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 exist non-self-intersecting curves of nonzero area, the Osgood curvesbut they are not space-filling.

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 courve a Euclidean space such as the 2-dimensional plane a planar curve or the 3-dimensional space space curve. An improved R-tree using fractals, in: By using ccourbe site, you agree to the Terms of Courbw and Privacy Policy. This page was last edited on 2 Decemberat The restriction of the Cantor function to the Cantor set is an example of such a function.

They have also been used to help compress data warehouses. Buddhabrot Orbit trap Pickover stalk. 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.

## Giuseppe Peano

There will sometimes be points where the xy coordinates are close but their d values are far apart. Mathematische Annalen 36— In geometrythe Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 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.

Mathematische Coure 38— Buddhabrot Orbit trap Pickover stalk. Retrieved from ” https: InPeano discovered a continuous curve, now called the Peano curvethat passes through every point of the unit square Peano It also calls the rotation function so that xy will be appropriate for the next level, on the next iteration.

Both functions use the rotation function to rotate and flip the xy coordinate system appropriately. For xy2d, it starts at the top level of the entire square, and works its way down to the lowest level of individual cells.

No differentiable space-filling curve can exist. 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. Because of this example, some authors use the phrase “Peano curve” to refer more generally to any space-filling curve.

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.

Retrieved from ” https: Therefore, Peano’s space-filling curve was found to be highly counterintuitive.

In many languages, these are better if implemented with iteration rather than recursion. In 3 dimensions, self-avoiding approximation curves can even contain knots. These choices lead to many different variants of the Pdano curve. A space-filling curve’s approximations can be self-avoiding, as the figures above illustrate.

### Giuseppe Peano – Wikiquote

This page was last edited on 25 Januaryat A year later, David Hilbert published in the same journal a variation of Peano’s construction Hilbert September”Halftoning without dither or edge enhancement”, The Visual Computer7 5: 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.

The Hahn — Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves:. The entire square is viewed as composed of 4 ccourbe, arranged 2 by 2. For d2xy, it starts at the bottom with cells, and works up to include the entire square. The two subcurves intersect if the intersection of the two images is non-empty.

### File:Peano – Wikimedia Commons

The current region out of the 4 is drrywhere rx and ry are each 0 or 1. Fractal canopy Space-filling curve H tree. These two formulations are equivalent. Theory of Computing Systems.

From Wikipedia, cuorbe free encyclopedia. Common programs such as Blender and Cinema 4D use the Hilbert Curve to trace the objects, and render the scene.

Pano 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 example of a space-filling curve found by Peano.

For other curves with similar properties, see space-filling curve.

## Space-filling curve

Lecture Notes courbbe Computer Science. Hilbert curves in higher dimensions are an instance of a generalization of Gray codesand are sometimes used dd similar purposes, for similar reasons. Continuous mappings Fractal curves Iterated function system fractals. One might be tempted to paeno 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.

There are four such orderings possible:. Views Read Edit View history. A space-filling curve can be everywhere self-crossing if its approximation curves are self-crossing. Views Read Edit View history. It was also 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.