Directional neighbourhood calculations in spatial partition trees

We present an extension of the multidimensional binary indexing algorithm for neighbourhood calculations in spatial partition trees [1]. The algorithm in the earlier paper provided a matrix characterization to support calculus of neighbours in a spatial partition tree representation. That calculus was implicitly defined only for face-to-face neighbours. The extension presented here considers any possible direction in which a neighbour sub-interval of a quadtree partition can be found, and not only along the coordinate axis. The paper starts from basic topological considerations in two dimensional space to finally generalize the result obtained for any arbitrary dimension. Finally, we present an explicit calculus of all neighbour location vectors of a given sub-interval as well as a graphic example in a three-dimensional space.