In mathematical analysis, a spacefilling curve is a curve whose range contains the entire. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen. If you want to use this page offline, you must also download and unzip diagram. Some reflections on the emergence of space filling curves. 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. Space filling curves universitext by hans sagan and a great selection of related books, art and collectibles available now at.
These citations may not conform precisely to your selected citation style. Hans sagans research works university of vienna, vienna. Wirth 9, by seeing the recursive nature of the curves, has given a recursive procedure to draw these curves. In this section, we give the surprising result that there is a continuous function from the interval 0,1 onto the unit square 0,1. The performance of space filling curves for dimension reduction. For that purpose, we adapt the general definition of exact selfsimilarity on sets, a typical property of fractals, to the. 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.
Space filling curves math 635 5 indexing that would like on top of the onedimensional retrieval system. Sag94 hans sagan, spacefilling curves, universitext series, springerverlag, 1994. However, formatting rules can vary widely between applications and fields of interest or study. The idea that a onedimensional curve may completely cover an area or a volume was, at that time, completely novel and counterintuitive. But worst is that the curve is no longer drawn as a single continuous line. We define exact selfsimilarity of space filling curves on the plane. Series universitext subject category mathematical physics and mathematics. A better way of flipping a section is to reverse all of the angles. He studied at the universities of jassy, gottingen, and berlin, and received his doctors degree from the university of jassy in 1926.
Spacefilling curves by hans sagan 1994, paperback at the best online prices at ebay. Space filling curve, pdf recurrence for pandimensional space filling functions arxiv. We will show in this chapter that it is still possible for such curves to have a positive twodimensional lebesgue measure. Hans sagan is the author of boundary and eigenvalue problems in mathematical physics 3. Triple points belong to a relatively simple set, while double points resist easy classification.
Space filling curves can be employed to accomplish this mapping. Computer generation of spacefilling curves several algorithms have been presented for construct ing a hilbert curve. Generation of spacefilling curves by stochastically independent. Hans sagan s 8 research works with 32 citations and 127 reads, including. Working in this area is like skating on the edge of reason. Jan 16, 2016 a montage of space filling curves, meant as a supplement to the hilbert curve video. Hans sagan space filling curves with 65 figures springerverlag new york berlin heidelberg london paris tokyo hong kong barcelona budapest. 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. Hans sagan author of boundary and eigenvalue problems in. Download download space filling curves hans sagan pdf read online read online space filling curves hans sagan pdf hilbert space space filling curve algorithm peano curve construction 3d space filling curve hilbert curve function download fulltext pdf. Stanford libraries official online search tool for books, media, journals, databases, government documents and more. Hilberts geometric generating process for the construction of a surface filling curve is generalized to lead to a space filling curve and an analyticarithmetic representation of that curve is found in terms of the eight similarity transformations that generate it. Download it once and read it on your kindle device, pc, phones or tablets. Must every continuous space filling curve be nowhere.
This theorem characterizes those subsets of euclidean space that are the image of the unit interval under a continuous space. On space filling curves and the hahnmazurkiewicz theorem alexander kupers abstract. Niedermeier, on multidimensional curves with hilbert property, theory comput. In this paper we seek to enhance the design of the scaffold architecture by using space filling curves within its interior space. Spacefilling curves, puzzling maps between dimensions, have surprising realworld applications. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire 2dimensional unit square or more generally an ndimensional unit hypercube. Librarything is a cataloging and social networking site for booklovers. Mathematical description in this text we will only deal with two dimensional sfcs. See all books authored by hans sagan, including introduction to the calculus of variations, and boundary and eigenvalue problems in mathematical physics, and more on. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant.
Hans sagan 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. Flipped means that instead of just drawing a segment, you instead. 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. 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. An elementary proof that schoenbergs spacefilling curve. Pushing lebesgue measure with peano curve mathematics. Spacefilling curves universitext kindle edition by sagan, hans.
Null 8 was the first to give a nonrecursive algorithm to generate hilbert and other curves. Space filling curves universitext kindle edition by sagan, hans. A cubefilling hilbert curve university of waterloo. Generation of space filling curves by stochastically independent functions 6. Download spacefilling curves universitext pdf by hans. What is the state of the art about spacefilling curves in. Although most space filling curves are not osgood curves they have positive area but often include infinitely many selfintersections, failing to be jordan curves it is possible to modify the recursive construction of space filling curves or other fractal curves to obtain an osgood curve.
Author of introduction to the calculus of variations, integral and differential calculus, boundary and eigenvalue problems in mathematical physics, beat the odds, calculus accompanied on the apple, introduction to the calculus of variations, boundary and eigenvalue problems in mathematical physics, advanced calculus. A more recent book may be of interest to followers of this thread. The subject of space filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Cantor, hilbert, moore, knopp, lebesgue, and polya are among the prominent mathematicians who have contributed to the field. An interesting book related to space filling curves and. An elementary proof that schoenbergs space filling curve is nowhere differentiable hans sagan north carolina state university raleigh, nc 27695 up to the end of the nineteenth century and in six, randomly selected, contemporary calculus books a plane curve was and is defined as the graph of a pair of parametric equations xft tee i. A montage of space filling curves, meant as a supplement to the hilbert curve video. To accompany this is a table of sierpinski indices of the points of a 100 x 100 grid pdf format, 22 pages, with which you can set up your own routing. Hans sagan spacefilling curves with 65 figures springerverlag. Some reflections on the emergence of spacefilling curves. Heres an excellent curve i first saw in the book brain filling curves the trick is that part of the curve is drawn flipped. Flipping by replacing left turns with right turns etc.
The subject of spacefilling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. On spacefilling curves and the hahnmazurkiewicz theorem 3 figure 1. Sagan, hans 1994, spacefilling curves, springerverlag, doi. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. 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. Use features like bookmarks, note taking and highlighting while reading space filling curves universitext. Shchepin the known serpinskyknopp curve 1 maps the segment to an isosceles rectangular triangle in such a way that one half of the segment is mapped to half of the triangle, and the other half the. The way it could have happened and should have happened, but did not happen by hans sagan north carolina state university, department of mathematics, raleigh, nc 276958205, u. And the whole idea of space filling curves is that you never lift the pen. Please use this display as a guideline and modify as needed. More editions of space filling curves universitext. Here is an ascii textgraphic of the cube filling hilbert curve that was posted to the fracl fractal discussion list. The concept of space filling curves has been extended to handle polygons. Download letters from the teacher volume 2 teachings of the order of christian mystics volume 9 pdf by mrs harriette augusta curtiss download moscow mathematical olympiads, 19931999 msri mathematical circles library pdf by roman fedorov, alexei.
Pdf an introduction to spacefilling curves kyle byrne. Jordan curves of positive lebesgue measure springer for. The nowhere differentiability of the schoenberg curve 45 46 49 49 51 56 58 60 62 67 69 69 71 74. The function visits double and triple points predicted by nettos theorem. Hans sagan, author of introduction to the calculus of variations, on librarything. Also, it introduces some jordan curve with positive lebesgue measure, hope it will help all of us. We use cookies to make interactions with our website easy and meaningful, to better. Vector replacement rules allow formulation of the flowsnake planefilling curve as a bijection z z2 and as a surjection 2. Hans sagan sagan, hans used books, rare books and new books. However, there have been no comprehensive treatments of the subject since siepinskys in 1912.
The best reference for spacefilling curves is sagans book. 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. Several procedures for drawing particular space filiing curves have been proposed in the literature 2, 7, 10, 12, 15, 16. Could someone provide a reference or a sketch of a proof that no differentiable spacefilling curve exists. Recurrence for pandimensional spacefilling functions. Use features like bookmarks, note taking and highlighting while reading spacefilling curves universitext. An elementary proof that schoenbergs spacefilling curve is.
These are notes on space lling curves, looking at a few examples and proving the hahnmazurkiewicz theorem. For the preface and content of the book, visit the following website. The algorithm, which draws the spacefilling curve, in short is the following for more details have a look at the source code of spacefilling. Hans sagan the subject of space filling curves has generated a great deal of interest since the first such curve was discovered by peano over a century ago. A spacefilling curve is a continuous mapping from a lowerdimensional space into a higherdimensional one. Besides peanos space filling curve, there are many other famous space filling curves, the book space filling curves wrote by hans sagan introduce several of them. Representation of a space filling curve by an analytic function 6. Proof that no differentiable spacefilling curve exists. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by w. Space filling curves, recursion, simultaneous grammar, logo, postscript. 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. Isbn 9780387942650 this book at amazon print version.
Hans sagan, space filling curves, springerverlag, new york 1994. These are notes on spacefilling curves, looking at a few examples and proving. A summary of the ideas, minus technical details but with pointers to technical literature, may be found in my class notes a routing system based on spacefilling curves pdf format, 22 pages. Schoenberg 19031990 was born in galatz, romania, and died in madison, wisconsin. Space filling curves started their lives as mathematical curiosities, at the end of the nineteenth century. In this paper we propose to describe space filling and related curves by means of simultaneous nonlinear grammars. Hans sagan books list of books by author hans sagan. Introduction 7 a common application of space filling curves is storage and retrieval of multidimensional data in a database 88. Contents preface vii acknowledgments ix illustration credits xi chapter 1.
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