Information Processing Letters
The interface between computational and combinatorial geometry
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Ray shooting amid balls, farthest point from a line, and range emptiness searching
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The voronoi diagram of three lines
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Information Processing Letters
Arrangements in geometry: recent advances and challenges
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Triangulations of line segment sets in the plane
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
On multiplicatively weighted Voronoi diagrams for lines in the plane
Transactions on computational science XIII
On topological changes in the delaunay triangulation of moving points
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We show that the combinatorial complexity of the Euclidean Voronoi diagram of $n$ lines in $\mathbb{R}^3$ that have at most c distinct orientations is $O(c^3n^{2+\varepsilon})$ for any $\varepsilon0$. This result is a step toward proving the long-standing conjecture that the Euclidean Voronoi diagram of lines in three dimensions has near-quadratic complexity. It provides the first natural instance in which this conjecture is shown to hold. In a broader context, our result adds a natural instance to the (rather small) pool of instances of general 3-dimensional Voronoi diagrams for which near-quadratic complexity bounds are known.