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Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions.

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Title: Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions.
Authors: Barequet, Gill, Papadopoulou, Evanthia, Suderland, Martin
Source: Discrete & Computational Geometry; Oct2024, Vol. 72 Issue 3, p1304-1332, 29p
Subject Terms: VORONOI polygons, POLYHEDRA, SPHERES, TUNNELS
Abstract: We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is O (min { k , n - k } n d - 1) , which is tight for n - k = O (1) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d - 1) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of n ≥ 2 lines in general position has exactly n (n - 1) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in O (n d - 1 α (n)) time, for d ≥ 4 , while if d = 3 , the time drops to worst-case optimal Θ (n 2) . We extend the obtained results to bounded polyhedra and clusters of points as sites. [ABSTRACT FROM AUTHOR]
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Abstract:We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is O (min { k , n - k } n d - 1) , which is tight for n - k = O (1) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d - 1) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of n ≥ 2 lines in general position has exactly n (n - 1) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in O (n d - 1 α (n)) time, for d ≥ 4 , while if d = 3 , the time drops to worst-case optimal Θ (n 2) . We extend the obtained results to bounded polyhedra and clusters of points as sites. [ABSTRACT FROM AUTHOR]
ISSN:01795376
DOI:10.1007/s00454-023-00492-2