Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids

Abstract

In this paper we use symmetric coordinate systems for the hexagonal and the triangular grids (that are dual of each other). We present new coordinate systems by extending the symmetric coordinate systems that are appropriate to address elements (cells) of cell complexes. Coordinate triplets are used to address the hexagon/triangle pixels, their sides (the edges between the border of neighbour pixels) and the points at the corners of the hexagon/triangle pixels. Properties of the coordinate systems are detailed, lines (zig-zag lines) and lanes (hexagonal stepping lanes) are defined on the triangular (resp. hexagonal) grid by fixing a coordinate value. The bounding relation of the cells can easily be captured by the coordinate values. To illustrate the utility of these coordinate systems some topological algorithms, namely collapses and cuts are presented.