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

Khaled Abuhmaidan and Benedek Nagy
Additional contact information
Khaled Abuhmaidan: Department of Computing, Global College for Engineering and Technology, Ruwi, P.C. 112, Muscat P.O. Box 2546, Oman
Benedek Nagy: Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, 99450 Famagusta, North Cyprus, via Mersin 10, Turkey

Mathematics, 2019, vol. 8, issue 1, 1-16

Abstract: 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.

Keywords: non-traditional grids; triangular grid; computer graphics; discretized translations; digital geometry; non-bijective mappings (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/1/29/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/1/29/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2019:i:1:p:29-:d:301228

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().