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A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.

Herbert Edelsbrunner¹, Georg Osang¹

¹IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria.

|January 23, 2023

Summary

This summary is machine-generated.

We developed a straightforward algorithm for calculating higher-order Delaunay mosaics in any dimension. This method efficiently constructs complex mosaics using combinatorial operations and provides open-source tools for broader application.

Keywords:

Algorithms Computational experiments Delaunay mosaics Software Voronoi tessellations

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Area of Science:

Computational Geometry
Topology
Data Analysis

Background:

Delaunay mosaics are fundamental structures in computational geometry.
Higher-order Delaunay mosaics generalize standard Delaunay triangulations to higher dimensions and orders.
Efficient algorithms are needed for their computation and application.

Purpose of the Study:

To present a simple, generalizable algorithm for computing higher-order Delaunay mosaics.
To extend the algorithm for computing higher-order shapes.
To provide open-source implementations and analyze mosaic properties.

Main Methods:

An incremental approach selecting vertices from lower-order mosaics.
Utilizing a black-box algorithm for weighted first-order Delaunay mosaics.
Employing combinatorial operations for constructing higher-order mosaics.

Main Results:

A computationally simple algorithm for higher-order Delaunay mosaics in any finite dimension.
Extension of the algorithm to compute higher-order shapes.
Open-source implementations are available for practical use.

Conclusions:

The presented algorithm offers an accessible method for generating higher-order Delaunay mosaics.
The combinatorial approach simplifies implementation and extends applicability.
Experimental results demonstrate the utility for analyzing random point sets.