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Harmonious Hilbert curves and other extradimensional spacefilling curves∗
, 2014
"... This paper introduces a new way of generalizing Hilbert’s twodimensional spacefilling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d ′ < d, the ddimensional curve is compatible with the d′dimensional curve with respect ..."
Abstract

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This paper introduces a new way of generalizing Hilbert’s twodimensional spacefilling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d ′ < d, the ddimensional curve is compatible with the d′dimensional curve with respect to the order in which the curves visit the points of any d′dimensional axisparallel 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 halfcoil curve, and the Meurthe curve). The ddimensional 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 halfcoil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of Rtrees, this paper shows how to construct a 2ddimensional generalized Hilbert or Peano curve that traverses the points of a certain ddimensional diagonally placed subspace in the order of a given ddimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper. 1