Theory of Neighborhood Sequences on Hexagonal Grids

In this paper distances based on neighborhood sequences in hexagonal grid are analysed. From every vertex of the grid one can step to various neighbors directed by a neighborhood sequence, which allows to vary the used neighborhood relations step by step. The distances of two vertices are defined as the lengths of shortest paths between them. Theoretic results, such as algorithm to provide a shortest path, computing the distance and properties of distances are shown. Necessary and sufficient condition to have metric distance is proven. Digital circles and discs are described as well.