Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations
Conference paper
Abstract
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.
Keywords
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.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Robinson, A., Morrison, J., Muehrcke, P., Kimerling, J., Guptill, S.: Elements of cartography. John Wiley & Sons (1995)Google Scholar
- 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.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.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.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.Langlois, P., Denain, J.C.: Cartographie en anamorphose. Cybergeo: European Journal of Geography (1996)Google Scholar
- 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.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.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.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.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.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.Wolff, A., Strijk, T.: The map labeling bibliography (2009), http://liinwww.ira.uka.de/bibliography/Theory/map.labeling.html
- 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.Strijk, T., van Kreveld, M.: Labeling a rectilinear map more efficiently. Information Processing Letters 69(1), 25–30 (1999)CrossRefMathSciNetGoogle Scholar
- 16.Saux, E., Daniel, M.: Data reduction of polygonal curves using B-splines. Computer-Aided Design 31(8), 507–515 (1999)CrossRefzbMATHGoogle Scholar
- 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.Halperin, D.: Arrangements. In: Handbook of Discrete and Computational Geometry. Chapman & Hall/CRC (2004)Google Scholar
- 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.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