, Volume 1, Issue 4, pp 345–367 | Cite as

Achieving Integrity in Geographic Information Systems—Maps and Nested Maps

  • Lutz Plümer
  • Gerhard Gröger


This article provides a formal data model which allows to establish geometrical-topological integrity of areal objects in a geographical information system (GIS). The data model leads to an automatic tool able to check consistency of a given set of data and to avoid inconsistencies caused by updates of the database. To this end we start from the mathematical notion of a map which provides an irregular tessellation, i.e., a partition of the plane which is non-overlapping and covering. From another perspective, a map is a plane graph with an explicit representation of faces as its atomic areal components. The concept of nested maps extends this standard notion by the specification of a hierarchical structure which aggregates the set of faces. Such aggregations are common in political and administrative structures. Whereas the mathematical notion of a map is familiar in GIS and the base for many tools supporting topological editing, there was a lack of effectively checkable integrity constraints which are correct and complete, i.e., equivalent, for maps. This article provides an axiomatic, effectively checkable characterization of maps which is equivalent to the standard mathematical one, extends it to nested maps and discusses how to use them in order to achieve and maintain integrity in a GIS.

spatial data models topology irregular tessellation combinatorial maps nested maps aggregation correctness completeness of axioms graph theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.A. Armstrong. Basic Topology, McGraw-Hill: Maidenhead, 1979.Google Scholar
  2. 2.
    J.P. Corbett. “Topological Principles in Cartography,” Technical Paper No. 48, US Bureau of the Census, US Government Printing Office: Washington, DC, USA, 1979.Google Scholar
  3. 3.
    M.J. Egenhofer, A. Frank, and J.P. Jackson. “A Topological Data Model for Spatial Databases,” Proc. of the International Symposium on Spatial Databases (SSD 89), LNCS No. 409, Springer-Verlag: Santa Barbara, CA, pp. 189–211, 1989.Google Scholar
  4. 4.
    M.J. Egenhofer and J. Sharma. “Topological Consistency,” in Proc. 5th Int. Symp. on Spatial Data Handling, P. Presnahan, E. Corwin, and D. Cowen (Eds.), Charleston, South Carolina, Vol. 1, pp. 335–343, 1992.Google Scholar
  5. 5.
    S. Even. Graph Theory, Computer Science Press, 1979.Google Scholar
  6. 6.
    A. Frank and W. Kuhn. “Cell Graphs: A provable correct method for the storage of geometry,” Proc. of the Second Int. Symp. on Spatial Data Handling, Seattle, 1986.Google Scholar
  7. 7.
    G. Gröger and L. Plümer. “Provably Correct and Complete Transaction Rules for GIS,” Proc. of the 5th ACM Workshop on Geographic Information Systems, Las Vegas, Nevada, USA, November 13–14, ACM Press, 1997.Google Scholar
  8. 8.
    D. Greene and F. Yao. “Finite resolution computational geometry,” Proc. of the 27 th IEEE Symp. on the Foundations of Computer Science, IEEE: New York, pp. 143–152, 1986.Google Scholar
  9. 9.
    R. Güting, “An Introduction to Spatial Database Systems,” The Int. Journal on Very Large Databases, Vol. 3,No. 4, October 1994.Google Scholar
  10. 10.
    R. Güting and M. Schneider. “Realms: A foundation for spatial data types in database systems,” Proc. of the Third Int. Symp. on Large Spatial Databases, Singapore, 1993.Google Scholar
  11. 11.
    F. Harary. Graph Theory, Addison-Wesley, 1969.Google Scholar
  12. 12.
    E.U. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979.Google Scholar
  13. 13.
    W. Kainz. “Spatial Relationships—topology versus order,” in Proc. Fourth Int. Symp. on Spatial Data Handling, edited by K. Brassel and H. Kishimoto, Zurich, Switzerland, Vol. 2, pp. 814–819, 1990.Google Scholar
  14. 14.
    W. Kainz. Logical Consistency, in S.C. Guptill and J.L. Morrison (Eds.), Elements of Spatial Data Quality, Elsevier, 1995.Google Scholar
  15. 15.
    O. Kufoniyi, M. Molenaar, and T. Bouloucos. Topologic Consistency Operations in Vector Maps, GIS 4, pp. 7–13, 1995.Google Scholar
  16. 16.
    O. Kufoniyi. Spatial coincidence modelling, automated database updating and data consistency in vector GIS, Int. Institute for Aerospace Survey and Earth Science (ITC), Publication No. 28, Enschede, The Netherlands, 1995.Google Scholar
  17. 17.
    R. Laurini and F. Milleret-Raffort. “Using integrity constraints for checking consistency of spatial databases,” GIS/LIS 1991 Proceedings, Atlanta, Georgia, USA, pp. 634–642, 1991.Google Scholar
  18. 18.
    R. Laurini and D. Thompson. Fundamentals of Spatial Information Systems, Academic Press, 1992.Google Scholar
  19. 19.
    R.W. Marx and A.J. Saalfeld. “Programs for assuring map quality at the Bureau of the Census.” Fourth Annual Research Conf., US Bureau of the Census, Arlington, Virginia, US Government Printing Office: Washington DC, USA, 1988.Google Scholar
  20. 20.
    M. Molenaar. Formal data structures, object dynamics and consistency rules, in (Eds.) Ebner, Fritsch, Heipke, Wichmann, Digital Photogrammetric Systems, 1991.Google Scholar
  21. 21.
    M. Molenaar. Spatial Concepts as Implemented in GIS, in A.U. Frank (Ed.) Geographic Information Systems—Materials for a Post-Graduate Course, Vol. 1: Spatial Information, Department of Geoinformation, Technical University Vienna, Austria, 1995.Google Scholar
  22. 22.
    J. Nunes. General Concepts of Space and Time, in A.U. Frank (Ed.) Geographic Information Systems—Materials for a Post-Graduate Course, Vol. 1: Spatial Information, Department of Geoinformation, Technical University Vienna, Austria, 1995.Google Scholar
  23. 23.
    L. Plümer and G. Gröger. “Nested Maps—a Formal, Provably Correct Object Model for Spatial Aggregates,” Proc. of the fourth ACM Workshop on Advances in Geographic Information Systems, pp. 77–84, Rockville, Maryland, November 15–16, ACM Press, 1996.Google Scholar
  24. 24.
    L. Plümer. “Achieving Integrity of Geometry and Topology in Geographical Information Systems,” Proc. of the “SAMOS” Int. Conf. on Geographic Information Systems in Urban, Environmental and Regional Planning, Island of Samos, Greece, April 19–21, 1996.Google Scholar
  25. 25.
    F.P. Preparata and M.I. Shamos. Computational Geometry, Springer, 1985.Google Scholar
  26. 26.
    J. Rumbaugh, M. Blaha, W. Premerlani, F. Eddy, and W. Lorensen. Object-Oriented Modeling and Design, Prentice Hall, 1991.Google Scholar
  27. 27.
    J.R. Shoenfield, Mathematical Logic, Addison-Wesley, 1979.Google Scholar
  28. 28.
    J.D. Ullman. Principles of Database and Knowledge-Base Systems, Volumes I and II, Computer Science Press, 1989.Google Scholar
  29. 29.
    J. Widom and S. Ceri. Active Database Systems, Morgan Kaufmann, San Francisco, 1996.Google Scholar
  30. 30.
    M.F. Worboys and P. Bofakos. A canonical model for a class of areal spatial objects, in D. Abel and B.C. Ooi (Eds.) Advances in Spatial Databases, Proc. of SSD'93, Singapore, LNCS No. 692, Springer-Verlag, pp. 36–52, 1993.Google Scholar
  31. 31.
    M.F. Worboys. GIS—A Computing Perspective, Taylor and Francis, 1995.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Lutz Plümer
    • 1
  • Gerhard Gröger
    • 2
  1. 1.Institut für Informatik IIIUniversität BonnGermany
  2. 2.Institut für Kartographie und TopographieUniversität BonnGermany

Personalised recommendations