Advertisement

GeoInformatica

, Volume 15, Issue 1, pp 137–165 | Cite as

How to achieve consistency for 3D city models

  • Gerhard GrögerEmail author
  • Lutz Plümer
Article

Abstract

Consistency is a crucial prerequisite for a large number of relevant applications of 3D city models, which have become more and more important in GIS. Users need efficient and reliable consistency checking tools in order to be able to assess the suitability of spatial data for their applications. In this paper we provide the theoretical foundations for such tools by defining an axiomatic characterization of 3D city models. These axioms are effective and efficiently supported by recent spatial database management systems and methods of Computational Geometry or Computer Graphics. They are equivalent to the topological concept of the 3D city model presented in this paper, thereby guaranteeing the reliability of the method. Hence, each error is detected by the axioms, and each violation of the axioms is in fact an error. This property, which is proven formally, is not guaranteed by existing approaches. The efficiency of the method stems from its locality: in most cases, consistency checks can safely be restricted to single components, which are defined topologically. We show how a 3D city model can be decomposed into such components which are either topologically equivalent to a disk, a sphere, or a torus, enabling the modeling of the terrain, of buildings and other constructions, and of bridges and tunnels, which are handles from a mathematical point of view. This enables a modular design of the axioms by defining axioms for each topological component and for the aggregation of the components. Finally, a sound, consistent concept for aggregating features, i.e. semantical objects like buildings or rooms, to complex features is presented.

Keywords

3D city models CityGML Consistency constraints 3D surfaces 2-manifolds Solids 

Notes

Acknowledgements

We thank Gerrit Kowatsch for assistance in preparing the illustrations and Jan Prinz for proof-reading.

References

  1. 1.
    Aleksandroff PS (1961) Elementary concepts of topology. Dover, New YorkGoogle Scholar
  2. 2.
    Armstrong MA (2005) Basic topology, 1st edn. Springer, New YorkGoogle Scholar
  3. 3.
    Bungartz HJ, Griebel M, Zenger C (2002) Einführung in die Computergraphik. Grundlagen, Geometrische Modellierung, Algorithmen. 2nd edn, Vieweg, (in German)Google Scholar
  4. 4.
    Booch G, Rumbaugh J, Jacobson I (1997) Unified modeling language user guide. Addison-Wesley, ReadingGoogle Scholar
  5. 5.
    Coors V (2003) 3D-GIS in networking environments. Comput Environ Urban Syst 27(4):345–357. doi: 10.1016/S0198-9715(02)00035-2 CrossRefGoogle Scholar
  6. 6.
    Dijkstra EW (1959) A note on two problems in connexion with graphs. Numerische Math 1:269–271. doi: 10.1007/BF01386390 CrossRefGoogle Scholar
  7. 7.
    ECMA (2005) Universal 3D File Format, Standard ECMA-363, 2nd edn. http://www.ecma-international.org/publications/files/ECMA-ST/ECMA-363.pdf
  8. 8.
    Egenhofer MJ, Franzosa RD (1991) Point-set topological spatial relations. Int J Geogr Inf Syst 5(2):161–174. doi: 10.1080/02693799108927841 CrossRefGoogle Scholar
  9. 9.
    Ellul C, Haklay M (2006) Requirements for topology in 3D GIS. Trans GIS 10(2):157–175. doi: 10.1111/j.1467-9671.2006.00251.x CrossRefGoogle Scholar
  10. 10.
    Emgård L, Zlatanova S (2008) Implementation alternatives for an integrated 3D Information Model. In: van Oosterom P, Zlatanova S, Penninga F, Fendel E (eds) Advances in 3D geoinformation systems, Lecture Notes in Geoinformation and Cartography, Chapter 17. Springer, pp 313–329Google Scholar
  11. 11.
    Flick S (1999) Konzeption eines adaptiven frameworks für 3D-GIS. PhD thesis. Fraunhofer IRB Verlag, Stuttgart. (in German)Google Scholar
  12. 12.
    Foley A, van Dam A, Feiner S, Hughes J (1995) Computer graphics: principles and practice, 2nd edn. Addison Wesley Longman, ReadingGoogle Scholar
  13. 13.
    Frank AU (2001) Tiers of ontology and consistency constraints in geographic information systems. Int J Geogr Inf Sci 15(7). doi: 10.1080/13658810110061144
  14. 14.
    Gold C (2003) But is it GIS? J Geospatial Eng 5(2):11–26Google Scholar
  15. 15.
    Gröger G (2000) Modellierung raumbezogener Objekte und Datenintegrität in GIS, Wichmann, (in German)Google Scholar
  16. 16.
    Gröger G (2006) Konsistente Modellierung virtueller Städte und Regionen. Habilitation Thesis, University of Bonn, (in German)Google Scholar
  17. 17.
    Gröger G, Kolbe TH, Czerwinski A, Nagel C (2008) OpenGIS city geography markup language (CityGML), Implementation Specification, Version 1.0.0, Implementation Specification, OGC Doc. No. 08-007r1Google Scholar
  18. 18.
    Gröger G, Plümer L (1997) Provably correct and complete transaction rules for GIS. In: Laurini R, Bergougnoux P, Pissinou N (eds) Proc of the 5th Int. ACM Workshop on Advances in Geographic Information Systems (ACM-GIS’97), Las Vegas, ACM PressGoogle Scholar
  19. 19.
    Gröger G, Plümer L (2003) Exploiting 2D concepts to achieve consistency in 3D GIS applications. In: Hoel E, Rigaux P (eds) Proc. of the 11th Int. Symposium on Advances in Geographic Information Systems (ACM-GIS’03), November 7–8, New Orleans, ACM PressGoogle Scholar
  20. 20.
    Gröger G, Plümer L (2005) How to get 3-D for the price of 2-D—topology and consistency of 3-D urban GIS. Geoinformatica 9(2):139–158 SpringerCrossRefGoogle Scholar
  21. 21.
    Google (2005) Google Earth Keyhole Markup Language KML 2.0. http://www.keyhole.com/kml/docs/Google_Earth_KML.pdf
  22. 22.
    Harary F (1969) Graph theory. Addison-Wesley, ReadingGoogle Scholar
  23. 23.
    Hatcher A (2001) Algebraic topology. Cambridge University Press, CambridgeGoogle Scholar
  24. 24.
    Herring J (ed) (2001) The OpenGIS abstract specification, topic 1: feature geometry (ISO 19107 Spatial Schema), Version 5. OGC Document Number 01-101Google Scholar
  25. 25.
    Herring J (ed) (2006a) OpenGIS implementation specification for geographic information—simple feature access—part 1: common architecture. Version: 1.2.0 Doc. No. OGC 06-103r3Google Scholar
  26. 26.
    Herring J (ed) (2006b) OpenGIS Implementation specification for geographic information—simple feature access—part 1: SQL option. Version: 1.2.0 Doc. No. OGC 06-104r3Google Scholar
  27. 27.
    ISO TC 211, ISO/FDIS 19109 (2005) Geographic information—rules for application schema. Final Draft International Standard. International Organization for Standardization, Technical Committee 211Google Scholar
  28. 28.
    Jungnickel D (2004) Graphs, networks and algorithms, 2nd edn. Springer, New YorkGoogle Scholar
  29. 29.
    Kainz W (1995) Logical consistency. In: Guptill SC, Morrison JL (eds) Elements of spatial data quality. Elsevier, OxfordGoogle Scholar
  30. 30.
    Kolbe TH, Gröger G, Plümer L (2005) CityGML—interoperable access to 3D city models. In: van Oosterom P, Zlatanova S, Fendel EM (eds) Geo-information for disaster management. Proc. of the 1st International Symposium on Geo-information for Disaster Management, Delft. SpringerGoogle Scholar
  31. 31.
    Kolbe TH, Plümer L (2004) Bridging the Gap between GIS and CAAD. GIM International 18(7):12–15Google Scholar
  32. 32.
    Kuhn W (2007) An image-schematic account of spatial categories. Spatial Information Theory, 8th International Conference, COSIT 2007. Melbourne, Australia. Lecture Notes in Computer Science 4736. Springer, pp 152–168Google Scholar
  33. 33.
    Löwner M-O (2005) Semantische Modellierung von steilen Hangbereichen in einem Geoinformationssystem unter besonderer Berücksichtigung von Wänden und steilen Hangbereichen. PhD thesis, University of Bonn, (in German)Google Scholar
  34. 34.
    Mäntylä M (1988) An introduction to solid modeling. Computer Science, RockvilleGoogle Scholar
  35. 35.
    Molenaar M (1992) A topology for 3D vector maps. ITC Journal 1992, No 1. The International Institute for Aerospace Survey and Earth Sciences, The Netherlands, pp. 25–33Google Scholar
  36. 36.
    Molenaar M (1991) Formal data structures, object dynamics and consistency rules. In: Ebner H, Fritsch D, Heipke C (eds) Digital photogrammetric systems, Wichmann, Karlsruhe, pp 262–273Google Scholar
  37. 37.
    Mortenson M (1997) Geometric modeling, 2nd edn. Wiley, New YorkGoogle Scholar
  38. 38.
    Okabe A, Boots B, Sugihara K (1992) Spatial tessellations. Wiley series in probability and mathematical statistics. Wiley, New YorkGoogle Scholar
  39. 39.
    Oracle (2009) Oracle spatial Developers Guide 11g release 1 (11.1). Oracle. http://www.oracle.com/pls/db111/to_pdf?pathname=appdev.111/b28400.pdf
  40. 40.
    Penninga F, van Oosterom P (2007) A compact topological DBMS data structure for 3D topography. In: Fabrikant S, Wachowicz M (eds) Proceedings of the Geographic Information Science and Systems in Europe, Agile Conference. Lecture notes in geoinformation and cartography. Springer, New York, pp 455–471Google Scholar
  41. 41.
    Pilouk M (1996) Integrated modelling for 3D GIS. PhD Thesis, ITC Publication Series No. 40, Enschede, The NetherlandsGoogle Scholar
  42. 42.
    Plümer L, Gröger G (1996) Nested maps—a formal, provably correct object model for spatial aggregates. Proc. of the fourth ACM Workshop on Advances in Geographic Information Systems. Rockville, Maryland, pp 77–84, November 15–16, 1996. ACMGoogle Scholar
  43. 43.
    Plümer L, Gröger G (1997) Achieving integrity in geographic information systems—maps and nested maps. GeoInformatica 1(4):345–366. doi: 10.1023/A:1009706411129 CrossRefGoogle Scholar
  44. 44.
    Portele C (2007) OpenGIS Geography Markup Language (GML) encoding standard, version 3.2.1, OGC Doc. No. 07-036Google Scholar
  45. 45.
    Preparata FP, Shamos MI (1985) Computational geometry. Springer, BerlinGoogle Scholar
  46. 46.
    Seifert H, Threlfall WA (1980) Textbook of topology. Academic, New YorkGoogle Scholar
  47. 47.
    Staab S, Studer R (2004) Handbook on ontologies. Springer, BerlinGoogle Scholar
  48. 48.
    Tse ROC, Gold C (2002) TIN meets CAD—extending the TIN concept to GIS. In: Sloot PMA et al. (eds) Computational Science - ICCS 2002, Lecture Notes in Computer Science 2331, pp 135–143. doi: 10.1007/3-540-47789-6
  49. 49.
    van Oosterom P, Stoter J, Jansen E (2006) Bridging the worlds of CAD and GIS. In: Zlatanova S, Prosperi D (eds) Large-scale 3D data integration: challenges and opportunities. CRC, Boca Raton, pp 9–36Google Scholar
  50. 50.
    VRML97 (1997) Information technology—computer graphics and image processing—the Virtual Reality Modeling Language (VRML)—part 1: functional specification and UTF-8 encoding. Part 1 of ISO/IEC Standard 14772-1Google Scholar
  51. 51.
    Weiler K (1988) The radial edge data structure: a topological representation for non-manifold geometric boundary modeling. In: Encarnacao JL, Wozny MJ, McLaughlin HW (eds) Geometric modeling for CAD applications. Elsevier Science, Amsterdam, pp 3–36Google Scholar
  52. 52.
    Web3D (2005) X3D international specification standards. http://www.web3d.org/x3d/specifications/x3d_specification.html
  53. 53.
    Zlatanova S, van Oosterom P (2004) GIS/CAD integration. GIM International 11(11):51–53Google Scholar
  54. 54.
    Zlatanova S (2000) 3D-GIS for urban environments. PhD thesis, ITC publication 69, Enschede, the Netherlands, The International Institute for Aerospace Survey and Earth SciencesGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Geodesy and GeoinformationUniversity of BonnBonnGermany

Personalised recommendations