A Quadtree for Hyperbolic Space

We propose a data structure in d-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. We demonstrate the usefulness of our hyperbolic quadtree data structure by giving an algorithm for constant-approximate closest pair and dynamic constant-approximate nearest neighbours in hyperbolic space of constant dimension d.