The recent interest in fractals has refocused interest on space filling curves, and the study of fractals has thrown some new light on this small but venerable part of mathematics. This paper presents and discusses a radically different approach to multidimensional indexing based on the concept of the space filling curve. Space filling curves, puzzling maps between dimensions, have surprising realworld applications. Living cities have intrinsically fractal space filling properties, similar to living systems like bronchial trees of lungs and evolve organically. The space filling curve shown above also is nowhere differentiable if we write hilberts curve in parametric form, xft, ygt, then the graph of yfxlooks like this. Generating hilberts spacefilling curve by recursion article pdf available in acm transactions on mathematical software 242. An efficient algorithm for the generation of hilberts space filling curve is given. Special focus is laid on the representation of sfc and on resulting algorithms. Is it true that a spacefilling curve cannot be injective.
The website is still a bit under construction but in any case, i will strive to constantly add further. Physically, a curve describes the motion of a particle in nspace, and the trace is the trajectory of the particle. Media in category space filling curves the following 27 files are in this category, out of 27 total. A space filling curve consists of a set of segments.
Algorithm for generating a 3d hilbert spacefilling curve. Space aggregation hilbert curve keeps better space aggregation than zorder curve, which can be seen from left figure. As t varies, the first two coordinates in all three functions trace out the points on the unit circle, starting with 1, 0 when t 0 and proceeding counterclockwise around the circle as t increases. Smooth hilbert space filling curve for minimizing traversal complexity in covering a monitoring area. If the particle follows the same trajectory, but with di. The following animation shows how the algorithm works. I have an image of size mn whose pixels coordinates has been flattened to a 1d array according to a space filling curve i. The chain assumption presented in the markov chains section can then be applied.
Mathematical description in this text we will only deal with two dimensional sfcs. This website collects additional course material and also errata for the text book space filling curves an introduction with applications in scientific computing by michael bader published in the series texts in computational science and engineering by springer. Scr is a windows screen saver which draws successive approximations to the hilbert curve on your inactive pc. Curves in space in previous chapters, we have seen that the curvature. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. An efficient algorithm for computing contextbased space filling curves is. Arithmetic definition an arithmetic description of the hilbert curve would allow us to calculate the coordinates of the image point of any t i using a form of parameter representation. Calculation complexity hilbert curve is more complicated in calculation, which aims to keep the space aggregation of data points. Using spacefilling curves for multi dimensional indexing. The construction shown above is due to david hilbert 18621943. My interest is to expand their range to the entirety of nspace. How to draw the hilbert curve the hilbert space filling curve is a one dimensional curve which visits every point within a two dimensional space. Pdf because they are continuous and selfsimilar, spacefilling curves have been widely used in mathematics to transform multidimensional problems. Because giuseppe peano 18581932 was the first to discover one, spacefilling curves in the 2.
This curve has been called the first space filling curve sfc. If we keep in mind that the geometric generation was based on a recursive division of the. A spacefilling curve is a parameterized, injective function which maps a unit line segment to a continuous curve in the unit square, cube, hypercube, etc, which gets arbitrarily close to a given point in the unit cube as the parameter increases. Pdf smooth hilbert space filling curve for minimizing. A closedform algorithm for converting hilbert spacefilling curve indices. Performance of multidimensional space filling curves. The hilbert curve is the limit of the infinite sequence of these approximations. In other words, the spacefilling curve turns a 2d problem into a onedimensional problem. Spacefilling curves math 635 2 image having positive jordan content. In fact, by similar reasoning, the curve can be made to fill completely an entire cube. Dynamic octree load balancing using spacefilling curves. Contextbased space filling curves stanford cs theory. Introduction to linear independence video khan academy.
Selecting the appropriate space filling curve for different applications. Jan 25, 2017 generating a space filling curve in python posted on january 25, 2017 by admin posted in python, tsp as mentioned in a previous post, i had taken an interest in space filling curves as a method for approaching the traveling salesman problem. Next we give a general introduction to space filling curves and discuss properties of them. For example, grammarbased techniques are introduced for traversals of cartesian and octreetype meshes, and. All structured data from the file and property namespaces is available under the creative commons cc0 license. The hilbert space filling curve the hilbert screen saver. The subject of space filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Spacefilling curves in geospatial applications dr dobbs. We begin with an example of a space filling curve and demonstrate how it can be used to find a short tour through a set of points. Another application is designing clock trees on vlsi chips to minimize clock skew. But at the same time, the control points p1, p2, p3, p4 are the coordinates of the curve in the bernstein basis. Mar 07, 2011 generating polyas spacefilling curve begins with a right triangle.
A space filling curve is a parameterized, injective function which maps a unit line segment to a continuous curve in the unit square, cube, hypercube, etc, which gets arbitrarily close to a given point in the unit cube as the parameter increases. The algorithm implements a recursive procedure that involves simple integer operations and quickly converges to. Each segment connects two consecutive multiilimensionalpoints. Roughly speaking, differentiability puts a bound on how fast the curve can turn. The dotted blue lines indicate how to connect the elements of the hilbert space filling curves. Exquisitely convoluted, space filling curves, are created by very simple recursive procedures. Space filling curves and hierarchical basis klaus speer. Nowhere differentiability of the hilbert curve 12 2.
A solutions manual for topology by james munkres 9beach. This spacefilling curve which completely fills up the unit square is the limit of a sequence of curves. These two famous examples were invented by david hilbert left, the hilbert curve, and wraclaw sierpinski right. Here, it is said that a space filling curve cannot be injective because that will make the curve a homeomorphism from the unit interval onto the unit square, since every continuous bijection from a compact space to a hausdorff space is a homeomorphism. Despite the creation of these other curves, sometimes spacefilling curves are still referred to as peano curves. Spacefilling curves are special cases of fractal constructions. Files are available under licenses specified on their description page.
But with so much potential variation, it doesnt take much genetic manipulation to turn a square filling curve into a wild and crazy beast. Space filling curves serve as a counterexample to lessthanrigorous notions of dimension. If so, here are two families of pictures that are made up from square tiles. In fact the e curve, fills a square, like the hilbert curve. However, there have been no comprehensive treatments of the subject since siepinskys in 1912. Spacefilling curve how is spacefilling curve abbreviated. Older cities were fractal, because they worked on all scales with small sqaure leading to larger one. A spacefilling curve can be thought of as a map from onedimensional space onto a higherdimensional space.
Github repository here, html versions here, and pdf version here contents. Geometric generation of the hilbert spacefilling curve 2. The hahn mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves. Posts about spacefilling curve written by jeffrey ventrella. 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 1891, as a variant of the space filling peano curves discovered by giuseppe peano in 1890. An algorithm for generating hilberts space filling curve in a byteoriented manner is presented. Spacefilling curves are not squares spacefilling curves. Explore anything with the first computational knowledge engine.
Generating a space filling curve in python third landing. A spacefilling curve can help index points on a map by placing the many points of the maps region into some suitable order, like beads on a string. It was also easy to extend peanos example to continuous curves without endpoints, which filled the entire ndimensional euclidean space where n is 2, 3, or any. Polyas spacefilling curve wolfram demonstrations project. For the same region of space, the hilbert curve has less falsepositives than zorder. This monograph is neither a textbook nor an encyclopedic treatment of the subject nor a historical account, but it is a little of each. Figure 2 shows three successively detailed approximations of the wellknown space filling curve invented by david hilbert. Hilbert space filling curves abend now that the nearest neighbor dependence of markov chains has been looked at, we need to look at a method which quickly and efficiently looks at every pixel within an image. A hilbert curve also known as a hilbert spacefilling curve is a continuous fractal spacefilling curve first described by the german mathematician david hilbert in 1891, as a variant of the space filling curves discovered by giuseppe peano in 1890.
Two definitions of a curve a continuous 1d set of points in 2d or 3d a mapping from an interval s onto the plane that is, pt is the point of the curve at parameter t big differences. In addition to their mathematical importance, space. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire 2dimensional unit square or more generally an n dimensional hypercube. From what i understand, a space filling curve such as the hilbert curve is a homoemorphism between the real line and a subset of the plane. Wikipedia in particular only provides example code for generating 2d curves. By now, i hope you will relish the challenge of working out how to draw pictures without much help from me. The subject of spacefilling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by peano. In mathematical analysis, a space filling curve is a curve whose range disambiguation needed contains the entire 2dimensional unit square or more generally an ndimensional unit hypercube. 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 1891, as a variant of the space filling peano curves discovered by giuseppe peano in 1890 because it is space filling, its hausdorff dimension is 2 precisely, its image is the unit square, whose dimension is 2 in any. The gosper curve is one of the most amazing and satisfying plane filling curves the gosper curve is a selfavoiding, plane filling fractal curve containing 7n segments where n is the order of the teragon. A montage of space filling curves, meant as a supplement to the hilbert curve video. This curve has been called the first spacefilling curve sfc. Hilbertcurve n returns a line primitive corresponding to a path that starts at 0, 0, then joins all integer points in the 2 n1 by 2 n1 square, and ends at 2 n1, 0. Using this ordering, you can translate points of the 2d region into values suitable for btree indexing.
Hilbertcurve is also known as hilbert spacefilling curve. Multilevel optimization by spacefilling curves in adaptive. Other spacefilling curves, such as those due to sierpinski and peano, also attain these rates, while upper bounds for the lebesgue curve are. A 2dimensional space filling curve, here, is a surjective continuous map. Take a square and subdivide it into four squares, then draw a path through the centers of the four squares. A spacefilling curve is a parameterized, injective function which maps a unit line segment to a continuous curve in the unit square, cube, hypercube, etc, which gets arbitrarily close to a given point in the unit cube as the parameter increases space filling curves serve as a counterexample to lessthanrigorous notions of dimension. Rapid graph layout using space filling curves chris muelder and kwanliu ma, senior member, ieee abstract network data frequently arises in a wide variety of. We can construct a mapping from a one dimensional interval to a finite higher dimensional interval.
In this work spacefilling curves are used on several levels of algorithmic abstraction in order to optimize an atmospheric modeling tool. It is fairly easy to see, using methods of calculus, that any space filling curve has to be nondifferentiable on an uncountable set. Cantor, hilbert, moore, knopp, lebesgue, and polya are among the prominent mathematicians who have contributed to the field. This website collects additional course material and also errata for the text book spacefilling curves an introduction with applications in scientific computing by michael bader published in the series texts in computational science and engineering by springer. Working in this area is like skating on the edge of. Calculation complexity hilbert curve is more complicated. Zorder curve and hilbert curve compared public zorder curve hilbert curve space aggregation hilbert curve keeps better space aggregation than zorder curve, which can be seen from left figure.
A space filling curve is a parameterized function which maps a unit line segment to a continuous curve in the unit square, cube, hypercube, etc, which gets arbitrarily close to a given point in the unit cube as the parameter increases. Because giuseppe peano 18581932 was the first to discover one, space filling curves in the 2dimensional plane are sometimes called peano curves, but that phrase also refers to the peano curve, the. However, we will also construct special spacefilling curves on an. Hilbertcurve takes a datarange option that can be used to specify the range the coordinates should be assumed. The first examples were constructed by giuseppe peano 18581932 and thus curves of this kind are known as peano curves. There are classical curves like the hilbert, the peano and the l ebesguecurve, see 21.
The influence of their topology is also considered. Spacefilling curves home welcome to spacefillingcurves. Nov 16, 2014 spacefilling curves are not squares wikipedia says. Space filling curves serve as a counterexample to lessthanrigorous notions of. Generating a space filling curve in python posted on january 25, 2017 by admin posted in python, tsp as mentioned in a previous post, i had taken an interest in space filling curves as a method for approaching the traveling salesman problem. Other articles where spacefilling curve is discussed. Other sfcs soon followed with hilberts in 1891, moores in 1900, lebesgues in 1904. Space filling curves, turning visual information into audio information, and the connection between infinite and finite math this is a reupload of an older video which had much worse audio. Because it is space filling, its hausdorff dimension is 2 precisely, its image is the. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen. The subject of space filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by peano. Jul 21, 2017 space filling curves, turning visual information into audio information, and the connection between infinite and finite math this is a reupload of an older video which had much worse audio. In this text we will only deal with two dimensional sfcs. Since partitions are formed from contiguous segments of this linearization, its form has a direct e.
This can easily be extended to any 2 m x2 m case by filling the grid with the result above and by looking at the grid in subgroups of 4x4 pixels. For each iteration, split each right triangle into two smaller right triangles and join the centers of the resulting smaller triangles. Using spacefilling curves for multidimensional indexing. Sep 26, 2016 henry segerman shows us some 3dprinted spacefilling curves, including the hilbert curve and dragon curve. A closedform algorithm for converting hilbert space. On the quality of spacefilling curve induced partitions gerhard. Nov 20, 2012 living cities have intrinsically fractal space filling properties, similar to living systems like bronchial trees of lungs and evolve organically. Spacefilling curves, puzzling maps between dimensions, have surprising realworld applications. 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 1891, as a variant of the space filling curves discovered by giuseppe peano in 1890. Since a composition of homeomorphisms is a homeomorphism, we could compose the surface patches of say a sphere with the hilbert curve to get a homeomorphism between the real line and the sphere.
One that immediately comes to mind is cachefriendly layout of poweroftwo textures, usually called texture swizzling. Contextbased space filling curves revital dafner, daniel cohenor and yossi matias department of computer science, telaviv university, israel abstract a contextbased scanning technique for images is presented. I believe using a 3d hilbert space filling curve would be a good way to do this, but ive searched and havent found very helpful resources for this problem. A practical use for spacefilling curves hacker news. Chihsheng chen, shenyi lin,minhsuanfan, and chuahuang huang abstractwe use the tensor product theory to for mulate a closedform algorithm for converting hilbert. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire 2dimensional unit square or more generally an n dimensional unit hypercube.
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