Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations
The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Omega (n(2)) lower bound and the O(n(3+epsilon)) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1') the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2') the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered "cross-sections" of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1') and (2') have recently been shown to have a nearly tight bound: both complexities are O(n(2+epsilon)) with lower bound Omega (n2). In this paper, we determine the combinatorial complexities of these four problems for some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n(2)) or O(n(2+epsilon)) and lower bounds of Omega (n(2)). (C) 2004 Elsevier B.V. All rights reserved.
voronoi diagrams combinatorial problems computational geometry