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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A grayscale photograph can be converted to a dithered black-and-white image using thresholding, with the leftover amount from each pixel added to the next pixel along the Hilbert curve. This subsequence is formed by grouping the nine smaller squares into three columns, ordering the centers contiguously within each column, and then ordering the columns from one side of the square to the other, in such a way that the distance between each consecutive pair of points in the subsequence equals the side length of the small squares.
Buddhabrot Orbit trap Pickover stalk. Retrieved from ” https: There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. There are four such orderings possible:. Continuous mappings Fractal curves Iterated function system fractals. Peano’s solution does not set up a continuous one-to-one correspondence between the unit interval and the unit square, and indeed such a correspondence does not exist see “Properties” below.
Theory of Computing Systems. Retrieved from ” https: In other projects Wikimedia Commons.
The following C code performs the mappings in both directions, using iteration and bit operations rather than recursion. Retrieved from ” https: The two mapping algorithms work in similar ways. A space-filling curve can be everywhere self-crossing if its approximation curves are self-crossing.
In geometrythe Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in The xy2d function works top down, starting with the most significant bits of x and yand building up the most significant bits of ee first.
Courbe de Peano (analyse) — Wikipédia
Each region is composed of 4 smaller regions, and so on, for a number of levels. Hilbert’s article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. 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. 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.
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. 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.
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. Conversely a compact metric space is second-countable. Fractal canopy Space-filling curve H tree. 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.
These two formulations are equivalent. These use the C conventions: A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square any continuous bijection from a compact coufbe onto a Hausdorff space is a homeomorphism.
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At that time the beginning of the foundation of general topologygraphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counterintuitive results. Both the true Hilbert curve and its discrete approximations are useful because they give a mapping between 1D and 2D space that preserves locality fairly well.
In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theoremsecond-countable then implies metrizable. 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. His purpose was to construct a continuous mapping from the unit interval onto the unit square. No differentiable space-filling curve can exist. In 3 dimensions, self-avoiding approximation curves can even contain knots. This page was last edited on 14 Decemberat For the classic Peano and Hilbert space-filling curves, where two subcurves intersect in the technical sensethere is self-contact without self-crossing.
To eliminate the inherent vagueness of this notion, Jordan in introduced the following rigorous definition, which has since been adopted as the precise description re the notion of courbbe continuous curve:. Peano was motivated by Georg Cantor ‘s earlier counterintuitive result that the infinite number of points in a cohrbe interval is the same cardinality as the infinite number pezno points in any finite-dimensional manifoldsuch as the unit square.
If c coyrbe the first point in its coirbe, then the first of these four orderings is chosen for the nine centers that replace c. For d2xy, it starts at the bottom with cells, and works up to include the entire square.
In many formulations of the Hahn—Mazurkiewicz theorem, second-countable is replaced by metrizable. Approximation curves remain within a bounded portion of n -dimensional space, but their lengths increase without bound. At level seach region is s by s cells.
Intuitively, a continuous curve in 2 or 3 or higher dimensions can be thought of as the path of a continuously moving point. The d2xy function works in the opposite order, starting with the least significant ;eano of dand building up x and y starting with the least significant bits. A Hilbert curve also known as a Hilbert space-filling 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