Geometriae Dedicata

, Volume 61, Issue 3, pp 315–327 | Cite as

Three distinct distances in the plane

  • Heiko Harborth
  • Lothar Piepmeyer


For n≥6 all sets of n points in the plane with three distinct distances are determined.

Mathematics Subject Classifications (1991)

52C10 51K99 

Key words

Erdös problems distance geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Altman, E.: On a problem of P. Erdös, Amer. Math. Monthly 70 (1963), 148–157.Google Scholar
  2. 2.
    Altman, E.: Some theorems on convex polygons, Canad. Math. Bull. 15 (1972), 329–340.Google Scholar
  3. 3.
    Chung, F. R. K., Szemerédi, E. and Trotter, W. T.: The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7 (1992), 1–11.Google Scholar
  4. 4.
    Croft, H. T., Falconer, K. J. and Guy, R. K.: Unsolved Problems in Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1991.Google Scholar
  5. 5.
    Einhorn, S. J. and Schoenberg, I. J.: On Euclidean sets having only two distances between points II, Indag. Math. 28 (1966), 479–504.Google Scholar
  6. 6.
    Erdös, P.: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.Google Scholar
  7. 7.
    Erdös, P. and Fishburn, P.: Maximum planar sets that determine k distances, Preprint, 1994.Google Scholar
  8. 8.
    Fishburn, P.: Convex polygons with few intervertex distances, DIMACS, Technical Report 92–18.Google Scholar
  9. 9.
    Moser, W. and Pach, J.: Recent developments in combinatorial geometry, in: J. Pach (ed.), New Trends in Discrete and Computational Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993, pp. 281–302.Google Scholar
  10. 10.
    Piepmeyer, L.: Punktmengen mit minimaler Anzahl verschiedener Abstände, Dissertation, Techn. Univ. Braunschweig, 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Heiko Harborth
    • 1
  • Lothar Piepmeyer
    • 1
  1. 1.Diskrete Mathematik Technische Universität BraunschweigBraunschweigGermany

Personalised recommendations