We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-$k$ mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-$k$ mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order $\alpha$-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets.

中文翻译：

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高阶 Delaunay 马赛克和 Alpha 形状的简单算法

我们提出了一种用于计算高阶 Delaunay 马赛克的简单算法，该算法适用于任何有限维度的欧几里得空间。该算法从增量构建的低阶马赛克中选择阶 $k$ 马赛克的顶点，并使用加权一阶 Delaunay 马赛克的算法作为黑盒从其顶点构建阶 $k$ 马赛克。除了这个黑盒之外，该算法仅使用组合运算，因此易于实现。我们扩展此算法以计算高阶 $\alpha$ 形状并提供开源实现。我们展示了随机点集的高阶 Delaunay 镶嵌特性的实验结果。