Honeycomb Geometry: Rigid Motions on the Hexagonal Grid

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

“…In [22] we proved that in the hexagonal grid case a similar result is obtained for rotations corresponding to Eisenstein integers. We shall say that a rotation U α is Eisenstein rational 1 if α is given by a Eisenstein triple and then we have the following result [22,Proposition 9].…”

“…We shall say that a rotation U α is Eisenstein rational 1 if α is given by a Eisenstein triple and then we have the following result [22,Proposition 9].…”

“…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 .…”

“…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 . Indeed, it is shown in [22] that, on average, digitized rigid motions defined on the hexagonal grid lose less information than their counterparts defined with respect to the square grid.…”

“…For the hexagonal lattice, we use Eisenstein integers instead of Gaussian integers [4,9]. First, we discuss the density of rational rotations on the hexagonal grid, which were characterized in [22]. That is similar to the study of rational rotations on the square grid -angles given by right triangles of integer sides [1].…”