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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The problem Peano solved was whether such a mapping could be continuous; i. The two mapping algorithms work in similar ways. Each region is composed of 4 smaller regions, and so on, for a number of levels. The Hilbert Curve is commonly used among rendering images or videos.

Peano’s ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. To eliminate the inherent vagueness of this notion, Jordan in introduced the following rigorous definition, which ee since been adopted as the precise description of the notion of a continuous curve:.

This page was last edited on 14 Decemberat These choices lead to many different variants of the Peano curve.

## Giuseppe Peano

As a base case, S 0 consists of the single unit square, and P 0 is the one-element sequence consisting of its center point. In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theoremsecond-countable then implies metrizable.

The analytic form of the Hilbert curvehowever, is more complicated than Peano’s. The following C code performs the mappings in both directions, using iteration and bit operations rather than recursion. 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. In the most general form, the range of such a function 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.

Therefore, Peano’s space-filling curve was found to be highly counterintuitive.

A Hilbert curve also known as a Hilbert space-filling curve is paeno 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 So it consumes 2 input bits, either 2 from d or 1 each from x and yand generates two output bits. The entire square is viewed as composed of 4 regions, arranged 2 by 2. This page was last edited on 2 Decemberat The Hahn — Mazurkiewicz theorem is the following peaon of spaces that are the continuous image of curves:.

## Hilbert curve

Mathematische Annalen 36— 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.

Lecture Notes in Computer Science. It also calls the rotation function so that xy will be appropriate for the next level, on the next iteration. Hilbert curves in higher dimensions are an instance of a generalization of Gray codesand are sometimes used for similar purposes, for similar reasons.

By using this site, you agree to the Terms of Use and Privacy Policy. For xy2d, it starts at the top level of the entire square, and works its way down to the lowest level of individual cells. One might be tempted to think that the meaning of curves intersecting is that they necessarily cross each other, like the intersection point of two non-parallel lines, from one side to the other.

### Space-filling curve – Wikipedia

Space-filling curves are special cases of fractal constructions. Buddhabrot Orbit trap Pickover stalk. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality.

No differentiable space-filling curve can exist. 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 nearby IP addresses close to each other in the picture. If c was the first point in its ordering, then the first of these four orderings is ee for the nine centers that replace c.

Buddhabrot Orbit trap Pickover stalk. An improved R-tree using fractals, in: Approximation curves remain within a bounded portion of n -dimensional space, but their lengths increase without bound.

There is a single FOR loop that iterates through levels. The handling of booleans in C means that in xy2d, the variable rx is set to 0 or 1 to match bit s of xand similarly for ry. In many formulations of the Hahn—Mazurkiewicz theorem, second-countable is replaced by metrizable.

The current region out of the 4 is rxrywhere rx and ry are each 0 or 1. 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.