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Voronoi Diagram of Orthogonal Polyhedra in Two and Three Dimensions

  • Ioannis Z. Emiris
  • Christina KatsamakiEmail author
Conference paper
  • 212 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11544)

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 \(O(n/ \varDelta + 1/\varDelta ^2)\) in 2D or \(O(n\alpha ^2/\varDelta ^2 +1/\varDelta ^3)\) in 3D, where n is the number of sites, namely edges or facets resp., \(\varDelta \) 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.

Keywords

Max norm Axis-aligned Rectilinear Straight skeleton Subdivision method Numeric implementation 

Notes

Acknowledgements

We thank Evanthia Papadopoulou for commenting on a preliminary version of the paper and Bernard Mourrain for collaborating on software. Both authors are members of AROMATH, a joint team between INRIA Sophia-Antipolis (France) and NKUA.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece
  2. 2.ATHENA Research and Innovation CenterMaroussiGreece

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