Voronoi diagrams of rigidly moving sets of points
Information Processing Letters
The upper bound theorem for polytopes: an easy proof of its asymptotic version
Computational Geometry: Theory and Applications
Voronoi diagrams in higher dimensions under certain polyhedral distance functions
Proceedings of the eleventh annual symposium on Computational geometry
Voronoi diagrams of lines in 3-space under polyhedral convex distance functions
Journal of Algorithms - Special issue on SODA '95 papers
A lower bound on Voronoi diagram complexity
Information Processing Letters
3-Dimensional Euclidean Voronoi Diagrams of Lines with a Fixed Number of Orientations
SIAM Journal on Computing
Voronoi Diagrams of Moving Points in the Plane
WG '91 Proceedings of the 17th International Workshop
Ready, set, go! the Voronoi diagram of moving points that start from a line
Information Processing Letters
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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 @W(n^2) lower bound and the O(n^3^+^@?) 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^+^@?) with lower bound @W(n^2). 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^+^@?) and lower bounds of @W(n^2).