Apollonius Solutions in Rd

TL;DR

This paper presents explicit equations for the vertices of Apollonius diagrams in multi-dimensional space, making their computation more accessible and practical for various scientific and engineering applications.

Contribution

It introduces simple, explicit vertex expressions for Apollonius diagrams in d-dimensional space, improving computational efficiency and stability compared to previous methods.

Findings

Vertices can be computed with standard vector and matrix libraries.

The method is as efficient as computing power diagrams.

Applications include molecular crystal shape analysis.

Abstract

Voronoi and related diagrams have technological applications, for example, in motion planning and surface reconstruction, and also find significant use in materials science, molecular biology, and crystallography. Apollonius diagrams arguably provide the most natural division of space for many materials and technology problems, but compared to Voronoi and power diagrams, their use has been limited, presumably by the complexity of their calculation. In this work, we report explicit equations for the vertices of the Apollonius diagram in a d-dimensional Euclidean space. We show that there are special lines that contain vertices of more than one type of diagram and this property can be exploited to develop simple vertex expressions for the Apollonius diagram. Finding the Apollonius vertices is not significantly more difficult or expensive than computing those of the power diagram and have application beyond their use in calculating the diagram. The expressions reported here lend themselves to the use of standard vector and matrix libraries and the stability and precision their use implies. They can also be used in algorithms with multi-precision numeric types and those adhering to the exact algorithms paradigm. The results have been coded in C++ for the 2-d and 3-d cases and an example of their use in characterizing the shape of a void in a molecular crystal is given.