Analysis of the Hilbert Curve for Representing Two-Dimensional Space

Abstract We give closed-form expressions for the average number of runs to cover an arbitrary square region in two dimensions using a Hilbert curve. The practical use of the derived formula is that it allows the estimation of the quality of the linearization obtained by using the Hilbert curve to map from a two-dimensional space to a one-dimensional space. Hilbert curves are used extensively as a basis for multi-dimensional indexing structures, and for declustering multi-dimensional data.