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.
[1]
Christos Faloutsos,et al.
On packing R-trees
,
1993,
CIKM '93.
[2]
H. Hahn.
Leçons sur l'intégration et la recherche des fonctions primitives
,
1904
.
[3]
D. Hilbert.
Über die stetige Abbildung einer Linie auf ein Flächenstück
,
1935
.
[4]
Harry W. Nelson.
Elemente der Mathematik
,
1911
.
[5]
G. Peano.
Sur une courbe, qui remplit toute une aire plane
,
1890
.
[6]
D. Hilbert.
Ueber die stetige Abbildung einer Line auf ein Flächenstück
,
1891
.
[7]
Arthur R. Butz,et al.
Alternative Algorithm for Hilbert's Space-Filling Curve
,
1971,
IEEE Transactions on Computers.
[8]
Herman J. Haverkort,et al.
Four-dimensional hilbert curves for R-trees
,
2008,
JEAL.
[9]
Rolf Niedermeier,et al.
On Multidimensional Curves with Hilbert Property
,
2000,
Theory of Computing Systems.
[10]
Michael Bader,et al.
Space-Filling Curves - An Introduction with Applications in Scientific Computing
,
2012,
Texts in Computational Science and Engineering.
[11]
Herman J. Haverkort,et al.
Recursive tilings and space-filling curves with little fragmentation
,
2010,
J. Comput. Geom..
[12]
Herman J. Haverkort.
An inventory of three-dimensional Hilbert space-filling curves
,
2011,
ArXiv.
[13]
Herman J. Haverkort,et al.
Locality and bounding-box quality of two-dimensional space-filling curves
,
2008,
Comput. Geom..
[14]
Walter Wunderlich.
Über Peano-Kurven.
,
1973
.