15,321 research outputs found
Sixteen space-filling curves and traversals for d-dimensional cubes and simplices
This article describes sixteen different ways to traverse d-dimensional space
recursively in a way that is well-defined for any number of dimensions. Each of
these traversals has distinct properties that may be beneficial for certain
applications. Some of the traversals are novel, some have been known in
principle but had not been described adequately for any number of dimensions,
some of the traversals have been known. This article is the first to present
them all in a consistent notation system. Furthermore, with this article, tools
are provided to enumerate points in a regular grid in the order in which they
are visited by each traversal. In particular, we cover: five discontinuous
traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton
indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and
Inside-out traversal; two discontinuous traversals based on subdividing
simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected
traversal; five continuous traversals based on subdividing cubes into 2^d
subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa
Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four
continuous traversals based on subdividing cubes into 3^d subcubes: the Peano
curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these
traversals are self-similar in the sense that the traversal in each of the
subcubes or subsimplices of a cube or simplex, on any level of recursive
subdivision, can be obtained by scaling, translating, rotating, reflecting
and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line
Harmonious Hilbert curves and other extradimensional space-filling curves
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.Comment: 40 pages, 10 figures, pseudocode include
Nutrients and Hydrology Indicate the Driving Mechanisms of Peatland Surface Patterning
Peatland surface patterning motivates studies that identify underlying structuring mechanisms. Theoretical studies so far suggest that different mechanisms may drive similar types of patterning. The long time span associated with peatland surface pattern formation, however, limits possibilities for empirically testing model predictions by field manipulations. Here, we present a model that describes spatial interactions between vegetation, nutrients, hydrology, and peat. We used this model to study pattern formation as driven by three different mechanisms: peat accumulation, water ponding, and nutrient accumulation. By on-and-off switching of each mechanism, we created a full-factorial design to see how these mechanisms affected surface patterning (pattern of vegetation and peat height) and underlying patterns in nutrients and hydrology. Results revealed that different combinations of structuring mechanisms lead to similar types of peatland surface patterning but contrasting underlying patterns in nutrients and hydrology. These contrasting underlying patterns suggest that the presence or absence of the structuring mechanisms can be identified by relatively simple short-term field measurements of nutrients and hydrology, meaning that longer-term field manipulations can be circumvented. Therefore, this study provides promising avenues for future empirical studies on peatland patternin
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