DOCSLIB.ORG

On Bounding Boxes of Iterated Function System Attractors Hsueh-Ting Chu, Chaur-Chin Chen*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Bounding Boxes of Iterated Function System Attractors Hsueh-Ting Chu, Chaur-Chin Chen* Computers & Graphics 27 (2003) 407–414 Technical section On bounding boxes of iterated function system attractors Hsueh-Ting Chu, Chaur-Chin Chen* Department of Computer Science, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Abstract Before rendering 2D or 3D fractals with iterated function systems, it is necessary to calculate the bounding extent of fractals. We develop a new algorithm to compute the bounding boxwhich closely contains the entire attractor of an iterated function system. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Fractals; Iterated function system; IFS; Bounding box 1. Introduction 1.1. Iterated function systems Barnsley [1] uses iterated function systems (IFS) to Definition 1. A transform f : X-X on a metric space provide a framework for the generation of fractals. ðX; dÞ is called a contractive mapping if there is a Fractals are seen as the attractors of iterated function constant 0pso1 such that systems. Based on the framework, there are many A algorithms to generate fractal pictures [1–4]. However, dðf ðxÞ; f ðyÞÞps Á dðx; yÞ8x; y X; ð1Þ in order to generate fractals, all of these algorithms have where s is called a contractivity factor for f : to estimate the bounding boxes of fractals in advance. For instance, in the program Fractint (http://spanky. Definition 2. In a complete metric space ðX; dÞ; an triumf.ca/www/fractint/fractint.html), we have to guess iterated function system (IFS) [1] consists of a finite set the parameters of ‘‘image corners’’ before the beginning of contractive mappings w ; for i ¼ 1; 2; y; n; which is of drawing, which may not be practical. i denoted as W ¼fX; w ; w ; y; w g: For this reason, this paper is devoted to develop a 1 2 n practical bounding boxalgorithm. A good bounding In this paper, we assume contractive affine transforms algorithm is also a priori for rendering 3D fractals. Hart in an iterated function system. A contractive affine and DeFanti [5] use bounding spheres in their system of transform w in R2 is 3D fractal rendering. Sometimes the bounding spheres "#!"#"# "# "# "# are very loose such that the rendering of common x ab x e x e fractals may be inefficient. Most of 3D computer w ¼ þ ¼ A þ ð2Þ y cd y f y f graphic systems employ bounding boxes for rendering objects. Thus, it is important to develop a tight and a contractive affine transform w in R3 is bounding boxalgorithm for rendering fractal objects 02 31 2 32 3 2 3 in a 3D system. x abc x t B6 7C 6 76 7 6 7 w@4 y 5A ¼ 4 def54 y 5 þ 4 u 5 z ijk z v 2 3 2 3 x t 6 7 6 7 *Corresponding author. Tel.:+11-886-3-573-1078; fax:+11- ¼ A4 y 5 þ 4 u 5; ð3Þ 886-3-572-3694. E-mail address: [email protected] (C.-C. Chen). z v 0097-8493/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0097-8493(03)00035-9 408 H.-T. Chu, C.-C. Chen / Computers & Graphics 27 (2003) 407–414 where the eigenvalues of the matrix A have magnitude Theorem 2 indicates that the entire attractor can be less than 1. generated by applying the IFS on some point zAA iteratively. Thus, it provides a formula to represent all the Theorem 1 (Barnsley [1]). Let f : : X-X be a contrac- points in the attractor as A ¼fpjp ¼ t1t2ytcðzÞ; tiAfw1; tive mapping in a complete metric space ðX; dÞ: Then f w2; y; wng and z is the fixed-point of some mapping wjg: possesses a unique fixed-point xf AX; and for any xAX; n the Cauchy sequence ff ðxÞg converges to xf ; where 1.2. Previous work f nðxÞ¼f nÀ1ðf ðxÞÞ; and f 0ðxÞ¼x: Let W ¼fX; w1; w2; y; wng be an iterated function In a Hausdorff metric space ðHðXÞ; hðdÞÞ; an IFS, W; system on X; and let A be the attractor associated with is also a contractive mapping, thus it possesses a W: We try to find the bounding volume of the attractor unique fixed-point, called the attractor of the IFS. A: In the literature, only bounding ball algorithms were Tables 1–3 list the 2D IFS codes of fractals given by developed [5–8] for IFSs. A bounding ball is also called a the program Fractint, which generate the ‘‘dragon’’, bounding circle in a 2D space or a bounding sphere in a ‘‘coral’’, and ‘‘fern’’ fractal objects [1], respectively. 3D space. Fig. 1 shows a picture of these fractal objects. We Canright [6] uses multiple bounding balls to envelop will compute the bounding boxes of these IFS codes the entire attractor of an IFS. Let z1; z2; y; zn be later. the fixed points of w1; w2; y; wn on ðX; dÞ; and let s1; s2; y; sn be the contractivity factors of w1; w2; y; wn; Lemma 1. Let W ¼fX; w ; w ; y; w g be an iterated 1 2 n respectively. They seek n balls SCi centered on zi; 1pipn function system. Let A be the attractor associated with W C C n such that wðAÞ Ci; then A i¼1 Ci: The radii frig are on ðHðXÞ; hðdÞÞ; and let z1; z2; y; zn be the fixed points of chosen to satisfy [6] w1; w2; y; wn; respectively, on ðX; dÞ: Then ziAA for ri ¼ si maxðrj þ dðzi; zjÞÞ; 1pipn: ð6Þ i ¼ 1; 2; y; n: jai N Fig. 2 illustrates Canright’s bounding balls for the fern. Proof. zi ¼ wi ðziÞ since zi is the fixed point of wi; hence N N In order to enclose Canright’s balls, a large bounding ziAW ðfzigÞ: By Theorem 1, W ðfzigÞ ¼ A: Hence N box B Boxðx ; x ; y ; y Þ¼ðÀ14:24; 19:16; ziAW ðfzigÞ ¼ A: & min max min max À9:36; 26:70Þ is necessary. We mark the minimum bounding boxof the fern with dotted lines in the figure, Theorem 2. Let W ¼fX; w1; w2; y; wng be an iterated function system. Let A be the attractor associated with then the computed bounding boxby Canright’s algo- W; and zAA: Then the attractor A is equal to rithm is too loose in practice. However, Canright fzg,fWðzÞg,fW 2ðzÞg,?,fW NðzÞg: provided a very simple and fast solution to compute a bound for drawing fractals. Proof. (a) By Theorem 1, fW NðzÞg ¼ A: Hence, Hart and DeFanti [5] introduced another bounding ball algorithm. Only one bounding ball is computed to ADfzg fWðzÞg fW 2ðzÞg ? fW NðzÞg: ð Þ , , , , 4 enclose the attractor of an IFS in their algorithm. Initially (b) We know zAA; then all the sets the ball C is the unit ball at the origin, then the ball is fzg; fWðzÞg; fW 2ðzÞg; y; fW NðzÞgDA: Thus, moved and enlarged to envelop the entire attractor n N iteratively. The next ball C in the sequence is found as fzg,fWðzÞg,fW 2ðzÞg,?,fW ðzÞgDA: ð5Þ Xn 2 n 1 From (a) and (b), A=fzg,fWðzÞg,fW ðzÞg o ¼ wiðoÞð7Þ N n ,?,fW ðzÞg: & i¼1 Fig. 1. The attractors of the 2D IFS codes in Tables 1–3. H.-T. Chu, C.-C. Chen / Computers & Graphics 27 (2003) 407–414 409 bounding boxalgorithm instead. We desire to acquire a very tight bounding volume instead of a loose one. C However, there are many of problems we have to 2 overcome. In the next section, we consider these problems and their solutions. For convenience, we consider the bounding problem in 2D cases. In Section 3, the new algorithm is presented. We show experimental results in Section 4. Finally, conclusions are drawn in Section 5. 2. Bounding boxes for IFS attractors C4 The advantage of a bounding ball algorithm is that it can handle rotations. It is a complexproblem we have to C3 C1 resolve when developing a bounding boxalgorithm instead. An affine transformation is composed of translation, scaling, and rotation. Thus a 2D contractive affine transform of Eq. (2) can be rewritten as "#!"#"# "# Fig. 2. Canright’s bounding balls for fern. x sx cos y sy sin y x e w ¼ þ ; ð9Þ y Àsx sin y sy cos y y f and where jsxjo1 and jsyjo1: n n For example, let y ¼ p=4; sx ¼ sy ¼ 0:9; ox ¼ oy ¼ 0; r ¼ max ðsir þ dðwiðoÞ; o ÞÞ; ð8Þ i¼1yn then "#!"#"# "# x 0:64 0:64 x e where o and r are the origin and radius of ball C and w ¼ þ : ð10Þ likewise for ball Cn: y À0:64 0:64 y f We show the bounding ball for the fern computed by In this case, we will always obtain a smaller ball after Hart’s algorithm in Fig. 3. The origin and radius of the applying w: Here the new radius is 0.9 of the original, ball Cn are (0.03, 1.44) and 9.28, respectively. Compared but we may need a larger boxto enclose the transformed with Canright’s algorithm, Hart’s algorithm obtains a box. Fig. 4 shows the transformation of a boxwith w: A better result in this case. Rice [8] tries to improve Hart’s square with the length of sides 1.28 is necessary to algorithm by optimizing the radius r: The effort made by enclose the transformed unit square. As a result, given Rice has little improvement in this case because the an arbitrary initial box B ; WðB Þ may exceed the computed radius for the fern, by Hart’s algorithm, is 0 0 extent of B ; for an IFS W: On the contrary, it would be close to the optimal radius.
Load more

Recommended publications
  • Object Oriented Programming
    No. 52 March-A pril'1990 $3.95 T H E M TEe H CAL J 0 URN A L COPIA Object Oriented Programming First it was BASIC, then it was structures, now it's objects. C++ afi<;ionados feel, of course, that objects are so powerful, so encompassing that anything could be so defined. I hope they're not placing bets, because if they are, money's no object. C++ 2.0 page 8 An objective view of the newest C++. Training A Neural Network Now that you have a neural network what do you do with it? Part two of a fascinating series. Debugging C page 21 Pointers Using MEM Keep C fro111 (C)rashing your system. An AT Keyboard Interface Use an AT keyboard with your latest project. And More ... Understanding Logic Families EPROM Programming Speeding Up Your AT Keyboard ((CHAOS MADE TO ORDER~ Explore the Magnificent and Infinite World of Fractals with FRAC LS™ AN ELECTRONIC KALEIDOSCOPE OF NATURES GEOMETRYTM With FracTools, you can modify and play with any of the included images, or easily create new ones by marking a region in an existing image or entering the coordinates directly. Filter out areas of the display, change colors in any area, and animate the fractal to create gorgeous and mesmerizing images. Special effects include Strobe, Kaleidoscope, Stained Glass, Horizontal, Vertical and Diagonal Panning, and Mouse Movies. The most spectacular application is the creation of self-running Slide Shows. Include any PCX file from any of the popular "paint" programs. FracTools also includes a Slide Show Programming Language, to bring a higher degree of control to your shows.
    [Show full text]
  • Iterated Function System Quasiarcs
    CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 21, Pages 78–100 (February 3, 2017) http://dx.doi.org/10.1090/ecgd/305 ITERATED FUNCTION SYSTEM QUASIARCS ANNINA ISELI AND KEVIN WILDRICK Abstract. We consider a class of iterated function systems (IFSs) of contract- ing similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural H¨older continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-H¨older) parameterization. We also give a related nec- essary condition for the invariant set of such an IFS to be homeomorphic to an interval. 1. Introduction Consider an iterated function system (IFS) of contracting similarities S = n {S1,...,SN } of R , N ≥ 2, n ≥ 1. For i =1,...,N, we will denote the scal- n n ing ratio of Si : R → R by 0 <ri < 1. In a brief remark in his influential work [18], Hutchinson introduced a class of such IFSs for which the invariant set is a Peano continuum, i.e., it possesses a continuous parameterization by the unit interval. There is a natural choice for this parameterization, which we call the Hutchinson parameterization. Definition 1.1 (Hutchinson, 1981). The pair (S,γ), where γ is the invariant set of an IFS S = {S1,...,SN } with scaling ratio list {r1,...,rN } is said to be an IFS path, if there exist points a, b ∈ Rn such that (i) S1(a)=a and SN (b)=b, (ii) Si(b)=Si+1(a), for any i ∈{1,...,N − 1}.
    [Show full text]
  • An Introduction to Apophysis © Clive Haynes MMXX
    Apophysis Fractal Generator An Introduction Clive Haynes Fractal ‘Flames’ The type of fractals generated are known as ‘Flame Fractals’ and for the curious, I append a note about their structure, gleaned from the internet, at the end of this piece. Please don’t ask me to explain it! Where to download Apophysis: go to https://sourceforge.net/projects/apophysis/ Sorry Mac users but it’s only available for Windows. To see examples of fractal images I’ve generated using Apophysis, I’ve made an Issuu e-book and here’s the URL. https://issuu.com/fotopix/docs/ordering_kaos Getting Started There’s not a defined ‘follow this method workflow’ for generating interesting fractals. It’s really a matter of considerable experimentation and the accumulation of a knowledge-base about general principles: what the numerous presets tend to do and what various options allow. Infinite combinations of variables ensure there’s also a huge serendipity factor. I’ve included a few screen-grabs to help you. The screen-grabs are detailed and you may need to enlarge them for better viewing. Once Apophysis has loaded, it will provide a Random Batch of fractal patterns. Some will be appealing whilst many others will be less favourable. To generate another set, go to File > Random Batch (shortcut Ctrl+B). Screen-grab 1 Choose a fractal pattern from the batch and it will appear in the main window (Screen-grab 1). Depending upon the complexity of the fractal and the processing power of your computer, there will be a ‘wait time’ every time you change a parameter.
    [Show full text]
  • Fractal Dimension of Self-Affine Sets: Some Examples
    FRACTAL DIMENSION OF SELF-AFFINE SETS: SOME EXAMPLES G. A. EDGAR One of the most common mathematical ways to construct a fractal is as a \self-similar" set. A similarity in Rd is a function f : Rd ! Rd satisfying kf(x) − f(y)k = r kx − yk for some constant r. We call r the ratio of the map f. If f1; f2; ··· ; fn is a finite list of similarities, then the invariant set or attractor of the iterated function system is the compact nonempty set K satisfying K = f1[K] [ f2[K] [···[ fn[K]: The set K obtained in this way is said to be self-similar. If fi has ratio ri < 1, then there is a unique attractor K. The similarity dimension of the attractor K is the solution s of the equation n X s (1) ri = 1: i=1 This theory is due to Hausdorff [13], Moran [16], and Hutchinson [14]. The similarity dimension defined by (1) is the Hausdorff dimension of K, provided there is not \too much" overlap, as specified by Moran's open set condition. See [14], [6], [10]. REPRINT From: Measure Theory, Oberwolfach 1990, in Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, numero 28, anno 1992, pp. 341{358. This research was supported in part by National Science Foundation grant DMS 87-01120. Typeset by AMS-TEX. G. A. EDGAR I will be interested here in a generalization of self-similar sets, called self-affine sets. In particular, I will be interested in the computation of the Hausdorff dimension of such sets.
    [Show full text]
  • Current Practices in Quantitative Literacy © 2006 by the Mathematical Association of America (Incorporated)
    Current Practices in Quantitative Literacy © 2006 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2005937262 Print edition ISBN: 978-0-88385-180-7 Electronic edition ISBN: 978-0-88385-978-0 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1 Current Practices in Quantitative Literacy edited by Rick Gillman Valparaiso University Published and Distributed by The Mathematical Association of America The MAA Notes Series, started in 1982, addresses a broad range of topics and themes of interest to all who are in- volved with undergraduate mathematics. The volumes in this series are readable, informative, and useful, and help the mathematical community keep up with developments of importance to mathematics. Council on Publications Roger Nelsen, Chair Notes Editorial Board Sr. Barbara E. Reynolds, Editor Stephen B Maurer, Editor-Elect Paul E. Fishback, Associate Editor Jack Bookman Annalisa Crannell Rosalie Dance William E. Fenton Michael K. May Mark Parker Susan F. Pustejovsky Sharon C. Ross David J. Sprows Andrius Tamulis MAA Notes 14. Mathematical Writing, by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts. 16. Using Writing to Teach Mathematics, Andrew Sterrett, Editor. 17. Priming the Calculus Pump: Innovations and Resources, Committee on Calculus Reform and the First Two Years, a subcomit- tee of the Committee on the Undergraduate Program in Mathematics, Thomas W. Tucker, Editor. 18. Models for Undergraduate Research in Mathematics, Lester Senechal, Editor. 19. Visualization in Teaching and Learning Mathematics, Committee on Computers in Mathematics Education, Steve Cunningham and Walter S.
    [Show full text]
  • Math Morphing Proximate and Evolutionary Mechanisms
    Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales.
    [Show full text]
  • Iterated Function System
    IARJSET ISSN (Online) 2393-8021 ISSN (Print) 2394-1588 International Advanced Research Journal in Science, Engineering and Technology ISO 3297:2007 Certified Vol. 3, Issue 8, August 2016 Iterated Function System S. C. Shrivastava Department of Applied Mathematics, Rungta College of Engineering and Technology, Bhilai C.G., India Abstract: Fractal image compression through IFS is very important for the efficient transmission and storage of digital data. Fractal is made up of the union of several copies of itself and IFS is defined by a finite number of affine transformation which characterized by Translation, scaling, shearing and rotat ion. In this paper we describe the necessary conditions to form an Iterated Function System and how fractals are generated through affine transformations. Keywords: Iterated Function System; Contraction Mapping. 1. INTRODUCTION The exploration of fractal geometry is usually traced back Metric Spaces definition: A space X with a real-valued to the publication of the book “The Fractal Geometry of function d: X × X → ℜ is called a metric space (X, d) if d Nature” [1] by the IBM mathematician Benoit B. possess the following properties: Mandelbrot. Iterated Function System is a method of constructing fractals, which consists of a set of maps that 1. d(x, y) ≥ 0 for ∀ x, y ∈ X explicitly list the similarities of the shape. Though the 2. d(x, y) = d(y, x) ∀ x, y ∈ X formal name Iterated Function Systems or IFS was coined 3. d x, y ≤ d x, z + d z, y ∀ x, y, z ∈ X . (triangle by Barnsley and Demko [2] in 1985, the basic concept is inequality).
    [Show full text]
  • Fast Visualisation and Interactive Design of Deterministic Fractals
    Computational Aesthetics in Graphics, Visualization, and Imaging (2008) P. Brown, D. W. Cunningham, V. Interrante, and J. McCormack (Editors) Fast Visualisation and Interactive Design of Deterministic Fractals Sven Banisch1 & Mateu Sbert2 1Faculty of Media, Bauhaus–University Weimar, D-99421 Weimar (GERMANY) 2Department of Informàtica i Matemàtica Aplicada, University of Girona, 17071 Girona (SPAIN) Abstract This paper describes an interactive software tool for the visualisation and the design of artistic fractal images. The software (called AttractOrAnalyst) implements a fast algorithm for the visualisation of basins of attraction of iterated function systems, many of which show fractal properties. It also presents an intuitive technique for fractal shape exploration. Interactive visualisation of fractals allows that parameter changes can be applied at run time. This enables real-time fractal animation. Moreover, an extended analysis of the discrete dynamical systems used to generate the fractal is possible. For a fast exploration of different fractal shapes, a procedure for the automatic generation of bifurcation sets, the generalizations of the Mandelbrot set, is implemented. This technique helps greatly in the design of fractal images. A number of application examples proves the usefulness of the approach, and the paper shows that, put into an interactive context, new applications of these fascinating objects become possible. The images presented show that the developed tool can be very useful for artistic work. 1. Introduction an interactive application was not really feasible till the ex- ploitation of the capabilities of new, fast graphics hardware. The developments in the research of dynamical systems and its attractors, chaos and fractals has already led some peo- The development of an interactive fractal visualisation ple to declare that god was a mathematician, because mathe- method, is the main objective of this paper.
    [Show full text]
  • An Introduction to the Mandelbrot Set
    An introduction to the Mandelbrot set Bastian Fredriksson January 2015 1 Purpose and content The purpose of this paper is to introduce the reader to the very useful subject of fractals. We will focus on the Mandelbrot set and the related Julia sets. I will show some ways of visualising these sets and how to make a program that renders them. Finally, I will explain a key exchange algorithm based on what we have learnt. 2 Introduction The Mandelbrot set and the Julia sets are sets of points in the complex plane. Julia sets were first studied by the French mathematicians Pierre Fatou and Gaston Julia in the early 20th century. However, at this point in time there were no computers, and this made it practically impossible to study the structure of the set more closely, since large amount of computational power was needed. Their discoveries was left in the dark until 1961, when a Jewish-Polish math- ematician named Benoit Mandelbrot began his research at IBM. His task was to reduce the white noise that disturbed the transmission on telephony lines [3]. It seemed like the noise came in bursts, sometimes there were a lot of distur- bance, and sometimes there was no disturbance at all. Also, if you examined a period of time with a lot of noise-problems, you could still find periods without noise [4]. Could it be possible to come up with a model that explains when there is noise or not? Mandelbrot took a quite radical approach to the problem at hand, and chose to visualise the data.
    [Show full text]
  • Iterated Function System Iterated Function System
    Iterated Function System Iterated Function System (or I.F.S. for short) is a program that explores the various aspects of the Iterated Function System. It contains three modes: 1. The Chaos Game, which implements variations of the chaos game popularized by Michael Barnsley, 2. Deterministic Mode, which creates fractals using iterated function system rules in a deterministic way, and 3. Random Mode, which creates fractals using iterated function system rules in a random way. Upon execution of I.F.S. we need to choose a mode. Let’s begin by choosing the Chaos Game. Upon selecting that mode, two windows will appear as in Figure 1. The blank one on the right is the graph window and is where the result of the execution of the chaos game will be displayed. The window on the left is the status window. It is set up to play the chaos game using three corners and a scale factor of 0.5. Thus, the next seed point will be calculated as the midpoint of the current seed point with the selected corner. It is also set up so that the corners will be graphed once they are selected, and a line is not drawn between the current seed point and the next selected corner. As a first attempt at playing the chaos game, let’s press the “Go!” button without changing any of these settings. We will first be asked to define corner #1, which can be accomplished either by typing in the xy-coordinates in the status window edit boxes, or by using the mouse to move the cross hair cursor appearing in the graph window.
    [Show full text]
  • Pointwise Regularity of Parametrized Affine Zipper Fractal Curves
    POINTWISE REGULARITY OF PARAMETERIZED AFFINE ZIPPER FRACTAL CURVES BALAZS´ BAR´ ANY,´ GERGELY KISS, AND ISTVAN´ KOLOSSVARY´ Abstract. We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise H¨olderexpo- nent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise H¨olderexponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, we apply our results for de Rham's curve. 1. Introduction and Statements Let us begin by recalling the general definition of fractal curves from Hutchin- son [22] and Barnsley [3]. d Definition 1.1. A system S = ff0; : : : ; fN−1g of contracting mappings of R to itself is called a zipper with vertices Z = fz0; : : : ; zN g and signature " = ("0;:::;"N−1), "i 2 f0; 1g, if the cross-condition fi(z0) = zi+"i and fi(zN ) = zi+1−"i holds for every i = 0;:::;N − 1. We call the system a self-affine zipper if the functions fi are affine contractive mappings of the form fi(x) = Aix + ti; for every i 2 f0; 1;:::;N − 1g; d×d d where Ai 2 R invertible and ti 2 R . The fractal curve generated from S is the unique non-empty compact set Γ, for which N−1 [ Γ = fi(Γ): i=0 If S is an affine zipper then we call Γ a self-affine curve.
    [Show full text]
  • Open Source Resources for Teaching and Research in Mathematics
    Open Source Resources for Teaching and Research in Mathematics Dr. Russell Herman Dr. Gabriel Lugo University of North Carolina Wilmington Open Source Resources, ICTCM 2008, San Antonio 1 Outline History Definition General Applications Open Source Mathematics Applications Environments Comments Open Source Resources, ICTCM 2008, San Antonio 2 In the Beginning ... then there were Unix, GNU, and Linux 1969 UNIX was born, Portable OS (PDP-7 to PDP-11) – in new “C” Ken Thompson, Dennis Ritchie, and J.F. Ossanna Mailed OS => Unix hackers Berkeley Unix - BSD (Berkeley Systems Distribution) 1970-80's MIT Hackers Public Domain projects => commercial RMS – Richard M. Stallman EMACS, GNU - GNU's Not Unix, GPL Open Source Resources, ICTCM 2008, San Antonio 3 History Free Software Movement – 1983 RMS - GNU Project – 1983 GNU GPL – GNU General Public License Free Software Foundation (FSF) – 1985 Free = “free speech not free beer” Open Source Software (OSS) – 1998 Netscape released Mozilla source code Open Source Initiative (OSI) – 1998 Eric S. Raymond and Bruce Perens The Cathedral and the Bazaar 1997 - Raymond Open Source Resources, ICTCM 2008, San Antonio 4 The Cathedral and the Bazaar The Cathedral model, source code is available with each software release, code developed between releases is restricted to an exclusive group of software developers. GNU Emacs and GCC are examples. The Bazaar model, code is developed over the Internet in public view Raymond credits Linus Torvalds, Linux leader, as the inventor of this process. http://en.wikipedia.org/wiki/The_Cathedral_and_the_Bazaar Open Source Resources, ICTCM 2008, San Antonio 5 Given Enough Eyeballs ... central thesis is that "given enough eyeballs, all bugs are shallow" the more widely available the source code is for public testing, scrutiny, and experimentation, the more rapidly all forms of bugs will be discovered.
    [Show full text]