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Isoperimetrically Optimal Polygons in the Triangular Grid

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Book cover Combinatorial Image Analysis (IWCIA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6636))

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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.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Faculty of Informatics, University of Debrecen, PO Box 12, 4010, Debrecen, Hungary

    Benedek Nagy

  2. Institute of Mathematics, University of Debrecen, PO Box 12, 4010, Debrecen, Hungary

    Krisztina Barczi

Authors
  1. Benedek Nagy
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  2. Krisztina Barczi
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, The University of Texas, 1 University Station C0803, 78712-0240, Austin, TX, USA

    Jake K. Aggarwal

  2. Department of Computer and Information Science, State University of New York at Fredonia, 280 Central Ave., 14063, Fredonia, NY, USA

    Reneta P. Barneva

  3. Mathematics Department, SUNY Buffalo State College, 1300 Elmwood Ave., 14222, Buffalo, NY, USA

    Valentin E. Brimkov

  4. Esuela Politécnica Superior, Departamento de Ingenería Informática, Universidad Autónoma de Madrid, c/Tomas y Valiente, 11, 28049, Cantoblanco, Madrid, Spain

    Kostadin N. Koroutchev

  5. Departamento de FĂ­sica Fundamental, Universidad Nacional de EducaciĂłn a Distancia, c/ Senda del Rey 9, 28080, Madrid, Spain

    Elka R. Korutcheva

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© 2011 Springer-Verlag Berlin Heidelberg

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Nagy, B., Barczi, K. (2011). Isoperimetrically Optimal Polygons in the Triangular Grid. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_19

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  • DOI: https://doi.org/10.1007/978-3-642-21073-0_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21072-3

  • Online ISBN: 978-3-642-21073-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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