Discrete & Computational Geometry

, Volume 34, Issue 1, pp 11–24

On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences


DOI: 10.1007/s00454-005-1165-3

Cite this article as:
Chan, T. Discrete Comput Geom (2005) 34: 11. doi:10.1007/s00454-005-1165-3


We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n2-1/2s) complexity. This answers one of the main open problems from the author’s previous paper [DCG 29, 375-393 (2003)], which provided a weaker upper bound for a restricted class of curves only (graphs of degree-s polynomials). When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n3/2 bound for parabolas.

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1Canada