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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These choices lead to many different variants of the Peano curve. This page was last edited on 25 Januaryat This article pano about a particular curve defined by Giuseppe Peano. Wikimedia Commons has media related to Hilbert curve.
Peano curve – Wikipedia
There exist non-self-intersecting curves of nonzero area, the Osgood curvesbut they are not space-filling. It was also easy to extend Peano’s example to continuous curves without endpoints, which filled the entire n -dimensional Euclidean ce where n is 2, 3, or any other positive integer.
This page was last edited on 14 Decemberat Theory of Computing Systems.
For multidimensional databases, Hilbert order has been proposed to be used instead of Z order because it has better locality-preserving behavior.
The Hilbert Curve is commonly used among rendering images or videos.
Courbe de Peano (analyse) — Wikipédia
For other curves with similar properties, see space-filling curve. But the graphical construction was perfectly clear to him—he made an ornamental tiling showing a picture of the curve in his home in Turin.
In the most general form, the range of such a function may lie in an arbitrary topological spacebut in the most commonly studied cases, psano range will lie in a Euclidean space such as the 2-dimensional plane a planar curve or the 3-dimensional space space curve. Common programs such as Blender and Cinema 4D use the Hilbert Curve to trace the objects, and render the scene.
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.
Views Read Edit View history. There is a single FOR loop that iterates through levels.
In 3 dimensions, self-avoiding approximation curves can even contain knots. Among these four orderings, the one for s is chosen in such a way that the distance between the first point of the ordering and its predecessor in P i also equals the side length of the small squares. Buddhabrot Orbit trap Pickover stalk. Mathematische Annalen 36— The two subcurves intersect if the intersection of the two images is non-empty.
These two formulations are equivalent.
Space-filling curves for domains with unequal side lengths”.
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.
Retrieved from ” https: For example, Hilbert curves have been used to compress and accelerate R-tree indexes  see Hilbert Ee. On each iteration, an amount is added to d or to x and ydetermined peanno which of the 4 regions it is in at the current level. 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.
Each region is composed of 4 smaller regions, and so on, for a number of levels. 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.
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. Approximation curves remain within a bounded portion of n -dimensional space, but their lengths increase without bound. A space-filling curve’s approximations can be self-avoiding, as the figures above illustrate.
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.