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Packing Disks by Flipping and Flowing

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Abstract

We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge \(e^-\) of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge \(e^-\) in G. This flow is parameterized by a single inversive distance.

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Acknowledgements

The authors would like to thank Louis Theran for many helpful discussions along the way.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, USA

    Robert Connelly

  2. School of Engineering and Applied Sciences, Harvard University, Cambridge, USA

    Steven J. Gortler

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  1. Robert Connelly
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  2. Steven J. Gortler
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Correspondence to Steven J. Gortler.

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R. Connelly: Partially supported by NSF Grant DMS-1564493. S. J. Gortler: Partially supported by NSF Grant DMS-1564473.

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Connelly, R., Gortler, S.J. Packing Disks by Flipping and Flowing. Discrete Comput Geom 66, 1262–1285 (2021). https://doi.org/10.1007/s00454-020-00242-8

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