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Open AccessArticle

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

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Department of Computing, Global College for Engineering and Technology, Ruwi, P.C. 112, Muscat P.O. Box 2546, Oman
2
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, 99450 Famagusta, North Cyprus, via Mersin 10, Turkey
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Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in 2011 IEEE 16th World Symposium on Applied Machine Intelligence and Informatics (SAMI 2018), Kosice, Slovakia, 7–10 February 2018; pp. 183–188.
Mathematics 2020, 8(1), 29; https://doi.org/10.3390/math8010029
Received: 18 November 2019 / Revised: 17 December 2019 / Accepted: 20 December 2019 / Published: 23 December 2019
(This article belongs to the Section Mathematics and Computer Science)
The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given. View Full-Text
Keywords: non-traditional grids; triangular grid; computer graphics; discretized translations; digital geometry; non-bijective mappings non-traditional grids; triangular grid; computer graphics; discretized translations; digital geometry; non-bijective mappings
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MDPI and ACS Style

Abuhmaidan, K.; Nagy, B. Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane. Mathematics 2020, 8, 29. https://doi.org/10.3390/math8010029

AMA Style

Abuhmaidan K, Nagy B. Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane. Mathematics. 2020; 8(1):29. https://doi.org/10.3390/math8010029

Chicago/Turabian Style

Abuhmaidan, Khaled; Nagy, Benedek. 2020. "Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane" Mathematics 8, no. 1: 29. https://doi.org/10.3390/math8010029

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