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Automatic Generation of 3-D Linear Fractal Shapes with Quadratic Map Basins Animation 芸術科学会論文誌 Vol.2 No.1 pp.61-70 Automatic Generation of 3-D Linear Fractal Shapes with Quadratic Map Basins Animation Hussein KARAM Masayuki NAKAJIMA Ain Shams University Graduate School of Information Sc. & Eng., Faculty of Science Tokyo Institute of Technology Math. & Computer Sc. Dept. 2-12-1 O-okayama, Meguro-ku, Tokyo, Abbassia, Cairo, Egypt 152-8552 Japan Abstract ment of a fractal shapes were originally developed in the complex plane C basedontheescape-timealgo- Quadratic map basins is a method used to generate rithm [19, 12]. The escape-time visualization method the images of Julia and Mandelbrot sets and has been was extended from Julia and Mandelbrot sets to more applied to visualize the attractors of iterated function higher order [21, 22], and several methods for classi- systems (IFS). Three dimensional extension of such fying divergent points in the complement of a fractal mapping with animation is an aspiring goal and a chal- shapes to generate fractal patterns in two dimension lenging task. In this paper an extension algorithm of were described in [18, 10, 8, 24]. These methods plot- quadratic map basin for classifying points in the com- ting simple discrete level sets by counting the number plement of a 3-D linear fractal shapes are discussed. of function applications required to transform a point Such extension produces fractal surfaces that exhibit outside a large circle. Although these methods gener- self-similarity and suggest smooth evolution under an- ate an aesthetically appealing 2-D fractal patterns, an imation. The proposed technique has fast computa- extension algorithm for visualizing points in the com- tions and simple representation whereby a computer plement of the 3-D fractal patterns still aspiring goals, can automatically select parameters and generate a challenging tasks and harder to be described. In this large collection of aesthetically appealing 3-D fractal paper we propose an extension algorithm for rendering patterns. The simple representation and the speed of the IFS escape-time behavior in three dimension with- computation of our algorithm make 3-D fractal pat- out inverting the IFS transformations which is com- terns reliable for interactive machine. putationally expensive, therefore reducing the time of complexity. The proposed algorithm has two basic 1 Introduction steps. The first is based on the way of approximat- ing the fractal attractor and the second is for creating Several methods for modeling and realistic visualiza- numerical data that characterize the space outside the tion of complex forms found in nature have been de- attractor. The advantages of the proposed method are veloped in the history of computer graphics. These that it is neither explicitly define regions, nor follows methods are more or less related to Mandelbrot’s frac- all point iteration possibilities. Moreover, it is sim- tal geometry of nature [15]. Fractal methods are quite ple to implement and its speed of computation allows popular in the modeling of natural phenomena in com- visible surface rendering to be performed simply and puter graphics ranging from random fractal models of efficiently. In addition makes rendering 3-D fractal terrain [15], to deterministic botanical models such as patterns reliable for interactive machines. L-systems [17], iterated function systems (IFS), recur- rent iterated function systems (RIFS) and language The paper is organized as follows. Section 2, sur- iterated function systems [1, 5, 3, 4, 16]. On the other veys the key principles and basic notions of escape- hand, methods for classifying points in the comple- time visualization method, iterated function systems 61 芸術科学会論文誌 Vol.2 No.1 pp.61-70 IFS, and recurrent iterated function systems RIFS. The dynamics of the escape-time classifications of p p quadratic fractals, IFS and RIFS are explained in sec- 11 12 tion 3. In section 4, the proposed algorithm for ex- 2 1 tending the IFS escape-time from 2-D to 3-D linear p 22 fractals is explained. Fractal rendering technique is p introduced in section 5. Some experimental results 31 with discussion are given in section 6. Finally, con- clusion and direction for future work are discussed in P section 7. 23 p 3 33 2 Background and Basic No- Figure 1. RIFS Directed Graph Example. tions The common ways to generate fractals is through it- erated function systems IFS, recurrent iterated func- tion system RIFS and quadratic map basins based on Quadratic map basins based on escape-time algo- escape-time algorithm. The mathematical theory of rithm was originally developed as a method for visu- both IFS and RIFS are based on the contraction map- alizing the dynamics of complex quadratic fractals and ping theorem which is described in details in [2, 6, 13]. has been extended to linear fractals modeled by IFS Mathematically, the IFS is usually defined as a pair and RIFS. An overview of the 2-D rendering methods {X; Tn,n=1, 2, ···,N},whereX is a complete met- classifications of the dynamics for the quadratic itera- ric space and each Tn are affine contractions, that is tion maps and its corresponding IFS escape-time are shown in figure 2(a) − (b). Ti(x)=Cix + Bi (1) where Ci is a square matrix with n rows and Bi is a vector with n elements. By a theorem of Hutchinson[13], there exist for each IFS a single com- pact non empty set Aˆ, called its attractor which is the Quadratic Iteration union of images of itself under the IFS maps, that is, z z * z + c N c belong to the Mandelbrot set Aˆ = i=1 Ti(Aˆ). On the other hand, recurrent model- ing is the process of partitioning an object into compo- nents and representing each component using smaller copies of itself and/or other components [2, 3]. The Filled in Complement of determination of which components are used is con- Julia set N Filled-in Julia set trolled by a graph. It consists of an IFS W = {Ti}i=1 as well as a control graph G consisting of N vertices Geometry Geometry and corresponding to the IFS maps and directed edges de- Only Dynamics noted by the ordered pair <i,j>. The notation <i,j>∈ G indicates that graph G contains a di- Integer Escape Distance Potential rected edge from vertex i to vertex j, and implies that Estimator Time Function transformation Tj may be applied directly after trans- Continous formation Ti. Figure 1, shows an example of a directed Escape Time graph with N =3andnumberspij ≥ 0 which repre- sents the probabilities of transfer among the vertices. (a) 62 芸術科学会論文誌 Vol.2 No.1 pp.61-70 { || t || ≥ } DT(z)=min t : Fc (z) R (2) 1+DT(Fc(z)) if ||z|| <R = 0 otherwise 3.2 Discrete Escape-time for IFS and RIFS The concept of the escape-time function can be ex- tended for IFS in which all transformations are invert- ible. In order to define the escape-time function for iterated function systems, let us review the analogous concept for Julia sets. The Julia set for the quadratic mapping Equation(1), can be viewed as the attractor Aˆ defined by the nonlinear IFS: √ √ T1(z)= z − c, T2(z)=− z − c (3) Equation (8), illustrate that inversion of an IFS is re- (b) quired to compute its escape-time classification. Figure 2. 2-D rendering methods classifications in Definition 3.2 (IFS Discrete Escape-time) (a) Quadratic maps and (b)IFSmaps. Given an IFS W = {T1,T2, ···,TN } with attractor Aˆ,letDR be a disk of radius R centered at the origin 3 Escape-time Classifications sufficiently large such that, W (DR) ⊂ DR, (4) Escape-time method has two forms: the first one is called discrete form and the other is called continu- then the IFS discrete escape-time is given by the 2 ous form[10, 24]. The discrete form of the escape- function DT : R → R which is defined recurrently time classification counts the number of iterations for as: a point to iterate outside the infinity circle. The con- 1+MAX if x ∈ DR DT(x)= tinuous form smoothly interpolates the area between 0 otherwise the discrete escape-time boundaries based on the loca- Where, tion of the point that escapes the infinity circle. This −1 MAX =maxi=1,2,..,N DT(T (x),DR) section briefly explains the different escape-time forms i for representing the 2-D fractal patterns using the IFS The discrete escape-time function is the maximum −1 ∈ −1 and their corresponding methods of the quadratic it- number of inverse transformations Ti W nec- eration function. Then, our extension algorithm of the essary to iterate x to a point outside DR. IFS escape-time computation from 2-D to 3-D fractal Definition 3.3 (RIFS Discrete Escape-time) patterns will be discussed in the next section. Given an RIFS W, G with attractor Aˆ,LetR>0 sat- isfy condition (10). Then the discrete escape-time is 3.1 Discrete Escape-time given by DT(x)=maxi=0,..,N DTi(x) where the func- 2 → Definition 3.1 (Quadratic Discrete Escape-time) tions DTi : R R are defined as follows 2 Given the function Fc(z)=z + c,adiskDR centered D if x ∈ DR DTi(x)= at the origin and of radius R sufficient such that the 0otherwise filled-in Julia set Kc ⊂ DR, then the discrete escape- Where, time DT : C → Z is given by −1 D =1+max<j,i>∈G DTj(Ti (x)) 63 芸術科学会論文誌 Vol.2 No.1 pp.61-70 3.3 Continuous Escape-time Function Where, −1 A continuous escape-time function requires a means to MAXX = maxi=1,···,N CT(Ti (x)) −1 interpolate smoothly between the boundaries of the Cond1=ifx ∈ DR,Ti (x) ∈ DR(∃i) −1 discrete escape-time level sets.
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