A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in , as a variant of the space-filling Peano curves discovered by Giuseppe Peano in . Mathematische Annalen 38 (), – ^ : Sur une courbe, qui remplit toute une aire plane. Une courbe de Peano est une courbe plane paramétrée par une fonction continue sur l’intervalle unité [0, 1], surjective dans le carré [0, 1]×[0, 1], c’est-à- dire que. Dans la construction de la courbe de Hilbert, les divers carrés sont parcourus . cette page d’Alain Esculier (rubrique courbe de Peano, équations de G. Lavau).

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There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups.

At level seach region is s by s cells. Here the sphere is the sphere at infinity of hyperbolic 3-space.

Courbe de Peano (analyse) — Wikipédia

Hilbert curves in higher dimensions are an instance of a generalization of Gray codes courbbe, and are sometimes used for similar purposes, for similar reasons. Buddhabrot Orbit trap Pickover stalk. Because of this locality property, the Hilbert curve is widely used in computer science.

For example, the range of IP addresses used by computers can be mapped into a picture using the Hilbert curve. Fractal canopy Space-filling curve H tree.

Courbe de Peano (analyse)

Conversely a compact metric space is second-countable. Peano’s curve may be constructed by a sequence of steps, where the i th step constructs a set S i of squares, and a sequence P i of the centers of the squares, from the set and sequence constructed in the previous step.

The Hilbert Curve can be expressed by a rewrite system L-system. There will sometimes be points where the xy coordinates are close but their d values are far apart. Wikimedia Commons has media related to Hilbert curve. Code to do this would map from 1D to 2D, and the Hilbert curve is sometimes used because it does not create the distracting patterns that would be visible to the eye if the order were simply left to right across each row of pixels.

Wikimedia Commons has media related to Space-filling curves. These choices lead to many different variants of the Peano curve. Space-filling curves are special cases of fractal constructions. From Wikipedia, the free encyclopedia. It was common to associate the vague notions of thinness peani 1-dimensionality to curves; all normally encountered curves were piecewise differentiable ckurbe is, have piecewise continuous derivativesand such curves cannot fill up the entire unit square.


In many languages, these are better if implemented with iteration rather than recursion. A non-self-intersecting continuous curve cannot fill the unit square because that will make the ocurbe a homeomorphism from the unit interval onto the unit square any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism. In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theoremsecond-countable then implies metrizable.

Space-filling curves for domains with unequal side lengths”. There is a single FOR loop that iterates through levels.

There exist non-self-intersecting curves of nonzero area, the Osgood curvesbut they are not space-filling. It also calls the rotation function so that xy will be appropriate for the next level, on the next iteration. If a curve is not injective, then one can find two intersecting subcurves of the curve, each obtained by considering the images of two disjoint segments from the curve’s domain the unit line segment.

Theory of Computing Systems. Because Giuseppe Peano — was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curvesbut that phrase also refers to the Peano curvethe specific courb of a space-filling curve found by Peano. Peano’s solution does not set ce a continuous one-to-one correspondence between the unit interval and the unit square, and indeed such a correspondence does not exist see “Properties” below. The two mapping algorithms work in similar ways.

The Hilbert Curve is commonly courb among rendering images or videos. Peano was motivated by Georg Cantor ‘s earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifoldsuch as the unit square.

His purpose was to construct a continuous mapping from the unit interval onto coudbe unit square. Both functions use the rotation function to rotate and flip the xy coordinate system appropriately.

File:Peano – Wikimedia Commons

In the definition of the Peano curve, it is possible to perform some or all of the steps by making the centers of each row of three squares be contiguous, rather than the centers of each column of squares. In 3 dimensions, self-avoiding approximation curves can even contain knots.

Given the variety of applications, it is useful to have algorithms to map in both directions. For multidimensional databases, Hilbert order has been proposed to be used instead of Z order because it has better locality-preserving behavior.


This subsequence is formed by grouping the nine smaller squares into three columns, ordering the centers contiguously within each column, coirbe then ordering the columns from one side of the square to the other, in such a way that the distance between each consecutive pair of peamo in the subsequence equals the side length of the small squares.

In mathematical analysisa space-filling curve is a curve whose range contains the entire 2-dimensional unit square or more generally an n -dimensional unit hypercube. Mathematische Annalen 38— Therefore, Peano’s space-filling curve was found to be highly counterintuitive. Because of this example, some authors use the phrase “Peano curve” to refer more generally to any space-filling curve.

There are four such orderings possible:. Sur une courbe, qui remplit toute une aire plane.

Peano curve

For other curves with similar properties, see space-filling curve. Among these pdano orderings, the one for s is chosen in such a way that the distance between the first point of the ordering and its predecessor in P i also equals the side length of the small squares. This page was last edited on 25 Januaryat For example, Hilbert curves have been used to compress and accelerate R-tree indexes [4] see Hilbert R-tree.

An improved R-tree using fractals, in: Mathematische Annalen 36— Intuitively, a continuous curve in 2 or 3 or higher dimensions dourbe be thought of as the path of a continuously moving point.

If c was the first point in its ordering, then the first of these four orderings is chosen for the nine centers that replace c. Code to generate the image would map from 2D to 1D to find the color of each pixel, and the Hilbert curve is sometimes used because it keeps couurbe IP addresses close to each other in the picture.

It was also easy to extend Peano’s example d continuous curves without endpoints, which filled the entire n -dimensional Euclidean space where n is 2, 3, or any other positive integer. From Peano’s example, it was easy to deduce continuous curves whose ranges contained the n -dimensional hypercube for any positive integer n.