence (IJCAI 1999), pp. 384–389. Morgan Kaufmann Publishers, Stockolm (1999)Google Scholar
  14. 14.
    Kulik, L., et al.: A graded approach to directions between extended objects. In: Proc. of the 2nd Int. Conf. on Geographic Information Science, pp. 119–131. Springer, Boulder (2002)Google Scholar
  15. 15.
    Kulik, L., Klippel, A.: Reasoning about Cardinal Directions Using Grids as Qualitative Geographic Coordinates. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 205–220. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    Moratz, R., Fischer, K.: Cognitively Adequate Modelling of Spatial Reference in Human-Robot Interaction. In: Proc. of the 12th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2000, Vancouver, BC, Canada, pp. 222–228 (2000)Google Scholar
  17. 17.
    OpenGIS Consortium, OpenGIS Simple Features Specification for SQL (1998)Google Scholar
  18. 18.
    Retz-Schmidt, G.: Various Views on Spatial Prepositions. AI Magazine 9(2), 95–105 (1988)Google Scholar
  19. 19.
    Schlieder, C.: Reasoning about ordering. In: Kuhn, W., Frank, A.U. (eds.) COSIT 1995. LNCS, vol. 988, pp. 341–349. Springer, Heidelberg (1995)Google Scholar
  20. 20.
    Schmidtke, H.R.: The house is north of the river: Relative localization of extended objects. In: Montello, D.R. (ed.) COSIT 2001. LNCS, vol. 2205, pp. 415–430. Springer, Heidelberg (2001)Google Scholar
  21. 21.
    Scivos, A., Nebel, B.: Double-Crossing: Decidability and Computational Complexity of a Qualitative Calculus for Navigation. In: Montello, D.R. (ed.) COSIT 2001. LNCS, vol. 2205, pp. 431–446. Springer, Heidelberg (2001)Google Scholar
  22. 22.
    Struik, D.J.: Projective Geometry. Addison-Wesley, London (1953)zbMATHGoogle Scholar
  23. 23.
    Vorwerg, C., et al.: Projective relations for 3D space: Computational model, application, and psychological evaluation. In: Proc. of the 14th National Conference on Artificial Intelligence and 9th Innovative Applications of Artificial Intelligence Conference, AAAI 1997, IAAI 1997, pp. 159–164. AAAI Press / The MIT Press, Providence, Rhode Island (1997)Google Scholar
  24. 24.
    Waller, D., et al.: Place learning in humans: The role of distance and direction information. Spatial Cognition and Computation 2, 333–354 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Roland Billen
    • 1
  • Eliseo Clementini
    • 2
  1. 1.Dept. of Geography and GeomaticsUniversity of GlasgowGlasgow, ScotlandUK
  2. 2.Dept. of Electrical EngineeringUniversity of L’AquilaPoggio di Roio (AQ)Italy

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