Wedges in Euclidean Arrangements

  • Jonathan Lenchner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)


Given an arrangement of n not all coincident lines in the Euclidean plane we show that there can be no more than \(\lfloor 4n/3\rfloor\) wedges (i.e. two-edged faces) and give explicit examples to show that this bound is tight. We describe the connection this problem has to the problem of obtaining lower bounds on the number of ordinary points in arrangements of not all coincident, not all parallel lines, and show that there must be at least \(\lfloor(5{\it n} + 6)/39\rfloor\) such points.


Convex Hull Intersection Point Extreme Point Parallel Line Euclidean Plane 
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  1. 1.
    Borwein, P., Moser, W.: A survey of Sylvester’s problem and its generalizations. Aequationes Mathematicae 40, 111–135 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ching, Y.T., Lee, D.T.: Finding the diameter of a set of lines. Pattern Recognition 18, 249–255 (1985)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Csima, J., Sawyer, E.: There exist 6n/13 ordinary points. Discrete and Computational Geometry 9, 187–202 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Devillers, O., Mukhopadhyay, A.: Finding an ordinary conic and an ordinary hyperplane. Nordic Journal of Computing 6, 462–468 (1999)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Devroye, L., Toussaint, G.: Convex hulls for random lines. Journal of Algorithms 14, 381–394 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Felsner, S.: Geometric Graphs and Arrangements. Vieweg and Sohn, Wiesbaden, Germany (2004)zbMATHGoogle Scholar
  7. 7.
    Golin, M., Langerman, S., Steiger, W.: The convex hull for random lines in the plane. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 172–175. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Keil, M.: A simple algorithm for determining the envelope of a set of lines. Information Processing Letters 39, 121–124 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kelly, L., Moser, W.: On the number of ordinary lines determined by n points. Canadian Journal of Mathematics 10, 210–219 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Lenchner, J.: On the dual and sharpened dual of Sylvester’s theorem in the plane. IBM Research Report, RC23411, W0409-066 (2004)Google Scholar
  11. 11.
    Melchior, E.: Über vielseite der projektiven eberne. Deutsche Math. 5, 461–475 (1940)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jonathan Lenchner
    • 1
  1. 1.IBM T.J. Watson Research InstituteYorktown Heights

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