That is, the multidimensional index takes the ndimensional key spaces and maps them onto a onedimensional space and then the onedimensional retrieval system is activated. In 2006 michael bader and christoph zenger of the technical university of munich showed that clustering the data with a spacefilling curve reduces memory traffic. Tensor product formulation for hilbert spacefilling curves. Lossless compression of medical images using hilbert space. Spacefilling curves in geospatial applications dr dobbs.
Anyone interested in or just curious about spacefilling curves will enjoy both books. Bader is also the author of an excellent recent book that discusses spacefilling curves from a computational point of view. 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. Of course i dont actually have a subroutine for filling in a quadrant. Ternary arithmetic is the basis for the required functional relationships in the discrete mapping. Proving that smooth curves cannot be spacefilling requires some sophisticated analysis, though. Space filling curves and mathematical programming sciencedirect. Lsystems come to mind as something you might want to look into.
The problem of finding if3141 in n dimensional euclidean space such that if3142, i 1, 2, n. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations. Using spacefilling curves for computation reordering. Spacefilling curves, simultaneous grammar, logo, postscript, recursion 1 introduction several procedures for drawing particular space.
On extending some primaldual interiorpoint algorithms from linear programming to semidefinite programming. Siam journal on applied mathematics volume 15, issue 3 10. Butz department of electrical engineering, northwestern university, evanston, illinois 60201 e x the problem of finding 2 in n dimensional euclidean space such that f2 781. In this paper we propose to describe spacefilling and related curves by means of simultaneous nonlinear grammars. Generally a spacefilling curve is defined as a curve that can pass through every point of a finite region of an ndimensional space. These properties made then very useful in computer science and scienti c computing. In mathematical analysis, a space filling curve is a curve whose range contains the entire 2dimensional unit square or more generally an ndimensional unit hypercube.
Space filling curves are mathematical entities that offer the possibility ofbuilding a wide range ofstructures, covering the surface ofa single layer with one continuous curve. A spacefilling curve is the image of a line, a fundamentally 1dimensional object, that fills a plane, a fundamentally 2dimensional object. The website is still a bit under construction but in any case, i will strive to constantly add further. Jagadish analyzes the clustering properties of hilbert space.
Global optimization using space filling curves semantic scholar. 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. Global optimization is a field of mathematical programming dealing with finding global absolute minima of multidimensional multiextremal functions. Hilbert and sierpinski spacefilling curves, and beyond. In the context of one application of spacefilling curves, the algorithm may be modified sothatthe results are correct for continua rather than. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen.
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 specific example of a. Another operation used in spatial databases is the spatial join that locates pairs of objects with some common spatial attribute for instance, they are within. Flipped means that instead of just drawing a segment, you instead. Hilbert curves is a unique app in the form of a book that shows, explains, and lets you explore and play with one of the most famous and elegant constructions in mathematics, hilberts spacefilling curve. Problems of this kind where the objective function is nondifferentiable, satisfies the lipschitz condition with an unknown lipschitz constant, and is given as a blackbox are very often encountered in engineering optimization applications. Several procedures for drawing particular spacefiliing curves have been proposed in the literature 2, 7, 10, 12, 15, 16. Spacefilling curves are useful even inside the rtree. An earlier volume by hans sagan is more mathematical. Recurrence for pandimensional spacefilling functions. Several space filling curves are surveyed, and peanos continuous mapping from the unit interval onto the unit square is used as a basis for defining a mapping from the unit quantized interval onto the unit quantized hypercube, and inversely.
The algorithm implements a recursive procedure that involves simple integer operations and quickly converges to the set of points that make the hilbert curve. Space filling curves and mathematical programming1. Also as nspace walsh series, abstract published in program. Algorithm for generating a 3d hilbert spacefilling curve. Contextbased space filling curves by revital dafner, daniel cohenor and yossi matias. Butz department of electrical engineering, northwestern university, evanston, illinois 60201 e x the problem of finding 2 in n dimensional euclidean space such that f2 curve has been called the first space filling curve sfc. Figure 2 lsystems are used in specifying and constructing graphical images of recursively defined mathematical objects such as fractal curves, recursive tiling and space filling curves. Walshfourier series for functions of n variables, with localization, j. You can construct spacefilling curves into the cube and the sphere, but neither one of those is smooth i.
Introduction the notion of using space filling curves to solve certain problems in mathematical. Spacefilling curves, recursion, simultaneous grammar, logo, postscript. 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. Pdf the subject of this paper is a means of converging to a set of numbers in certain mathematical programming problems where a. Explicit descriptions of such curves were proposedbypeano28andhilbert15. They provide simple examples of continuous, nondifferentiable functions. For instance, they have been used for data reordering 26, 31, dynamic partitioning 29. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. Wikipedia in particular only provides example code for generating 2d curves. Barnsley used iterated systems of contracting functions to define fractals. In addition to their mathematical importance, space. Information and control 12, 314330 1968 space filling curves and mathematical programming arthur r. It shows how to map the points from a line segment to every point inside a. 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.
A summary of the subject has beer given by hobson 1927. Sierpinski, mathematicians were forced to realize that curves of potentially infinite length could be constructed by iterating simple geometric transformation rules, that these curves could be approximated by fractured lines, and that some of these curves were spacefilling, as they. Mathematical description in this text we will only deal with two dimensional sfcs. The existence of space filling curves opens the way to reducing multivariate optimization problems to the minimization of univariate functions. A variant of the rtree called the hilbert rtree uses the hilbert curve to help maintain spatial organization within the tree. Short notes alternativealgorithmfor hilberts spacefilling curve arthurr. Heres an excellent curve i first saw in the book brain filling curves the trick is that part of the curve is drawn flipped. Spacefilling curves challenge my intuition dimension. 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 spacefilling peano curves discovered by giuseppe peano in 1890 because it is spacefilling, its hausdorff dimension is 2 precisely, its image is the unit square, whose dimension is 2 in any. Space filling curves and mathematical programming core. Spacefilling curves, the finite approximations to space filling functions, have found application in global optimization. Space lling curves sfcs are mathematical objects that enjoy nice proximity in space properties. Domain decomposition and space filling curves in toolpath. Baders book nicely complements sagans and provides an introduction to the algorithmics of spacefilling curves.
Space filling curves there is a mathematical curious thing called space filling curves that caught my attention several years ago, and ever since ive been studying its mathematical properties. Generating hilberts spacefilling curve by recursion article pdf available in acm transactions on mathematical software 242. Butz, member, ieee abstractan algorithm for generating hilberts spacefilling curve in a byteoriented manner is presented. The mathematical philosopher imre lakatos explored the role of examples, and counterexamples, including. Such spacefilling curves were originally developed as conceptual mathematical monsters, counterexamples to weierstrassian and reimannian treatments of calculus and continuity. I believe using a 3d hilbert spacefilling curve would be a good way to do this, but ive searched and havent found very helpful resources for this problem. Other sfcs soon followed with hilberts in 1891, moores in 1900, lebesgues in 1904, sierpinskas in 1912, and polyas in 19. An efficient algorithm for the generation of hilberts spacefilling curve is given. Another search word might be tiling, if thats what youre after, implying such things as penrose tiles, et al. Programming examples of spacefilling curves the role of.
It is a method to construct a spacefilling curve that is optimal in regards to the changes in underlying image data. Slicelevel manipulation in the current study domain decomposition was demonstrated using a. The wellknown book spacefilling curves, by hans sagan springer, 1994 provides an excellent introduction to the mathematics and history of spacefilling curves. We briefly explore potential applications to computational design, as well as connections to space filling. Convergence with hilberts space filling curve core. Space filling curves sfcs represent an efficient and straightforward method for sparse space indexing to transform an ndimensional space into a onedimensional representation. Despite the creation of these other curves, sometimes spacefilling curves are still referred to as peano curves.
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