On the union of cylinders in three dimensions

We show that the combinatorial complexity of the union of n infinite cylinders in ℝ³, having arbitrary radii, is O(n ^2+ε), for any ε > 0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir [3], who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir [3], in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in [3]. Finally, we extend our technique to the case of "cigars" of arbitrary radii (that is, Minkowski sums of line-segments and balls), and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in [3] for the restricted case where all cigars are (nearly) equal-radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders, and is significantly simpler than the proof presented in [3].