Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations

  • Kevin Buchin
  • Arthur van Goethem
  • Michael Hoffmann
  • Marc van Kreveld
  • Bettina Speckmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8728)


Linear cartograms visualize travel times between locations, usually by deforming the underlying map such that Euclidean distance corresponds to travel time. We introduce an alternative model, where the map and the locations remain fixed, but edges are drawn as sinusoid curves. Now the travel time over a road corresponds to the length of the curve. Of course the curves might intersect if not placed carefully. We study the corresponding algorithmic problem and show that suitable placements can be computed efficiently. However, the problem of placing as many curves as possible in an ideal, centered position is NP-hard. We introduce three heuristics to optimize the number of centered curves and show how to create animated visualizations.


Planar Graph Edge Length Isosceles Triangle Highway Network Edge Width 
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.
    Robinson, A., Morrison, J., Muehrcke, P., Kimerling, J., Guptill, S.: Elements of cartography. John Wiley & Sons (1995)Google Scholar
  2. 2.
    Bies, S., van Kreveld, M.: Time-space maps from triangulations. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 511–516. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Cabello, S., Demaine, E., Rote, G.: Planar embeddings of graphs with specified edge lengths. Journal of Graph Algorithms and Applications 11(1), 259–276 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kaiser, C., Walsh, F., Farmer, C., Pozdnoukhov, A.: User-centric time-distance representation of road networks. In: Fabrikant, S.I., Reichenbacher, T., van Kreveld, M., Schlieder, C. (eds.) GIScience 2010. LNCS, vol. 6292, pp. 85–99. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Shimizu, E., Inoue, R.: A new algorithm for distance cartogram construction. International Journal of Geographical Information Science 23(11), 1453–1470 (2009)CrossRefGoogle Scholar
  6. 6.
    Langlois, P., Denain, J.C.: Cartographie en anamorphose. Cybergeo: European Journal of Geography (1996)Google Scholar
  7. 7.
    Bouts, Q., Dwyer, T., Dykes, J., Speckmann, B., Riche, N., Carpendale, S., Goodwin, S., Liebman, A.: Visual encoding of dissimilarity data via topology preserving map deformation (in preparation, 2014)Google Scholar
  8. 8.
    Barequet, G., Goodrich, M., Riley, C.: Drawing planar graphs with large vertices and thick edges. Journal of Graph Algorithms and Applications 8(1), 3–20 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Goodrich, M., Wagner, C.: A framework for drawing planar graphs with curves and polylines. Journal of Algorithms 37(2), 399–421 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Efrat, A., Erten, C., Kobourov, S.: Fixed-location circular-arc drawing of planar graphs. Journal of Graph Algorithms and Applications 11(1), 145–164 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Duncan, C., Eppstein, D., Goodrich, M., Kobourov, S., Nöllenburg, M.: Lombardi drawings of graphs. Journal of Graph Algorithms and Applications 16(1), 37–83 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Nielsen, C., Jackman, S., Birol, I., Jones, S.: Abyss-explorer: visualizing genome sequence assemblies. IEEE Transactions on Visualization and Computer Graphics 15(6), 881–888 (2009)CrossRefGoogle Scholar
  13. 13.
    Wolff, A., Strijk, T.: The map labeling bibliography (2009),
  14. 14.
    Poon, C.K., Zhu, B., Chin, F.: A polynomial time solution for labeling a rectilinear map. Information Processing Letters 65(4), 201–207 (1998)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Strijk, T., van Kreveld, M.: Labeling a rectilinear map more efficiently. Information Processing Letters 69(1), 25–30 (1999)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Saux, E., Daniel, M.: Data reduction of polygonal curves using B-splines. Computer-Aided Design 31(8), 507–515 (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Aspvall, B., Plass, M., Tarjan, R.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters 8(3), 121–123 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Halperin, D.: Arrangements. In: Handbook of Discrete and Computational Geometry. Chapman & Hall/CRC (2004)Google Scholar
  19. 19.
    Guibas, L., Hershberger, J., Mitchell, J., Snoeyink, J.: Approximating polygons and subdivisions with minimum-link paths. International Journal of Computational Geometry & Applications 3(4), 383–415 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Fabrikant, S., Montello, D., Ruocco, M., Middleton, R.: The distance–similarity metaphor in network-display spatializations. Cartography and Geographic Information Science 31(4), 237–252 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kevin Buchin
    • 1
  • Arthur van Goethem
    • 1
  • Michael Hoffmann
    • 2
  • Marc van Kreveld
    • 3
  • Bettina Speckmann
    • 1
  1. 1.Technical University EindhovenEindhovenThe Netherlands
  2. 2.ETH ZürichZürichSwitzerland
  3. 3.Utrecht UniversityUtrechtThe Netherlands

Personalised recommendations