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# Topic: Spacefilling curve

 Bartholdi on spacefilling curves A famous spacefilling curve is that due to Sierpinski, which is formed by repeatedly copying and shrinking a simple pattern (the convoluted tour in Figure 1). A useful property of a spacefilling curve is that it tends to visit all the points in a region once it has entered that region. Vertex-labelling algorithms for the Hilbert spacefilling curve by J. Bartholdi and P. Goldsman. www2.isye.gatech.edu /people/faculty/John_Bartholdi/mow/mow.html   (1050 words)

 Space-Filling Curves   (Site not responding. Last check: 2007-11-05) A space-filling curve is typically defined as the limit of a sequence of curves. For example, the name Hilbert space-filling curve should properly be used only for the limit curve reached as the level parameter of the Hilbert curves tends to infinity. The Sierpinski_C curve and the Sierpinski curve are, in a sense, complements of each other: the shape of the areas excluded by one curve is the shape included by the other. www.cs.utexas.edu /users/vbb/misc/sfc/Oindex.html   (511 words)

 The Hilbert Space Filling Curve   (Site not responding. Last check: 2007-11-05) The actual Hilbert Curve is not a member of this family, it is the limit that the sequence approaches. The next curve in the sequence is a refinement of this, we consider each of the quarters to be a box with the appropriate orientation, so that the curve is entering and leaving from the bottom left and leaving in the bottom right. This apparent riddle is solved as although none of the family of curves leading to the Hilbert curve crosses itself the final curve does cross itself all over the place. www.dcs.napier.ac.uk /~andrew/hilbert.html   (363 words)

 Space-filling curve Summary   (Site not responding. Last check: 2007-11-05) 3 iterations of the Peano curve, a space-filling curve A curve (with endpoints) is a continuous function whose domain is the unit interval [0,1]. Space-filling curves are curves whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube, or n-dimensional hypercube). www.bookrags.com /Space-filling_curve   (1442 words)

 FRACTINT L-systems Spacefilling Curves This curve looks like a church on which a mural is painted reminiscent of the painting of God giving life to Adam. All of the spacefilling curves described so far revisit certain points, but there are many spacefilling curves which never revisit any points. There is another famous spacefilling curve that does not revisit any points due to Hilbert. spanky.fractint.org /www/fractint/lsys/spacefilling.html   (848 words)

 thoughts… » Blog Archive » Hilbert Curve I’ve been reading a bit about the spacefilling curves for my wavelet image compression (take a look here and here). There is a very nice way to convert from the Hilbert derived key to (multi-dimensional coordinates) described by John J. Bartholdi, III and Paul Goldsman in “Vertex-Labeling Algorithms for the Hilbert Spacefilling Curve”. * This implementation is based on the paper "Vertex-Labeling Algorithms for the * Hilbert Spacefilling Curve" by John J. Bartholdi, III and Paul Goldsman. www.maven.de /2005/12/hilbert-curve   (370 words)

 FRACTINT L-systems Spacefilling Curves   (Site not responding. Last check: 2007-11-05) This curve looks like a church on which a mural is painted reminiscent of the painting of God giving life to Adam. All of the spacefilling curves described so far revisit certain points, but there are many spacefilling curves which never revisit any points. There is another famous spacefilling curve that does not revisit any points due to Hilbert. spanky.triumf.ca /www/fractint/lsys/spacefilling.html   (848 words)

 Plane Filling Curves The curves become longer and longer but, since it's always possible to move from one step to another, the process will never end. Disregarding the pieces of the curve that cross small square borders we see (starting with the second stage) that those small squares contain small replicas of the curve drawn on the previous stage, i.e. As before, when placing curves in the smaller squares two of the staples are placed in the direction of the bigger one they replace whereas one is turned left and another right quarter turn relative to the position of the parent staple. www.cut-the-knot.org /do_you_know/hilbert.shtml   (1149 words)

 [No title]   (Site not responding. Last check: 2007-11-05) Then divide this up into four smaller squares with an edge such that the designated edges of the smaller squares form a curve starting and ending at the same points as the edge of the large one. But the image of a curve is closed, and so it must be all of the square. However, the curve itself is a function from an interval to the square. www.math.niu.edu /~rusin/known-math/98/spacefill   (459 words)

 FRACTINT L-systems Variations   (Site not responding. Last check: 2007-11-05) To create a curve that does not touch itself, he suggested connecting the midpoints of the segments in the Dragon with line segments. If you now examine the order 3 curve, you'll see that the same traversing plan is used, begining with the upper left 16 squares and ending with the upper right 16 squares, each group of squares traversed according as the order 2 curve dictates. As a final example of a median curve based on a dissection, here is one using a dissection of a 30-60-90 degree triangle dissected into three such triangles. spanky.triumf.ca /www/fractint/lsys/variations.html   (1084 words)

 Citations: Linear clustering of objects with multiple attributes - Jagadish (ResearchIndex)   (Site not responding. Last check: 2007-11-05) ....curves have shown to be of high value [2] 4] 9] 11] 17] 20] In this paper we study Hilbert curves [10] perhaps the most popular space filling indexing schemes. The desirable features of the Hilbert curve are that the points close on the Hilbert curve are close in the domain space, and the points close in the domain space are likely to be close on the Hilbert curve. The Hilbert curves of degree 1 (H 1) degree 2 (H 2) and degree 3(H 3) on a.... citeseer.ifi.unizh.ch /context/9063/0   (5713 words)

 Please see PDF version The sum of squares of the edge lengths of the tour provided by the spacefilling curve heuristic applied to a random sample of n points from the unit square is proved to be asymptotically equal to a periodic function of the logarithm of the sample size. The spacefilling curve heuristic rests on the existence of a surjective mapping 0 : [0, 1], [0, 112 such that for each X E [0, 1]2 we can quickly compute a t E [0, 11 such that V)(t) = x. We will establish the possibly surprising fact that for a large class of spacefilling curves the value of this random variable is well approximated by a periodic function of the logarithm of the sample size. www-stat.wharton.upenn.edu /~steele/Publications/HTML/SoSoEL.html   (2987 words)

 HAKMEM -- FLOWS AND ITERATED FUNCTIONS -- DRAFT, NOT YET PROOFED Answer: It is a cousin to both the dragon and snowflake curves (and arose as a bug in a spacefilling curve). As the process continues, the curve crosses itself more and more often, eventually taking on the shape of a wildly curly letter C which forms the envelope of a myriad of epicyclic octagons. There are other pictures besides the C curve which are preserved by this process, but they are of infinite size. www.inwap.com /pdp10/hbaker/hakmem/flows.html   (1178 words)

 [No title]   (Site not responding. Last check: 2007-11-05) The spacefilling curve heuristic takes only $O(N \log N)$ time and generates a solution about 25 percent longer than optimal (for cities distributed uniformly in a square). The heuristic is based on a spacefilling curve $\phi$, which is a continuous (but not one-to-one) mapping from the unit interval C = [0,1] onto the unit square $S = [0,1]^2$. The key to generating multiple solutions, as required for the initial population for the GA, is the observation that the spacefilling curve heuristic is {\it not} invariant with respect to these operations. www.aic.nrl.navy.mil /galist/digests/v4n7   (1632 words)

 Please see PDF version The tour given by the spacefilling curve heuristic applied to a random sample of points from the unit square differs in subtle ways from the shortest tour through those points. The theory is developed further by detailing the selfsimilarity properties of the spacefilling curve that are useful for obtaining the asymptotics of the expected length of the heuristic path. Many of the classical spacefilling curves are Lipschitz of order onehalf, which is to say there exists a constant cp so that for any 0  s, t  1 one has 110(s)  0(01 5 cpl S  t1112. www-stat.wharton.upenn.edu /~steele/Publications/HTML/GSCHaL.html   (3801 words)

 [No title] It is hoped that random matrix theory can be used to predict the frequency with which curves in a family have a given rank. One possibility is that the current model makes incorrect assumptions about the distribution of heights of generating points of rank 1 curves. You can also see that you do not have to worry about the floor reflection if the woofer is placed low to the ground and the midrange is placed high and the crossover point is placed between the floor knees of the two drivers. www.lycos.com /info/curves--points.html?page=3   (570 words)

 Hilbert Space Filling Curve Abstract Geometric Art I have been experimenting with space filling curves now for some time, this space filling curve art generator (opens in a new page, shockwave 35k) in particular has been in the works since Summer 2005. It is an example of a "space-filling" curve: it literally covers every point in a square. Cityscapes with Ladders and Helipads(requires shockwave plug-in 123k) is a new Hilbert curve based prototype that generates abstract geometric art. www.donrelyea.com /hilbert_algorithmic_art_menu.htm   (902 words)

 SLIB - Hashing The returned procedures is a function that takes the x and y coordinates of a point, (non-negative integers) and returns an integer corresponding to the relative position of that point along a Sierpinski curve. Note that locations (24, 78) and (23, 80) are near in index and therefore, because the Sierpinski spacefilling curve is continuous, we know they must also be near in the plane. For details of performance, see L. Platzman and J. Bartholdi, "Spacefilling curves and the Euclidean travelling salesman problem", JACM 36(4):719--737 (October 1989) and references therein. www.student.cs.uwaterloo.ca /~isg/res/scheme/scm/slib_14.html   (441 words)

 HAKMEM -- FLOWS AND ITERATED FUNCTIONS -- DRAFT, NOT YET PROOFED   (Site not responding. Last check: 2007-11-05) Answer: It is a cousin to both the dragon and snowflake curves (and arose as a bug in a spacefilling curve). As the process continues, the curve crosses itself more and more often, eventually taking on the shape of a wildly curly letter C which forms the envelope of a myriad of epicyclic octagons. There are other pictures besides the C curve which are preserved by this process, but they are of infinite size. www.cs.utk.edu /~vose/math/hbaker/hakmem/flows.html   (1178 words)

 [No title]   (Site not responding. Last check: 2007-11-05) It may not be the actual values of the parameters that are of interest, so much as their distribution: whether they are well scattered, or clustered, or confined to one or two small regions of the parameter space. A spacefilling curve $\phi$ is a continuous mapping from the unit interval to a higher dimensional region, for example the unit $k$-cube. To use this technique, a pseudoinverse of the spacefilling curve is needed. www.aic.nrl.navy.mil /galist/digests/v4n15   (1198 words)

 The Supply Chain & Logistics Institute at Georgia Tech - Research   (Site not responding. Last check: 2007-11-05) The SFC algorithm is fast: Only O(n log n) effort to construct a tour of n points and only O(log n) effort to update the solution by adding or removing points. Technical details about algorithm performance, along with citations to related work, may be found in "Spacefilling curves and the planar travelling salesman problem" with L. Platzman, Journal of the Association for Computing Machinery 36(4):719-737 (1989). "Vertex-labelling algorithms for the Hilbert spacefilling curve" by J. Bartholdi and P. Goldsman, submitted (2000). www.scl.gatech.edu /research/casestudies/spacefilling_curves   (735 words)

 On Multi-Dimensional Hilbert Indexings - Alber, Niedermeier (ResearchIndex)   (Site not responding. Last check: 2007-11-05) Hilbert curves are the most simple and popular space-filling indexing scheme. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first... 8 A three-dimensional Hilbert curve (context) - Sagan - 1993 sherry.ifi.unizh.ch /29051.html   (582 words)

 [No title] For our purposes, we will define a SFC (space-filling curve), as a curve whose ranges contains the entire 2-dimensional unit square (as defined on wikipedia). If the curve is followed from beginning to end, then all of the points in the TSP will be encountered. Technically a better way of doing this would be to apply a higher iterate of the space filling curve to those regions with multiple points, but that is unneccessary for this examination. www.idynamix.org /_pmath370FinalProj/index.html   (879 words)

 Citations: Partitioning with spacefilling curves - Pilkington, Baden (ResearchIndex)   (Site not responding. Last check: 2007-11-05) Citations: Partitioning with spacefilling curves - Pilkington, Baden (ResearchIndex) Estimate 4 also holds for a space filling curve partitioning of a (quasi) uniform mesh by superposition of f or mesh dependent construction of f. Ou and Ranka [35] have applied the Hilbert curve to a free particle method in which a chaining mesh is not used to organize the particles. citeseer.ist.psu.edu /context/383471/367247   (1633 words)

 HAKMEM -- SERIES -- DRAFT, NOT YET PROOFED For another property of the parity number, see the spacefilling curve item in the TOPOLOGY section. The third hand rotates around the end of the second at four times this rate; etc. It would seem that the end of the "last" hand (really there are infinitely many) would sweep through space very fast, tracing out an (infinitely) long curve in the time the first hand rotates once. Thus it is unclear whether the curve's arc length is really infinite. www.inwap.com /pdp10/hbaker/hakmem/series.html   (1129 words)

 On the Spacefilling Curve Heuristic for the Euclidean Traveling Salesman Problem - Bertsimas, Grigni (ResearchIndex)   (Site not responding. Last check: 2007-11-05) Abstract: Bartholdi and Platzman [3] proposed the spacefilling curve heuristic for the Euclidean Traveling Salesman Problem and proved that their heuristic returns a tour within an O(lg n) factor of optimal length. ...that the Spacefilling Curve heuristic can never produce tours that are worse than O(log N) times optimum. Bertsimas and M. Grigni, On the spacefilling curve heuristic for the Euclidean traveling salesman problem, tech. citeseer.ist.psu.edu /23831.html   (434 words)

 draw a space-filling curve of arbitrary size   (Site not responding. Last check: 2007-11-05) If you want to use this page offline, you must also download and unzip diagram.zip and copy this file (draw_sfc.html) and the file spacefilling.html into the diagram directory. The algorithm, which draws the spacefilling curve, in short is the following (for more details have a look at the source code of spacefilling.html). It was tested up to a size of 33x33 to work without bugs. www.lutanho.net /pic2html/draw_sfc.html   (65 words)

 BarrosH’s Blog in English :: June :: 2005   (Site not responding. Last check: 2007-11-05) And there is a homepage, Some combinatorial applications of spacefilling curves. Therefore, intuitively, I thought the well-designed SFC should be a pattern that are recursively constructed, and the structure of such pattern will be reserved anywhere in the big one. But when I tried to find out such SFC, I found that the Sierpinski space-filling curve is near what I want when I visited the homepage for spacefilling curves and applications. barrosh.blogsome.com /2005/06/09   (418 words)

 Analysis of the Clustering Properties of Hilbert Space-filling Curve - Moon, Jagadish, Faloutsos, Saltz (ResearchIndex)   (Site not responding. Last check: 2007-11-05) In all these applications, one of the most desired properties from such linear mappings is clustering, which means the locality between objects in the multidimensional space is preserved in the linear space. Analysis of the clustering properties of hilbert spacefilling curve. 14 Analysis of the Hilbert curve for representing two-dimension.. sherry.ifi.unizh.ch /56660.html   (603 words)

 Abstracts for Charles J. Alpert's Publications We begin with a d-dimensional spectral embedding from which a 1-dimensional ordering of the modules is obtained using a spacefilling curve. Thus, we also present a new partitioning algorithm that exploits both the geometric embedding and netlist information, as well as a Restricted Partitioning formulation that requires each cluster of the k-way partitioning to be contiguous in a given linear ordering. We begin with a d-dimensional spectral embedding and construct a heuristic 1-dimensional ordering of the modules (combining spacefilling curve with 3-Opt approaches originally proposed for the traveling salesman problem). vlsicad.ucsd.edu /UCLAWeb/cheese/abstracts.html   (2118 words)

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