Team, Visitors, External Collaborators
Overall Objectives
Research Program
Application Domains
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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Section: New Results

Voronoï diagram of orthogonal polyhedra in two and three dimensions

Participants : Ioannis Emiris, Christina Katsamaki.

In [20], we study Voronoï diagrams, which 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 Voronoï diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoï 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/D) in 2D or O(na2/D2) in 3D, where n is the number of sites, namely edges or facets resp., D is the maximum cell size for the subdivision to stop, and a bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm.

The software was developed during Katsamaki's internship in 2018 at Sophia-Antipolis under the supervision of Bernard Mourrain. The problem has been proposed by our industrial collaborator ANSYS Hellas. The paper is based on Katsamaki's MSc thesis.