Shape deformation in continuous map generalization
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Given a collection of regions on a map, we seek a method of continuously altering the regions as the scale is varied. This is formalized and brought to rigor as well-defined problems in homotopic deformation. We ask the regions to preserve topology, area-ratios, and relative position as they change over time. A solution is presented using differential methods and computational geometric techniques. Most notably, an application of this method is used to provide an algorithm to obtain cartograms.
KeywordsContinuous generalization Scale change Homotopy Cartograms
We are grateful to the NSF for partially supporting this project with grants DMS-0353634 and CARGO DMS-0310354. We thank Tamal Dey, Chris Jones, Marc van Kreveld, Alan Saalfeld, Jim Stasheff and Robert Weibel for helpful conversations. Satyan Devadoss also wishes to thank Jörg-Rüdiger Sack, Monika Sester, Peter van Oosterom, and Michael Worboys for organizing the Dagstuhl workshop on Spatial Data in 2006, which motivated and solidified several concepts in this work. Finally, the authors thank the anonymous reviewers for their insightful comments and corrections.
- 1.de Boor C (2001) A practical guide to splines, revised edn. Springer, New YorkGoogle Scholar
- 4.Goodman J, O’Rourke J (eds) (2004) Handbook of discrete and computational geometry, 2nd edn. CRC, New YorkGoogle Scholar
- 5.Hatcher A (2002) Algebraic topology. Cambridge University Press, New YorkGoogle Scholar
- 6.Hoff K, Culver T, Keyser J, Lin M, Manocha D (1999) Fast computation of generalized Voronoi diagrams using graphics hardware. In: International conference on computer graphics and interactive techniques, pp 277–286Google Scholar
- 8.Lam N, Catts D, Quattrochi D, Brown D, McMaster R (2004) Chapter 4: scale. In: McMaster R, Lynn Usery E (eds) A research agenda for geographic information science. CRC Press, pp 93–128Google Scholar
- 9.Okabe A, Boots B, Sugihara K (1992) Spatial tessellations: concepts and applications of Voronoi diagrams. John Wiley and Sons, New YorkGoogle Scholar
- 10.Saalfeld A (1996) Consistent map generalization via white space management (preprint)Google Scholar
- 11.Sester M, Brenner C (2006) Continuous generalization for small mobile displays. In: Next generation geospatial information. Taylor and Francis Group, London, pp 33–41Google Scholar
- 13.van Kreveld M (2001) Smooth generalization for continuous zooming. In: Proceedings 20th international geographic conference, pp 2180–2185Google Scholar
- 15.Ramanathan M, Gurumoorthy B (2003) A tracing algorithm for constructing medial axis transform of 3D objects bound by free-form surfaces. CAD 35:619–632Google Scholar
- 16.Yang W, Gold C (1997) A system approach to automated map generalization. In: Proceedings: international workshop on dynamic and multi-dimensional GIS, pp 229–235Google Scholar