Mathematically provable correct implementation of integrated 2D and 3D representations

  • Rodney Thompson
  • Peter van Oosterom
Part of the Lecture Notes in Geoinformation and Cartography book series (LNGC)


The concept of the ‘Regular Polytope’ has been designed to facilitate the search for a rigorous closed algebra for the query and manipulation of the representations of spatial objects within the finite precision of a computer implementation. It has been shown to support a closed, complete and useful algebra of connectivity, and support a topology, without assuming the availability of infinite precision arithmetic. This paper explores the practicalities of implementing this approach both in terms of the database schema and in terms of the algorithmic implementation of the connectivity and topological predicates and functions. The problem domains of Cadastre and Topography have been chosen to illustrate the issues.


Half Space Half Plane Spatial Object Convex Polytopes Regular Polytope 
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. Arens, C., J. Stoter and P. van Oosterom (2003). Modelling 3D Spatial Objects in a Geo-DBMS Using A 3D Primitive. Association Geographic Information Laboritories Europe, Lyon, France.Google Scholar
  2. Borgo, S., N. Guarino and C. Masolo (1996). A Pointless Theory of Space Based On Strong Connection and Congruence. 6th International Conference on Principles of Knowledge Representation and Reasoning (KR96), Morgan Kaufmann.Google Scholar
  3. Castellanos, D. (1988). ‘The Ubiquitous pi (Part II)’. Mathematics Magazine 61(3): 148–161.CrossRefGoogle Scholar
  4. Clementini, E., P. Di Felice and P. van Oosterom (1993). A Small Set of Formal Topological Relationships Suitable for End-User Interaction. Third International Symposium on Advances in Spatial Databases, Singapore.Google Scholar
  5. Düntsch, I. and M. Winter (2004). ‘Algebraization and Representation of Mereotopological Structures.’ Relational Methods in Computer Science 1: 161–180.Google Scholar
  6. Egenhofer, M. J. and J. R. Herring (1994). Categorising binary topological relations between regions, lines, and points in geographic databases. The nine intersection: formalism and its use for naturallanguage spatial predicates. M. J. Egenhofer, D. M. Mark and J. R. Herring, University of California.Google Scholar
  7. Ellul, C. and M. Haklay (2005). Deriving a Generic Topological Data Structure for 3D Data. Topology and Spatial Databases Workshop, Glasgow, UK.Google Scholar
  8. Franklin, W. R. (1984). ‘Cartographic errors symptomatic of underlying algebra problems’. International Symposium on Spatial Data Handling, Zurich, Switzerland: 190–208.Google Scholar
  9. Guttman, A. (1984). ‘R-Trees: A Dynamic Index Structure for Spatial Searching’ ACM SIGMOD 13: 47–57.CrossRefGoogle Scholar
  10. Hölbling, W., W. Kuhn and A. U. Frank (1998). ‘Finite-Resolution Simplical Complexes.’ Geoinformatica 2:3: 281–298.CrossRefGoogle Scholar
  11. Kazar, B. M., R. Kothuri, P. van Oosterom and S. Ravada (2007). On Valid and Invalid Three-Dimensional Geometries. In this book ‘2nd International Workshop on 3D Geo-Information: Requirements, Acquisition, Modelling, Analysis, Visualisation, 12–14 December 2007, Delft, the Netherlands’.Google Scholar
  12. Naimpally, S. A. and B. D. Warrack (1970). Proximity Spaces. University Press, Cambridge.Google Scholar
  13. OMG. (1997). ‘UML 1.5’. Retrieved 2004 from
  14. Randell, D. A., Z. Cui and A. G. Cohn (1992). A spatial logic based on regions and connection. 3rd International Conference on Principles of Knowledge Representation and Reasoning, Cambridge MA, USA, Morgan Kaufmann.Google Scholar
  15. Smith, B. (1997). Boundaries: An Essay in Mereotopology. The Philosophy of Roderick Chisholm. L. Hahn, LaSalle: Open Court: 534–561.Google Scholar
  16. Stoter, J. (2004). 3D Cadastre. PhD Thesis. Delft, Delft University of Technology.Google Scholar
  17. Stoter, J. and P. van Oosterom (2006). 3D Cadastre in an International Context. Taylor & Francis, Boca Raton FL.Google Scholar
  18. Tarbit, S. and R. J. Thompson (2006). Future Trends for Modern DCDB’s, a new Vision for an Existing Infrastructure. Combined 5th Trans Tasman Survey Conference and 2nd Queensland Spatial Industry Conference. Cairns, Queensland, Australia.Google Scholar
  19. Thompson, R. J. (2004). 3D Topological Framework for Robust Digital Spatial Models. Directions Magazine.Google Scholar
  20. Thompson, R. J. (2005a). 3D Framework for Robust Digital Spatial Models. Large-Scale 3D Data Integration. S. Zlatanova and D. Prosperi. Boca Raton, FL, Taylor & Francis.Google Scholar
  21. Thompson, R. J. (2005b). 3D Cadastral Issues Within NR&M. Brisbane, Department of Natural Resources and Mines (Internal Report).Google Scholar
  22. Thompson, R. J. (2005c). ‘Proofs of Assertions in the Investigation of the Regular Polytope’. Retrieved 2 Feb 2007 from
  23. Thompson, R. J. (2007). Towards a Rigorous Logic for Spatial Data Representation. Geo Database Management Centre. Delft, Delft University of Technology. PhD Thesis.Google Scholar
  24. Thompson, R. J. and P. van Oosterom (2007). ‘Connectivity in the Regular Polytope Representation.’ submitted to GeoInformatica.Google Scholar
  25. van Oosterom, P., W. Quak and T. Tijssen (2004). About Invalid, Valid and Clean Polygons. Developments In Spatial Data Handling. P. F. Fisher. New York, Springer-Verlag: 1–16.Google Scholar
  26. Verbree, E., A. van der Most, W. Quak and P. van Oosterom (2005). Overlay of 3D features within a tetrahedral mesh: A complex algorithm made simple. Auto Carto 2005, Las Vegas.Google Scholar
  27. Weisstein, E. W. (1999). ‘Boolean Algebra’. MathWorld — A Wolfram Web Resource Retrieved 20 Jan 2007 from
  28. Weisstein, E. W. (2005). ‘Rational Number’. MathWorld — A Wolfram Web Resource Retrieved 23 May 2005 from
  29. Zlatanova, S. (2000). 3D GIS for Urban Development. Graz, Graz University of Technology.Google Scholar
  30. Zlatanova, S., A. A. Rahman and W. Shi (2004). ‘Topological models and frameworks for 3D spatial objects’. Journal of Computers & Geosciences 30(4): 419–428.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Rodney Thompson
    • 1
    • 2
  • Peter van Oosterom
    • 1
  1. 1.OTB, section GIS TechnologyDelft University of Technologythe Netherlands
  2. 2.Department of Natural Resources and WaterAustralia

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