Hilbert curve

Figure 2 also shows the Hilbert curves of order two and three.

The following provides a brief introduction to the Hilbert curve.

It is possible to implement Hilbert curves efficiently even when the data space does not form a square.

Because of this locality property, the Hilbert curve is widely used in computer science.

Moreover there are several possible generalizations of Hilbert curves to higher dimensions.

But the Hilbert curve does a good job of keeping those d values close together much of the time.

Quaternary numbers are used in the representation of 2D Hilbert curves.

The Hilbert curve can be generalized for higher dimensionalities.

The analytic form of the Hilbert curve, however, is more complicated than Peano's.

It is a fractal object similar in its construction to the dragon curve and the Hilbert curve.

The Hilbert Curve can be expressed by a rewrite system (L-system).

Like the Hilbert curve, the Moore curve can be extended to three dimensions:

For example, the range of IP addresses used by computers can be mapped into a picture using the Hilbert curve.

Arthur Butz provided an algorithm for calculating the Hilbert curve in multidimensions.

It has been noted that this dimensional requirement is not met by fractal space-filling curves such as the Hilbert curve.

JSHilbert - Calculate normalised distances along Hilbert curve for points in the unit square.

Graphics Gems II discusses Hilbert Curve coherency, and provides implementation.

(The Hilbert value of a point is the length of the Hilbert curve from the origin to the point.)

Hilbert curves in higher dimensions are an instance of a generalization of Gray codes, and are sometimes used for similar purposes, for similar reasons.

XKCD cartoon using the locality properties of the Hilbert curve to create a "map of the internet"

Hilbert R-trees use space-filling curves, and specifically the Hilbert curve, to impose a linear ordering on the data rectangles.

The Hilbert curve runs in a unique pattern through the whole image, it traverses every pixel without visits any of them twice and keeps a continuous curve.

Both the true Hilbert curve and its discrete approximations are useful because they give a mapping between 1D and 2D space that fairly well preserves locality.

For example, Hilbert curves have been used to compress and accelerate R-tree indexes (see Hilbert R-tree).

As an alternative, the Hilbert curve has been suggested as it has a better order-preserving behaviour, but here the calculations are much more complicated, leading to significant processor overhead.