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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Peano’s ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. For d2xy, it starts at the bottom with cells, and works up to include the entire square.

From Wikipedia, the free encyclopedia. The xy2d function works top down, starting with the most significant bits of x and yand building up the most significant bits of d first. Lecture Notes in Computer Science. In 3 dimensions, self-avoiding approximation curves can even contain knots. In other projects Wikimedia Courve.

In other projects Wikimedia Commons. A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism.

Space-filling curves are special cases courb fractal constructions.

Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. For example, the range of IP addresses used by computers can be mapped into a picture using the Hilbert curve. In the most general form, the range of such a function coure lie in an arbitrary topological spacebut in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane a planar curve or the 3-dimensional space space curve.

Here the sphere coube the sphere at infinity of hyperbolic 3-space. For xy2d, it starts at the top level of the entire square, and works its way down to the lowest level of individual cells. In many languages, these are better if implemented with iteration rather than recursion. If c was the first point in its coureb, then the first of these four orderings is chosen for the nine centers that replace c.

This page was last edited on 14 Decemberat It was cokrbe easy to extend Peano’s example to continuous curves without endpoints, which filled the entire n -dimensional Euclidean space where n is 2, 3, or any other positive integer.

Intuitively, a continuous curve in 2 or 3 or higher dimensions can be thought of as the path of a continuously moving point. Because Giuseppe Peano — was the first to discover one, space-filling curves epano the 2-dimensional plane are sometimes called Peano curvesbut that phrase also refers to the Peano curvethe specific example of a space-filling curve found by Peano.

Because of this locality property, ve Hilbert curve is widely used in computer science. Articles containing video clips Articles with example C code. Roughly speaking, differentiability puts a bound on how fast the curve can turn.

From Peano’s example, it was easy to deduce continuous curves whose ranges contained the n -dimensional hypercube for any positive integer n.

But a unit square has no cut-pointand so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points. The Hilbert curve is a prano variant of the same idea, based on subdividing squares into four equal smaller squares instead of into nine equal smaller squares. There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups.

Sometimes, the curve is identified with the range or image of the function the set of all possible values of the functioninstead of the function itself. The Hahn — Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves:.

## Hilbert curve

For the classic Peano and Hilbert space-filling curves, where two subcurves intersect in the technical sensethere is self-contact without self-crossing.

In many formulations of the Hahn—Mazurkiewicz theorem, second-countable is replaced by metrizable. Theory of Computing Systems. Ciurbe are four such orderings fe.

Both the true Hilbert curve and its discrete approximations are useful because they give a mapping between 1D and 2D space that preserves locality fairly well. From Wikipedia, the free encyclopedia. It is also possible to define curves without endpoints to be a continuous function on the real line or on the open unit interval 0, 1.

### Hilbert curve – Wikipedia

There will sometimes be points where the xy coordinates are close but their d values are far apart. The restriction of the Cantor function to the Cantor set is an example of such a function.

However, two curves or two subcurves of one curve may contact one another without crossing, as, for example, a line tangent to a circle does. A Hilbert curve also known as a Hilbert space-filling curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in[1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in The Peano curve itself is the limit of the curves through the sequences of square centers, as i goes to infinity.

A space-filling curve can be everywhere self-crossing if its approximation curves are self-crossing.

## File:Peano curve.png

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. Fractal canopy Space-filling curve H tree. Spaces that are the psano image of a unit interval are sometimes called Peano spaces. Space-Filling Curves and a Measure of Coherence, pp.