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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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.

The problem Peano solved was whether such a mapping could be continuous; i. The two subcurves intersect if the intersection of the two images is non-empty. But courbbe unit square has no cut-pointand so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points.

For xy2d, it starts at the top level of the entire square, and works its way down to the lowest level of individual cells. Code to do this would map from 1D to 2D, and the Hilbert curve is sometimes used because it does not create the ciurbe patterns that would be visible to the eye if the order were simply left to right across each row of pixels.

Buddhabrot Orbit trap Pickover stalk. Articles containing video clips Articles with example C code. Fe une courbe, qui remplit toute une aire plane. The two mapping algorithms work in similar ways.

### File:Peano – Wikimedia Commons

Lecture Notes in Computer Science. Hilbert’s article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. Each region is composed of 4 smaller regions, and so on, for a number of levels.

On each iteration, an amount is added corube d or to x and ydetermined by which of the 4 regions it is in at the current level.

Most well-known space-filling curves are constructed iteratively as the limit of a sequence of piecewise linear continuous curves, each one more closely approximating the space-filling limit. Cokrbe speaking, differentiability puts a bound on how fast the curve can turn. Fractal canopy Space-filling curve H tree. The Hilbert Curve can be expressed by a rewrite system L-system.

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

Fractal canopy Space-filling curve H tree.

Wiener pointed out in The Fourier Integral and Certain of its Applications that space filling curves could be used to reduce Lebesgue integration in higher dimensions to Lebesgue integration in one dimension. His purpose was to construct a continuous mapping from the unit interval onto the unit square.

## Courbe de Peano (analyse)

In geometrythe Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 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 coudbe infinite number of points in any finite-dimensional manifoldsuch as the unit square.

So it consumes 2 input bits, either 2 from d or 1 each from x and yand generates two output bits.

By using this site, you agree to the Terms of Use and Privacy Policy. In the most general form, the range of such a ;eano may 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.

The Hilbert curve is a simpler variant of the same idea, based on subdividing squares into four equal smaller squares instead of into nine equal smaller squares. Here the sphere is the sphere at infinity of hyperbolic peao.

There are four such orderings possible:.

Space-filling curves are special cases of fractal constructions. It is possible to implement Hilbert curves efficiently even when the data space does not form a square. Retrieved from ” https: Views Read Edit View history.