### Abstract

Original language | English |
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Publisher | s.n. |

Number of pages | 40 |

Publication status | Published - 2012 |

### Publication series

Name | arXiv.org |
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Volume | 1211.0175 [cs.CG] |

### Fingerprint

### Cite this

*Harmonious Hilbert curves and other extradimensional space-filling curves*. (arXiv.org; Vol. 1211.0175 [cs.CG]). s.n.

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*Harmonious Hilbert curves and other extradimensional space-filling curves*. arXiv.org, vol. 1211.0175 [cs.CG], s.n.

**Harmonious Hilbert curves and other extradimensional space-filling curves.** / Haverkort, H.J.

Research output: Book/Report › Report › Academic

TY - BOOK

T1 - Harmonious Hilbert curves and other extradimensional space-filling curves

AU - Haverkort, H.J.

PY - 2012

Y1 - 2012

N2 - 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

AB - 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

M3 - Report

T3 - arXiv.org

BT - Harmonious Hilbert curves and other extradimensional space-filling curves

PB - s.n.

ER -