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Isoperimetrically Optimal Polygons in the Triangular Grid

  • Benedek Nagy
  • Krisztina Barczi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6636)

Abstract

It is well known that a digitized circle doesn’t have the smallest (digital arc length) perimeter of all objects having a given area. There are various measures of perimeter and area in digital geometry, and so there can be various definitions of digital circles using the isoperimetric inequality (or its digital form). Usually the square grid is used as digital plane. In this paper we use the triangular grid and search for those (digital) objects that have optimal measures. We show that special hexagons are Pareto optimal, i.e., they fulfill both versions of the isoperimetric inequality: they have maximal area among objects that have the same perimeter; and they have minimal perimeter among objects that have the same area.

Keywords

Discrete isoperimetric problem digital geometry digital circles triangular grid 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Benedek Nagy
    • 1
  • Krisztina Barczi
    • 2
  1. 1.Department of Computer Science, Faculty of InformaticsUniversity of DebrecenDebrecenHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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