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Crossings between Curves with Many Tangencies

  • Jacob Fox
  • Fabrizio Frati
  • János Pach
  • Rom Pinchasi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

Abstract

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jacob Fox
    • 1
  • Fabrizio Frati
    • 2
  • János Pach
    • 3
  • Rom Pinchasi
    • 4
  1. 1.Department of MathematicsPrinceton UniversityPrinceton
  2. 2.Dipartimento di Informatica e AutomazioneRoma Tre UniversityItaly
  3. 3.EPFL LausanneSwitzerland and Rényi Institute BudapestHungary
  4. 4.Mathematics DepartmentTechnion – Israel Institute of TechnologyHaifaIsrael

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