Explicit descriptions of such curves were proposedbypeano28andhilbert15. 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. Programming examples of spacefilling curves the role of. Slicelevel manipulation in the current study domain decomposition was demonstrated using a. 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.
We briefly explore potential applications to computational design, as well as connections to space filling. Bader is also the author of an excellent recent book that discusses spacefilling curves from a computational point of view. An efficient algorithm for the generation of hilberts spacefilling curve is given. Pdf convergence with hilberts space filling curve researchgate. Hilbert and sierpinski spacefilling curves, and beyond. 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. 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. Siam journal on applied mathematics volume 15, issue 3 10. The website is still a bit under construction but in any case, i will strive to constantly add further. Baders book nicely complements sagans and provides an introduction to the algorithmics of spacefilling curves. The existence of space filling curves opens the way to reducing multivariate optimization problems to the minimization of univariate functions. Spacefilling curves challenge my intuition dimension. In addition to their mathematical importance, space.
A spacefilling curve is the image of a line, a fundamentally 1dimensional object, that fills a plane, a fundamentally 2dimensional object. Then, if i just draw three line segments to link the four pieces into one long curve, ill be finished. Other sfcs soon followed with hilberts in 1891, moores in 1900, lebesgues in 1904, sierpinskas in 1912, and polyas in 19. 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. Spacefilling curves math 635 5 indexing that would like on top of the onedimensional retrieval system. An earlier volume by hans sagan is more mathematical. On extending some primaldual interiorpoint algorithms from linear programming to semidefinite programming. Such spacefilling curves were originally developed as conceptual mathematical monsters, counterexamples to weierstrassian and reimannian treatments of calculus and continuity. 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. Global optimization is a field of mathematical programming dealing with finding global absolute minima of multidimensional multiextremal functions. 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. Despite the creation of these other curves, sometimes spacefilling curves are still referred to as peano curves. Wikipedia in particular only provides example code for generating 2d curves. Spacefilling curves can be employed to accomplish this mapping.
Butz, member, ieee abstractan algorithm for generating hilberts spacefilling curve in a byteoriented manner is presented. The wellknown book spacefilling curves, by hans sagan springer, 1994 provides an excellent introduction to the mathematics and history of spacefilling curves. Space filling curves and mathematical programming sciencedirect. Generating hilberts spacefilling curve by recursion article pdf available in acm transactions on mathematical software 242. It is a method to construct a spacefilling curve that is optimal in regards to the changes in underlying image data. Heres an excellent curve i first saw in the book brain filling curves the trick is that part of the curve is drawn flipped. The problem of finding if3141 in n dimensional euclidean space such that if3142, i 1, 2, n. Proving that smooth curves cannot be spacefilling requires some sophisticated analysis, though. 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. It shows how to map the points from a line segment to every point inside a. Flipped means that instead of just drawing a segment, you instead.
A variant of the rtree called the hilbert rtree uses the hilbert curve to help maintain spatial organization within the tree. These properties made then very useful in computer science and scienti c computing. 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. Anyone interested in or just curious about spacefilling curves will enjoy both books. Algorithm for generating a 3d hilbert spacefilling curve.
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. In this case, n must be greater than or equal to 2. 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. A summary of the subject has beer given by hobson 1927.
Contextbased space filling curves by revital dafner, daniel cohenor and yossi matias. Space lling curves sfcs are mathematical objects that enjoy nice proximity in space properties. Spacefilling curves, recursion, simultaneous grammar, logo, postscript. Space filling curves and mathematical programming core. 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. Tensor product formulation for hilbert spacefilling curves. Of course i dont actually have a subroutine for filling in a quadrant. Space filling curves sfcs represent an efficient and straightforward method for sparse space indexing to transform an ndimensional space into a onedimensional representation. Another search word might be tiling, if thats what youre after, implying such things as penrose tiles, et al. Also as nspace walsh series, abstract published in program. The mathematical philosopher imre lakatos explored the role of examples, and counterexamples, including.
Information and control 12, 314330 1968 space filling curves and mathematical programming arthur r. Convergence with hilberts space filling curve core. Spacefilling curves, simultaneous grammar, logo, postscript, recursion 1 introduction several procedures for drawing particular space. Lossless compression of medical images using hilbert space. Introduction the notion of using space filling curves to solve certain problems in mathematical. Using spacefilling curves for computation reordering. Jagadish analyzes the clustering properties of hilbert space.
Mathematical description in this text we will only deal with two dimensional sfcs. Spacefilling curves in geospatial applications dr dobbs. The algorithm implements a recursive procedure that involves simple integer operations and quickly converges to the set of points that make the hilbert curve. Pdf the subject of this paper is a means of converging to a set of numbers in certain mathematical programming problems where a. In this paper we propose to describe spacefilling and related curves by means of simultaneous nonlinear grammars. Spacefilling curves are useful even inside the rtree. Space filling curves and mathematical programming1. Short notes alternativealgorithmfor hilberts spacefilling curve arthurr. 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. Since the mid 1990s books appeared to detail crucial defining mathematical properties of these curves. 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.
Ternary arithmetic is the basis for the required functional relationships in the discrete mapping. They provide simple examples of continuous, nondifferentiable functions. Barnsley used iterated systems of contracting functions to define fractals. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. Lsystems come to mind as something you might want to look into.
Generally a spacefilling curve is defined as a curve that can pass through every point of a finite region of an ndimensional space. Several procedures for drawing particular spacefiliing curves have been proposed in the literature 2, 7, 10, 12, 15, 16. Walshfourier series for functions of n variables, with localization, j. 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. Global optimization using space filling curves semantic scholar. Spacefilling curves, the finite approximations to space filling functions, have found application in global optimization. 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. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen. You can construct spacefilling curves into the cube and the sphere, but neither one of those is smooth i. In the context of one application of spacefilling curves, the algorithm may be modified sothatthe results are correct for continua rather than. Domain decomposition and space filling curves in toolpath. For instance, they have been used for data reordering 26, 31, dynamic partitioning 29.
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