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 Sd-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}nd-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(nd-1α(n)) time, for d ≥ 4 , while if d = 3 , the time drops to worst-case optimal Θ(n2) . We extend the obtained results to bounded polyhedra and clusters of points as sites.