Harmonious Hilbert curves and other extradimensional space-filling curves

TL;DR

This paper generalizes Hilbert and other space-filling curves to arbitrary dimensions, introducing harmonious Hilbert curves with statistical invariance and compatibility properties, along with constructions for specific subspace traversals.

Contribution

It presents a novel generalization of Hilbert curves called harmonious Hilbert curves, with properties like statistical invariance and compatibility across dimensions, and extends these ideas to other space-filling curves.

Findings

Harmonious Hilbert curves are compatible across dimensions.

Curves exhibit statistical invariance under rotation.

Construction methods for specific subspace traversals are provided.

Abstract

This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d' < d, the d-dimensional curve is compatible with the d'-dimensional curve with respect to the order in which the curves visit the points of any d'-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation---unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper.