The spatial U-order and some of its mathematical characteristics

A new spatial order, called the U-order, is analyzed; the order is closely related to the Z-order or Morton-order which induces linear quadtrees. A number of mathematical relationships and algorithms are presented, in particular the functional relationships between the Euclidean coordinates of a pixel in the U-domain and its location code, the translation of a pixel by an arbitrary vector, finding the eight neighbours of a pixel, mirroring and discrete rotations of pixels. The operations are generalised to non-pixel nodes.