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

TL;DR

This paper introduces a straightforward, dimension-independent algorithm for computing higher-order Delaunay mosaics and alpha shapes, leveraging combinatorial operations and black-box weighted mosaics, with open-source implementations and experimental validation.

Contribution

The paper presents a simple, combinatorial algorithm for higher-order Delaunay mosaics and alpha shapes that is easy to implement and works in any finite dimension.

Findings

Algorithm effectively computes higher-order mosaics in various dimensions.

Experimental results demonstrate properties of mosaics for random point sets.

Open-source implementation facilitates adoption and further research.

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.