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

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-1α (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.