A Simple Algorithm for Higher-order Delaunay Mosaics and Alpha Shapes

Abstract: 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.