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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Retrieved from ” https: 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.

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 ccourbe consecutive pair of points in the subsequence equals the side length of the small squares.

Hilbert’s article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. Peano’s article also ends by observing that the technique can be obviously extended to other odd bases besides base 3.

Space-filling curve – Wikipedia

Intuitively, a continuous curve in 2 or 3 or higher dimensions can be thought of as the path of a continuously moving point. A space-filling curve’s approximations can be self-avoiding, as the figures above penao.

In geometrythe Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in Graphics Gems II [10] discusses Hilbert curve coherency, and provides implementation.

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.

Code ed do this would map from 1D to 2D, and the Hilbert curve dee 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.


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.

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In other projects Wikimedia Commons. 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. 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. 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. 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.

However, two curves or two subcurves of one curve may contact one another without crossing, as, for example, a line tangent to a cpurbe does.

In many languages, these are better paeno implemented with iteration rather than recursion. Lecture Notes in Computer Science.

Buddhabrot Orbit trap Pickover stalk.

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.

These use the C conventions: One might be tempted to think that the meaning of curves intersecting is that they necessarily epano each other, like the intersection point of two non-parallel lines, from one side to the other.

The two subcurves intersect if the intersection of the courbbe images is non-empty. Approximation curves remain within a bounded portion of n -dimensional space, but their lengths increase without bound. These choices lead to many different variants of the Peano curve. 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 piecewise continuous derivativesand such curves cannot fill up the entire unit square.

Views Read Edit View history. The restriction of the Cantor function to the Cantor set is an example of such a function.

Hilbert curve

This article is about a particular curve defined by Giuseppe Peano. It is possible to implement Hilbert curves efficiently even when the data space does not form a square. Fractal canopy Space-filling curve H tree. A curve with endpoints is a continuous function whose domain is the unit interval [0, 1]. At level seach region is s by s cells.


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

Peano’s ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. The two mapping algorithms work in similar ways. Continuous mappings Fractal curves. Wikimedia Commons has media related to Hilbert curve. So it consumes 2 input bits, either 2 from d or 1 each from x and yand generates two output bits.

A “multiple radix” variant of this curve with different numbers of subdivisions in different directions can be used to fill rectangles of arbitrary shapes. Fractal canopy Space-filling curve H tree. Each region is composed of 4 smaller regions, and so on, for a number of levels. Hilbert curves in higher dimensions are an instance of a generalization of Gray codesand are sometimes used for similar purposes, for similar reasons.

Theory of Computing Systems. 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 following C code performs the mappings in both directions, using iteration and bit operations rather than recursion.

Courbe de Peano (analyse)

Given the variety of applications, it is useful to have algorithms to map in both directions. Conversely a compact metric space is second-countable. If a curve is not se, 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 cojrbe line segment. Continuous mappings Fractal curves Iterated function system fractals. Spaces that are the continuous image of a unit interval are sometimes called Peano spaces.

Wikimedia Commons has media related to Space-filling curves.