Harmonious Hilbert curves and other extradimensional space-filling curves
Herman Haverkort

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…
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Taxonomy
TopicsData Management and Algorithms · Soil Geostatistics and Mapping · Data Visualization and Analytics
