双曲沃罗诺伊图的简化
Hyperbolic Voronoi diagrams made easy
摘要：
我们提出了一个简单的框架来计算双曲沃罗诺伊图的有限点集仿射图。我们证明，在克莱因的非共形磁盘模型的平分线是超平面，可以解释为权力平分的欧几里德球。因此，我们的方法简单地包括计算等效的裁剪幂图，然后根据双曲空间的所选表示进行映射变换（例如，庞加莱共形圆盘或上平面表示）。我们讨论了这种方法的扩展加权和$k$顺序图，并描述其对偶三角剖分。最后，我们考虑两个有用的原语双曲沃罗诺伊图设计定制的用户界面的图像目录浏览应用程序中的双曲磁盘：（1）寻找最近的邻居，和（2）计算最小的封闭球。
英文摘要：
We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincaré conformal disk or upper-plane representations). We discuss on extensions of this approach to weighted and $k$-order diagrams, and describe their dual triangulations. Finally, we consider two useful primitives on the hyperbolic Voronoi diagrams for designing tailored user interfaces of an image catalog browsing application in the hyperbolic disk: (1) finding nearest neighbors, and (2) computing smallest enclosing balls.
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