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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Space-filling curves are special penao of fractal constructions. There will sometimes pesno points where the xy coordinates are close but their d values are far apart. His purpose was to construct a continuous mapping from the unit interval onto the unit square. They have also been used to help compress data warehouses. To eliminate the inherent vagueness of this notion, Jordan in introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a continuous curve:.

The entire square is viewed as composed of 4 regions, arranged 2 by 2.

Buddhabrot Orbit trap Pickover stalk. By using this site, you agree to the Terms dee Use and Privacy Policy. These choices lead to many different variants of the Peano curve. From Peano’s example, it was easy to deduce continuous curves whose ranges contained the n -dimensional hypercube for any positive integer n. Here the sphere is the sphere at infinity of hyperbolic 3-space. Views Read Edit View history. Fractal canopy Space-filling curve H tree. The problem Peano solved was whether such a mapping could be continuous; i.


A space-filling curve’s approximations can be self-avoiding, as the ocurbe above illustrate. The current region out of the 4 is rxrywhere rx and ry are each 0 or 1. In other projects Wikimedia Commons. By using this site, you agree to the Terms of Use and Privacy Policy. Hilbert curves in higher dimensions are an instance of a generalization of Gray codesand are sometimes used for similar purposes, for similar reasons.

The two subcurves intersect if the intersection of the two images is non-empty. It also calls the rotation function so that xy will be appropriate for the next level, on the next iteration. 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.

Views Read Edit View history. Graphics Gems II [10] discusses Hilbert curve coherency, and provides implementation. In 3 dimensions, self-avoiding approximation curves can even contain knots. This article is about a particular curve defined by Giuseppe Peano.

Peano’s ground-breaking article contained vourbe illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator.

Wikimedia Commons has media related to Space-filling curves. Views Read Edit View history. From Wikipedia, the free encyclopedia. Peano’s article also ends by observing that the technique can be obviously extended to other odd bases besides base 3.


File:Peano curve.png

Retrieved from ” https: It was common to associate the vague notions of thinness and 1-dimensionality to curves; all normally encountered curves were piecewise differentiable that is, have courhe continuous derivativesand such curves cannot fill up the entire unit square. A “multiple radix” variant of this curve with different numbers of subdivisions in different directions can be used to fill rectangles of arbitrary shapes.

The converse can’t always be true. For xy2d, it starts at the top level of the entire square, and works its way down to the lowest level of individual cells. Retrieved from ” https: The Hilbert Curve can be expressed by a rewrite system L-system.

Giuseppe Peano – Wikiquote

Space-filling curves for domains with unequal side lengths”. 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 languages, these are better if implemented with iteration rather than recursion. 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.