Digital geometry on a cubic stair-case mesh

Abstract

In this paper, we investigate digital geometry on the rhombille tiling, D(6,3,6,3), that is the dual of the semi-regular tiling called hexadeltille T(6,3,6,3) tiling and also known as trihexagonal tiling. In fact, this tiling can be seen as an oblique mesh of the cubic grid giving practical importance to this specific grid both in image processing and graphics. The properties of the coordinate systems used to address the tiles are playing crucial roles in the simplicity of various algorithms and mathematical formulae of digital geometry that allow to work on the grid in image processing, image analysis and computer graphics, thus we present a symmetric coordinate system. This coordinate system has a strong relation to topological/combinatorial coordinate system of the cubic grid. It is an interesting fact that greedy shortest path algorithm may not be used on this grid, despite to this, we present algorithm to provide a minimal-length path between each pair of tiles, where paths are defined as sequences of neighbor tiles (those are considered to be neighbors which share a side). We also prove closed formula for computing the digital, i.e., path-based distance, the length (the number of steps) of a/the shortest path(s). Some example pictures on this grid are also presented, as well as its possible application as pixel geometry for color images and videos on the hexagonal grid. (c) 2022 Elsevier B.V. All rights reserved.