• Corpus ID: 17888931

Harmonious Hilbert curves and other extradimensional space-filling curves

@article{Haverkort2012HarmoniousHC,
  title={Harmonious Hilbert curves and other extradimensional space-filling curves},
  author={Herman J. Haverkort},
  journal={ArXiv},
  year={2012},
  volume={abs/1211.0175}
}

• H. Haverkort
• Published 1 November 2012
• Mathematics
• ArXiv

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.