Nowhere differentiability of the hilbert curve 12 2. We now turn to a general means of mapping an ndimensional space onto a 1d space. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen. For that purpose, we adapt the general definition of exact selfsimilarity on sets, a typical property of fractals, to the. Spacefilling curves, recursion, simultaneous grammar, logo, postscript. We then consider the space lling curve heuristic for the traveling. Sagan, spacefilling curves, springerverlag, new york, 1994. We begin with an example of a space lling curve and demonstrate how it can be used to nd a short tour through a set of points. Spacefilling curves are special cases of fractal constructions. Hans sagan 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. Spacefilling curves math 635 5 indexing that would like on top of the onedimensional retrieval system. A montage of space filling curves, meant as a supplement to the hilbert curve video. 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.
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. The construction shown above is due to david hilbert 18621943. Wirth 9, by seeing the recursive nature of the curves. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by w. These are notes for a stanford sumo talk on space lling curves. Spacefilling curves and applications in electronic and. The contents throughout is heavily influenced by what is one of the only books to really study the subject, hans sagans space filling curves 1.
Spacefilling curves space lling curves sagan 94 are maps from a 1d interval to a region in ndimensional space. Partitioning of grids for parallel computation with spacefilling curves hilbert, peano, and nested hilbert mpeano spacefilling curves is investigated, and numerical tests on atmosphere flows. Pdf partitioning with spacefilling curves on the cubed. A spacefilling curve is a continuous mapping from a lowerdimensional space into a higherdimensional one.
The first examples were constructed by giuseppe peano 18581932 and thus curves of this kind are known as peano curves. Ive read that spacefilling curves such as the peano curve are useful for maintaining cachefriendly data structures in a linear address space, since they maintain physical spatial locality. 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. Isbn 9780387942650 this book at amazon print version 9781461208716 this book at amazon electronic version doi ebook. This spacefilling curve which completely fills up the unit square is the. A spacefilling curves approximations can be selfavoiding, as the figures above illustrate. Spacefilling curves, puzzling maps between dimensions, have surprising realworld applications. These citations may not conform precisely to your selected citation style. Numerous and frequentlyupdated resource results are available from this search. Cantor, hilbert, moore, knopp, lebesgue, and polya are. Visualization of a spacefilling curve to see a spacefilling curve, we need to visualize or to approximate a spacefilling curve.
Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. Please use this display as a guideline and modify as needed. Hilbert spacefilling curve spacefilling curves sagan, 1994 map points in ndimensional space into a 1d linear order. Geometric generation of the hilbert spacefilling curve. Approximation curves remain within a bounded portion of ndimensional space, but their lengths increase without bound. Introduction 7 a common application of spacefilling curves is storage and retrieval of multidimensional data in a database 88. Next we give a general introduction to space lling curves and discuss properties of them.
Schoenberg 19031990 was born in galatz, romania, and died in madison, wisconsin. We begin with an example of a spacefilling curve and demon. Spacefilling curves and applications in electronic and information engineering. Cantor, hilbert, moore, knopp, lebesgue, and polya are among the prominent mathematicians who have contributed to the field. We will look at a few examples and then prove the hahnmazurkiewicz theorem. A nice collection of such spacefilling curves can be found on the website 30.
The means of realizing or approximating the lebesgue spacefilling curve sfc with binary arithmetic on a uniformly spaced binary grid are not obvious, one problem being its formulation in terms of ternary representations. Computation of lebesgues spacefilling curve computer. The website is still a bit under construction but in any case, i will strive to constantly add further. Heres an excellent curve i first saw in the book brain filling curves the trick is that part of the curve is drawn flipped. In 3 dimensions, selfavoiding approximation curves can even contain knots. Flipped means that instead of just drawing a segment, you instead. Some reflections on the emergence of space filling curves. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire 2dimensional unit square or more generally an ndimensional unit hypercube. The subject of spacefilling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. In this paper we propose to describe spacefilling and related curves by means of simultaneous nonlinear grammars.
Sagan establish a formula to calculate the exact coordinates of an image point if use complex representation, and affine transformations to wich. Generation of spacefilling curves by stochastically independent functions 6. This theorem characterizes those subsets of euclidean that are the image of the unit interval. 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 spacefilling curves serve as a counterexample to lessthanrigorous notions of dimension. Working in this area is like skating on the edge of reason. On spacefilling curves and the hahnmazurkiewicz theorem 3 figure 1. Series universitext subject category mathematical physics and mathematics. Proximity problems for which spacefilling curves have been used frequently are approximate nearest neighbors search and finding closest pairs see 99 and the references therein. Spacefilling curves serve as a counterexample to lessthanrigorous notions of. When vertices are sorted according to this index, they automatically assume the layout of a morton space lling curve. Download it once and read it on your kindle device, pc, phones or tablets. In addition to their mathematical importance, space. Mathematical description in this text we will only deal with two dimensional sfcs. We define exact selfsimilarity of space filling curves on the plane.
Several recent computer network visualizations have used a space. Intuitively, they can be thought of as curves that traverse all points of the ndimensional space, thus inducing an order on those points. 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. Recurrence for pandimensional spacefilling functions. We give a general introduction to spacefilling curves and discuss properties of.
A spacefilling 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. Peano 1890 first presented a spacefilling curve to efficiently and consecutively map every node for a closed 2 n square, appropriately. Null 8 was the first to give a nonrecursive algorithm to generate hilbert and other curves. The ith interval is then mapped into the ith square in a piecewise. Because giuseppe peano 18581932 was the first to discover one, spacefilling curves in the 2dimensional plane are sometimes called peano curves, but that phrase also refers to the peano curve, the specific example of a. The nowhere differentiability of the schoenberg curve 45 46 49 49 51 56 58 60 62 67 69 69 71 74. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. The curve visits each point in space only one time in a certain order usually points that are close on the curve are close in space. Spacefilling curves can be employed to accomplish this mapping. Because giuseppe peano 18581932 was the first to discover one, spacefilling curves in the 2dimensional plane are sometimes called peano curves, but that phrase also refers to the peano curve, the specific example of a spacefilling curve found by peano. In this paper we seek to enhance the design of the scaffold architecture by using space filling curves within its interior space. The subject of spacefilling curves has fascinated mathematicians for over a century. Fast solution of geophysical inversion using adaptive mesh. Pdf analysis of multidimensional spacefilling curves.
Use features like bookmarks, note taking and highlighting while reading spacefilling curves universitext. However, currently the interior macro pore design of scaffolds have been limited to simple regular shapes of either squares or circles due to limited cad capability. Pdf an introduction to spacefilling curves kyle byrne. Several procedures for drawing particular spacefiliing curves have been proposed in the literature 2, 7, 10, 12, 15, 16. Pdf a spacefilling curve is a way of mapping the multidimensional space into the. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant. Spacefilling curves universitext kindle edition by sagan, hans. Representation of a spacefilling curve by an analytic function 6.
1286 572 1015 1196 977 1615 441 1498 1471 327 729 1225 146 422 998 524 178 74 1290 976 507 927 1405 1168 1168 1084 631 718 857 1620 379 798 119 1307 1042 763 1152 1229 570 640 202 139 1108 586 53