In this paper we analyse some properties of the triangular and hexagonal grids in the 2D digital space. We define distances based on the neighbouring relations that can be introduced in these grids. On the triangular grid, this can be done by the help of neighbourhood sequences. We construct a shortest path in the hexagonal grid, in a natural way. We present an algorithm, which produces for a given neighbourhood sequence a shortest path between arbitrary two points of the triangular grid, and calculate the distance of these two points, as well.