Projective Relations in a 3D Environment

  • Roland Billen
  • Eliseo Clementini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4197)


This paper presents a model for positional relations among bodies of arbitrary shape in three dimensions. It is based on an existing model for projective relations among regions in two dimensions. The motivation is to provide a formal qualitative spatial relations model for emerging 3D applications. Two sets of relations are defined: ternary projective relations based on the concept of collinearity between a primary object and two reference objects and quaternary projective relations based on the concept of coplanarity between a primary object and three reference objects. Four sets of JEPD relations are defined for points and bodies in R 3.


Convex Hull Augmented Reality Tangent Plane Projective Geometry Reference 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bartie, P.J., Mackaness, W.A.: Development of a Speech-Based Augmented Reality System to Support Exploration of Cityscape. Transactions in GIS 10(1), 63–86 (2006)CrossRefGoogle Scholar
  2. 2.
    Bennett, B., et al.: Region-based qualitative geometry, University of Leeds, School of Computer Studies, LS2 9JT, UK. Technical Report 2000.07 (2000)Google Scholar
  3. 3.
    Billen, R., Clementini, E.: Introducing a reasoning system based on ternary projective relations. In: Developments in Spatial Data Handling, 11th International Symposium on Spatial Data Handling, Leicester, UK, pp. 381–394. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Billen, R., Clementini, E.: A model for ternary projective relations between regions. In: Bertino, E., Christodoulakis, S., Plexousakis, D., Christophides, V., Koubarakis, M., Böhm, K., Ferrari, E. (eds.) EDBT 2004. LNCS, vol. 2992, pp. 310–328. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Billen, R., Clementini, E.: Semantics of collinearity among regions. In: Meersman, R., Tari, Z., Herrero, P. (eds.) OTM-WS 2005. LNCS, vol. 3762, pp. 1066–1076. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Clementini, E., Billen, R.: Modeling and computing ternary projective relations between regions. IEEE Transactions on Knowledge and Data Engineering 18(6), 799–814 (2006)CrossRefGoogle Scholar
  7. 7.
    Coxeter, H.S.M.: Projective Geometry, 2nd edn. Springer, New York (1987)MATHGoogle Scholar
  8. 8.
    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
  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.
    Hernández, D.: Qualitative Representation of Spatial Knowledge. LNCS (LNAI), vol. 804. Springer, Heidelberg (1994)MATHCrossRefGoogle Scholar
  11. 11.
    Klein, F.: Vergleichende Betrachtungen über neuere geometrische Forschungen. Bulletin of the New York Mathematical Society 2, 215–249 (1893)CrossRefGoogle Scholar
  12. 12.
    Lewis, T., von Hohenbalken, B., Klee, V.: Common supports as fixed points. Geometriae Dedicata 60(3), 277–281 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ligozat, G.F.: Qualitative Triangulation for Spatial Reasoning. In: Campari, I., Frank, A.U. (eds.) COSIT 1993. LNCS, vol. 716, pp. 54–68. Springer, Heidelberg (1993)Google Scholar
  14. 14.
    Scivos, A., Nebel, B.: The Finest of its Class: The Natural Point-Based Ternary Calculus for Qualitative Spatial Reasoning. In: Int. Conf. Spatial Cognition, pp. 283–303. Springer, Heidelberg (2004)Google Scholar
  15. 15.
    Wenger, R.: Progress in geometric transversal theory. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry, pp. 375–393. Amer. Math. Soc., Providence (1998)Google Scholar
  16. 16.
    Zlatanova, S.: 3D GIS for Urban Development, University of Graz - ITC (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Roland Billen
    • 1
  • Eliseo Clementini
    • 2
  1. 1.Geomatics UnitUniversity of LiegeLiegeBelgium
  2. 2.Dept. of Electrical and Information EngineeringUniversity of L’AquilaL’AquilaItaly

Personalised recommendations