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.
Max norm Axis-aligned Rectilinear Straight skeleton Subdivision method Numeric implementation
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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.
Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. J. Univ. Comput. Sci. 1, 752–761 (1995)MathSciNetzbMATHGoogle Scholar
Aurenhammer, F., Walzl, G.: Straight skeletons and mitered offsets of nonconvex polytopes. Discrete Comput. Geom. 56(3), 743–801 (2016)MathSciNetCrossRefGoogle Scholar
Barequet, G., Eppstein, D., Goodrich, M., Vaxman, A.: Straight skeletons of three-dimensional polyhedra. In: Proceedings of the Twenty-fifth ACM Annual Symposium on Computational Geometry, pp. 100–101. ACM Press, Aarhus, Denmark, (2009). https://doi.org/10.1145/1542362.1542384.
Bennett, H., Papadopoulou, E., Yap, C.: Planar minimization diagrams via subdivision with applications to anisotropic Voronoi diagrams. Comput. Graph. Forum 35(5), 229–247 (2016)CrossRefGoogle Scholar
Eder, G., Held, M., Palfrader, P.: Computing the straight skeleton of an orthogonal monotone polygon in linear time. In: European Workshop on Computational Geometry, Utrecht, March 2019. www.eurocg2019.uu.nl/papers/16.pdf
Emiris, I.Z., Mantzaflaris, A., Mourrain, B.: Voronoi diagrams of algebraic distance fields. J. Comput. Aided Des. 45(2), 511–516 (2013). Symposium on Solid Physical Modeling 2012MathSciNetCrossRefGoogle Scholar
Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions. Discrete Comput. Geom. 22, 58–67 (1998)MathSciNetzbMATHGoogle Scholar
Papadopoulou, E., Lee, D.: The L\(_\infty \) Voronoi diagram of segments and VLSI applications. Int. J. Comput. Geom. Appl. 11(05), 503–528 (2001)MathSciNetCrossRefGoogle Scholar
Schaefer, S., Warren, J.: Dual marching cubes: primal contouring of dual grids. Comput. Graph. Forum 24(2), 195–201 (2005)CrossRefGoogle Scholar
Yap, C., Sharma, V., Jyh-Ming, L.: Towards exact numerical Voronoi diagrams. In: IEEE International Symposium on Voronoi Diagrams in Science and Engineering (ISVD), New Brunswick, NJ, June 2012Google Scholar