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Shape deformation in continuous map generalization

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Abstract

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.

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Notes

  1. This function might be non-differentiable at a few places, and thus needs to be replaced by an arbitrarily close differentiable one. Geometrically, this can be viewed as smoothing of the corners of the medial axis.

  2. Unlike the Voronoi regions defined in Eq. 1, we are not interested in the boundary \({\partial} {\Omega}\) now.

  3. It is necessary to separate the regions with some space. To do this we remove from each set A i a small δ-neighborhood of \(\partial A_i\). The resulting regions (still denoted as A i ) leaves every pair of regions separated by at least 2δ. As always, we smooth out corners of the regions so that their boundaries are differentiable.

  4. Two regions are arbitrarily close if the distance between the corresponding points of the regions are within some small ε > 0 of each other.

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Acknowledgements

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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Authors and Affiliations

  1. Stanford University, Stanford, CA, 94305, USA

    Jeff Danciger

  2. Williams College, Williamstown, MA, 01267, USA

    Satyan L. Devadoss

  3. University of Maryland, College Park, MD, 20742, USA

    John Mugno

  4. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Don Sheehy

  5. Princeton University, Princeton, NJ, 08544, USA

    Rachel Ward

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  1. Jeff Danciger
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  2. Satyan L. Devadoss
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  3. John Mugno
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  4. Don Sheehy
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  5. Rachel Ward
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Correspondence to Satyan L. Devadoss.

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Danciger, J., Devadoss, S.L., Mugno, J. et al. Shape deformation in continuous map generalization. Geoinformatica 13, 203–221 (2009). https://doi.org/10.1007/s10707-008-0049-0

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  • DOI: https://doi.org/10.1007/s10707-008-0049-0

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