3-dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations

We show that the combinatorial complexity of the Euclidean Voronoi diagram of n lines in ℝ³ that have at most c distinct orientations is 0(c³n^2+ε) for any ε > 0. 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.