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Trajectories and Traces on Non-traditional Regular Tessellations of the Plane

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Book cover Combinatorial Image Analysis (IWCIA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10256))

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Abstract

In this paper, shortest paths on two regular tessellations, on the hexagonal and on the triangular grids, are investigated. The shortest paths (built by steps to neighbor pixels) between any two points (cells, pixels) are described as traces and generalized traces on these grids, respectively. In the hexagonal grid, there is only one type of usual neighborhood and at most two directions of the steps are used in any shortest paths, and thus, the number of linearizations of these traces is easily computed by a binomial coefficient based on the coordinate differences of the pixels. Opposite to this, in the triangular grid the neighborhood is inhomogeneous (there are three types of neighborhood), moreover this grid is not a lattice, therefore, the possible shortest paths form more complicated sets, a kind of generalized traces. The linearizations of these sets are described by associative rewriting systems, and, as a main combinatorial result, the number of the shortest paths are computed between two triangles, where two cells are considered adjacent if they share at least one vertex.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Mersin-10, Famagusta, North Cyprus, Turkey

    Benedek Nagy & Arif Akkeleş

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  1. Benedek Nagy
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  2. Arif Akkeleş
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Correspondence to Benedek Nagy .

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Editors and Affiliations

  1. SUNY Buffalo State, Buffalo, New York, USA

    Valentin E. Brimkov

  2. Department of Applied Professional Studies, SUNY Fredonia, Fredonia, New York, USA

    Reneta P. Barneva

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Nagy, B., Akkeleş, A. (2017). Trajectories and Traces on Non-traditional Regular Tessellations of the Plane. In: Brimkov, V., Barneva, R. (eds) Combinatorial Image Analysis. IWCIA 2017. Lecture Notes in Computer Science(), vol 10256. Springer, Cham. https://doi.org/10.1007/978-3-319-59108-7_2

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  • DOI: https://doi.org/10.1007/978-3-319-59108-7_2

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-59108-7

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