Crossings between Curves with Many Tangencies

Let A and B be two families of two-way infinite x-monotone curves, no three of which pass through the same point. Assume that every curve in A lies above every curve in B and that there are m pairs of curves, one from A and the other from B, that are tangent to each other. Then the number of proper crossings among the members of A∪B is at least (1/2 − o(1))m ln m. This bound is almost tight.

• Type: Conference
• Year: 2010
• Where: WALCOM
• Authors: Jacob Fox, Fabrizio Frati, János Pach, Rom Pinchasi
• Keywords: Algorithms, Curve, Proper Crossings, Two-way Infinite X-monotone, WALCOM 2010