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

Mathematics

Khaled Abuhmaidan¹  Benedek Nagy² 

[1] Department of Computing, Global College for Engineering and Technology, Ruwi, P.C. 112, Muscat P.O. Box 2546, Oman;Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, 99450 Famagusta, North Cyprus, via Mersin 10, Turkey;

关键词: non-traditional grids;    triangular grid;    computer graphics;    discretized translations;    digital geometry;    non-bijective mappings;   

DOI  :  10.3390/math8010029

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown