Incidences between points and circles in three and higher dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
Lenses in arrangements of pseudo-circles and their applications
Proceedings of the eighteenth annual symposium on Computational geometry
Pseudo-line arrangements: duality, algorithms, and applications
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
On locally Delaunay geometric graphs
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
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Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk/sup 1-2/3/*)) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s=1 and s=2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk/sup 7/9/log/sup 2/3/ k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.