# Voronoi diagram of orthogonal polyhedra in two and three dimensions

## TL;DR

This paper presents a novel subdivision-based algorithm for constructing exact Voronoi diagrams of orthogonal polyhedra in 2D and 3D under the max-norm, avoiding full-dimensional elements and providing a skeletal representation.

## Contribution

It introduces an original subdivision paradigm with a bounding-volume hierarchy for efficient, exact Voronoi diagram computation of orthogonal polyhedra, including an open-source Julia implementation.

## Key findings

- Algorithm is adaptive and comparable in complexity to previous methods.
- Provides a numerically stable, open-source implementation in Julia.
- Complexity depends on site number, cell size, and vertex bounds.

## Abstract

Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axis-aligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly useful scenario in certain application domains. We construct the exact Voronoi diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoi diagram and yields a skeletal representation of the input object. We introduce a complete algorithm in 2D and 3D that follows the subdivision paradigm relying on a bounding-volume hierarchy; this is an original approach to the problem. The complexity is adaptive and comparable to that of previous methods. Under a mild assumption it is $O(n / \Delta + 1 / \Delta^2)$ in 2D or $O(n\alpha^2 / \Delta^2 +1 / \Delta^3)$ in 3D, where $n$ is the number of sites, namely edges or facets resp., $\Delta$ is the maximum cell size for the subdivision to stop, and $\alpha$ bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm.