Geometric transformations on the hexagonal grid

Abstract

The hexagonal grid has long been known to be superior to the more traditional rectangular grid system in many aspects in image processing and machine vision related fields. However, systematic developments of the mathematical backgrounds for the hexagonal grid are conspicuously lacking. The purpose of this paper is to study geometric transformations on the hexagonal grid. Formulations of the transformation matrices are carried out in a symmetrical hexagonal coordinate frame. A trio of new trigonometric functions are defined in this paper to facilitate the rotation transformations. A fast algorithm for rounding an arbitrary point to the nearest hexagonal grid point is also presented.

Isoperimetrically optimal polygons in the triangular grid with Jordan-type neighbourhood on the boundary

“…The pixels of the hexagonal grid can be addressed with pairs of integers [14]. There is a more elegant solution using coordinate triplets, where the sum of the values is zero in a triplet, reflecting the symmetry of the grid [12]. We note here that the digital topology of the hexagonal grid works well: the usual neighbourhood has no such a disadvantage as the neighbourhood relations of the square grid.…”

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

“…Studies related to digitized rotations on the hexagonal grid are less numerous: Her in [10]while working with the hexagonal grid represented by cube coordinate system in 3D-showed how to derive a rotation matrix such that it is simpler than a 3D rotation matrix obtained in a direct way ; Pluta et al in [22] provided a framework to study local alterations of discrete points on the hexagonal grid under digitized rigid motions. They also characterized Eisenstein rational rotations on the hexagonal grid which, together with their framework, allowed them to compare the loss of information induced by digitized rigid motions with the both grids .…”

Hexagonal Structure for Intelligent Vision

“…14), theories and equations previously developed for the oblique coordinate frame can directly be used in * R 3 . Moreover, in [32], the use of this symmetrical hexagonal coordinate frame is demonstrated to derive various affine transformations. Due to the physical relationships between the symmetrical hexagonal coordinate frame and the 3-dimensional Cartesian frame R 3 , geometric transformations on the hexagonal grid are conveniently simplified and the beautiful symmetry property of the hexagonal grid is successfully preserved.…”