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, such 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.

中文翻译：

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通过翻转和流动包装盘

我们为基于组合边缘翻转的 Koebe-Andreev-Thurston (KAT) 平面圆堆积定理提供了一种新型证明。特别是，我们展示了从具有最大平面接触图 $G$ 的圆盘包装开始，可以移除该图的任何可翻转边 $e^-$，然后在平面中连续流动圆盘，使得在最后在流程中，人们获得了一个新的磁盘包装，其接触图是在 $G$ 中翻转边 $e^-$ 所产生的图。该流由单个逆距离参数化。