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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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. Because 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.
Intuitively, a continuous curve in 2 or 3 or higher dimensions can be thought of as the path of a continuously moving point. 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:.
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. 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. 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.
InPeano discovered a continuous curve, now called the Peano curvethat passes through every point of the unit square Peano His purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was motivated by Georg Cantor ‘s earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifoldsuch as the unit square.
The problem Peano solved was whether such a mapping could be continuous; i. 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. 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.
Therefore, Peano’s space-filling curve was found to be highly counterintuitive. From Peano’s example, it was easy to deduce continuous curves whose ranges contained the n -dimensional hypercube for any positive integer n. 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.
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. Peano’s ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator.
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. Peano’s article also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to graphical visualization was, no doubt, motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures.
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. A year later, David Hilbert published in the same journal a psano of Peano’s construction Hilbert Hilbert’s article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here.
Space-filling curve – Wikipedia
The analytic form of the Hilbert curvehowever, is more complicated than Peano’s. The restriction of the Cantor function to the Cantor set is an example dee such a function. 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.
The two subcurves pfano if the intersection of the two images is non-empty. 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.
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. 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 space onto a Hausdorff space is a homeomorphism.
But a unit square has no cut-pointand so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points.
Peano curve – Wikipedia
There exist non-self-intersecting curves of nonzero area, the Osgood peaanobut they are not space-filling. For the classic Peano and Hilbert space-filling curves, where two subcurves intersect in the technical sensethere is self-contact without self-crossing. A space-filling curve can be everywhere self-crossing if its approximation curves are self-crossing. A space-filling curve’s approximations can be self-avoiding, as the figures above illustrate. In 3 dimensions, self-avoiding approximation curves can even contain knots.
Approximation curves remain within a bounded portion of n -dimensional space, but their lengths increase without bound. Space-filling curves are special cases of fractal constructions.
No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn. The Hahn — Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves:. Spaces that are the continuous image of a unit interval are sometimes called Peano spaces. In many formulations of the Hahn—Mazurkiewicz theorem, second-countable is replaced by metrizable.
These two formulations are equivalent. In one direction peamo compact Hausdorff space is a normal space and, by the Urysohn metrization theoremsecond-countable then implies metrizable. Conversely a compact metric space is second-countable. There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. Here the sphere is the sphere at infinity of hyperbolic 3-space.
Wiener pointed out in The Fourier Integral and Certain of its Applications that space filling curves could be coirbe to reduce Lebesgue integration in higher dimensions to Lebesgue integration in one dimension. From Wikipedia, the free encyclopedia. A curve with endpoints is a continuous function whose domain is the unit interval [0, 1]. Fractal canopy Space-filling curve H tree.