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.
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