Area-Preserving Subdivision Schematization

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6292)


We describe an area-preserving subdivision schematization algorithm: the area of each region in the input equals the area of the corresponding region in the output. Our schematization is axis-aligned, the final output is a rectilinear subdivision. We first describe how to convert a given subdivision into an area-equivalent rectilinear subdivision. Then we define two area-preserving contraction operations and prove that at least one of these operations can always be applied to any given simple rectilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff- and Fréchet-distance) that show that better schematizations might result in worse shapes.


Schematization polygonal subdivisions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cabello, S., de Berg, M., van Kreveld, M.: Schematization of networks. Computational Geometry: Theory and Applications 30(3), 223–238 (2005)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Nöllenburg, M., Wolff, A.: A mixed-integer program for drawing high-quality metro maps. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 321–333. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Swan, J., Anand, S., Ware, M., Jackson, M.: Automated schematization for web service applications. In: Ware, J.M., Taylor, G.E. (eds.) W2GIS 2007. LNCS, vol. 4857, pp. 216–226. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Damen, J., van Kreveld, M., Spaan, B.: High quality building generalization by extending morphological operators. In: 11th ICA Workshop on Generalization and Multiple Representation, Montpellier, France (2008)Google Scholar
  5. 5.
    Lamy, S., Ruas, A., Demazeau, Y., Jackson, M., Mackaness, W., Weibel, R.: The application of agents in automated map generalisation. In: 19th International Cartographic Conference (1999)Google Scholar
  6. 6.
    Sester, M.: Generalization based on least squares adjustment. ISPRS - International Archives of Photogrammetry and Remote Sensing 13, 931–938 (2000)Google Scholar
  7. 7.
    Yan, H., Weibel, R., Yang, B.: A multi-parameter approach to automated building grouping and generalization. GeoInformatica 12, 73–89 (2008)CrossRefGoogle Scholar
  8. 8.
    Mayer, H.: Scale-spaces for generalization of 3d buildings. International Journal of Geographical Information Science 19, 975–997 (2005)CrossRefGoogle Scholar
  9. 9.
    Regnauld, N., Edwardes, A., Barrault, M.: Strategies in building generalisation: modelling the sequence, constraining the choice. In: ICA Workshop on Progress in Automated Map Generalization, Ottawa, Canada (1999)Google Scholar
  10. 10.
    Ruas, A.: Modèle de généralisation de données géographiques à base de contraintes et d’autonomie. PhD thesis, Université de Marne la Vallée (1999)Google Scholar
  11. 11.
    Haunert, J.H., Wolff, A.: Optimal simplification of building ground plans. In: Proceedings of XXIst ISPRS Congress Beijing 2008. IAPRS, vol. XXXVII (Part B2), pp. 372–378 (2008)Google Scholar
  12. 12.
    Sester, M.: Optimization approaches for generalization and data abstraction. International Journal of Geographical Information Science 19, 871–897 (2005)CrossRefGoogle Scholar
  13. 13.
    Bose, P., Cabello, S., Cheong, O., Gudmundsson, J., van Kreveld, M., Speckmann, B.: Area-preserving approximations of polygonal paths. Journal of Discrete Algorithms 4(4), 554–566 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    de Berg, M., van Kreveld, M., Schirra, S.: A new approach to subdivision simplification. Proceedings of Auto-Carto 12, 79–88 (1995)Google Scholar
  15. 15.
    de Berg, M., van Kreveld, M., Schirra, S.: Topologically correct subdivision simplification using the bandwidth criterion. Cartography and Geographic Information Science 25(4), 243–257 (1998)CrossRefGoogle Scholar
  16. 16.
    Estkowski, R., Mitchell, J.S.B.: Simplifying a polygonal subdivision while keeping it simple. In: 17th Symposium on Computational Geometry, pp. 40–49 (2001)Google Scholar
  17. 17.
    van de Kraats, B., van Kreveld, M., Overmars, M.: Printed circuit board simplification: simplifying subdivisions in practice. In: 11th Symposium on Computational Geometry, pp. 430–431 (1995)Google Scholar
  18. 18.
    Guibas, L., Hershberger, J., Mitchell, J., Snoeyink, J.: Approximating polygons and subdivisions with minimum-link paths. International Journal of Computational Geometry and Applications 3, 383–415 (1993)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Dep. of Mathematics and Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations