Distances with neighbourhood sequences in cubic and triangular grids

In this paper we compute distances with neighbourhood sequences in the cubic and in the triangular grids. First we give a formula which computes the distance with arbitrary neighbourhood sequence in the three-dimensional digital space. After this, using the injection of the triangular grid to the cubic grid, we modify the formula for Z^3 to the triangular plane. The distances in the triangular grid have some properties which are not present on the square and cubic grids. It may be non-symmetric, and it is possible that the distance depends on the ordering of elements of the initial part of the neighbourhood sequence. The distance depends on the ordering of the initial part (up to the kth element) of the neighbourhood sequence if and only if there is a permutation of these elements such that the distance (up to value k) is non-symmetric using the elements in this new order. This dependence means somehow more flexibility of the distances based on neighbourhood sequences on the triangular grid than in Z^n.