Calculation of Mappings Between One and n-dimensional Values Using the Hilbert Space-filling Curve ⋆

Abstract

. This report reproduces and briefly discusses an algorithm proposed by Butz [2] for calculating a mapping between one-dimensional values and n-dimensional values regarded as being the coordinates of points lying on Hilbert Curves. It suggests some practical improvements to the algorithm and presents an algorithm for calculating the inverse of the mapping, from n-dimensional values to one-dimensional values. 1 Introduction The concept of a space-filling curve emerged in the 19-th Century and is originally attributed to Peano [5] who expressed it in mathematical terms. The first graphical, or geometrical, representation of the space-filling curve is attributed to David Hilbert [3]. He illustrates the the concept in 2 dimensions but it is applicable in any number of dimensions. Hilbert Curves pass through every point in an n-dimensional space once and once only in some particular order according to some algorithm. As such they graphically express a mapping between one-dimensional values...