The Distance 4-Sector of Two Points Is Unique

  • Robert Fraser
  • Meng He
  • Akitoshi Kawamura
  • Alejandro López-Ortiz
  • J. Ian Munro
  • Patrick K. Nicholson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


The (distance) k-sector is a generalization of the concept of bisectors proposed by Asano, Matoušek and Tokuyama. We prove the uniqueness of the 4-sector of two points in the Euclidean plane. Despite the simplicity of the unique 4-sector (which consists of a line and two parabolas), our proof is quite non-trivial. We begin by establishing uniqueness in a small region of the plane, which we show may be perpetually expanded afterward.


distance k-sector Tarski fixed point uniqueness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asano, T., Kirkpatrick, D.: Distance trisector curves in regular convex distance metrics. In: Proc. 3rd International Symposium on Voronoi Diagrams in Science and Engineering, pp. 8–17 (2006)Google Scholar
  2. 2.
    Asano, T., Matoušek, J., Tokuyama, T.: Zone diagrams: Existence, uniqueness, and algorithmic challenge. SIAM Journal on Computing 37(4), 1182–1198 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asano, T., Matoušek, J., Tokuyama, T.: The distance trisector curve. Advances in Mathematics 212(1), 338–360 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Asano, T., Tokuyama, T.: Drawing equally-spaced curves between two points. In: Proc. 14th Fall Workshop on Computational Geometry, pp. 24–25 (2004)Google Scholar
  5. 5.
    Chun, J., Okada, Y., Tokuyama, T.: Distance trisector of a segment and a point. Interdisciplinary Information Sciences 16(1), 119–125 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Imai, K., Kawamura, A., Matoušek, J., Reem, D., Tokuyama, T.: Distance k-sectors exist. Computational Geometry 43(9), 713–720 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kawamura, A., Matoušek, J., Tokuyama, T.: Zone diagrams in Euclidean spaces and in other normed spaces. Mathematische Annalen 354(4), 1201–1221 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Monterde, J., Ongay, F.: The distance trisector curve is transcendental. Geometriae Dedicata (in press)Google Scholar
  9. 9.
    Reem, D., Reich, S.: Zone and double zone diagrams in abstract spaces. Colloquium Mathematicum 115(1), 129–145 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Reem, D.: On the computation of zone and double zone diagrams. arXiv:1208.3124 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Robert Fraser
    • 1
  • Meng He
    • 2
  • Akitoshi Kawamura
    • 3
  • Alejandro López-Ortiz
    • 4
  • J. Ian Munro
    • 4
  • Patrick K. Nicholson
    • 5
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  2. 2.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada
  3. 3.Department of Computer ScienceUniversity of TokyoTokyoJapan
  4. 4.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  5. 5.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations