The ReadBajraktarevic Functional Equation and Selfsimilarity
Wieslaw Kotarski (kotarski@ux2.math.us.edu.pl)
& Agnieszka Lisowska (alisow@ux2.math.us.edu.pl)
Institute of Computer Science
Silesian University
Bedzinska 3941200 Sosnowiec, Poland
(Abstract)
The aim of this worksheet is to show to Maple users, following [8], [10], that fractal curves may be obtained as solutions to some functional equations with the socalled ReadBajraktarevic operator. Additionally, it will be demonstrated the power of Maple to handle with the problem that needs to its solving a symbolic approach.
1. Introduction
Fractals have been discovered by Mandelbrott in 1970s [1], [9]. They are objects having very complicated geometrical shapes impossible to describe with the help of classical Euclidean geometry. Fractals are selfsimilar objects. It means that parts of them are similar to the whole objects. One can generate fractals iteratively using very simple set of rules defined by the socalled IFS (Iterated Function System). More information about fractals can be found eg. in classical books [1], [9]. Below, following [8], [10], we present an approach for rendering fractal curves that are selfsimilar solutions to some functional equation. That functional equation has been investigated in 1950s independently by two mathematicians and, to honour them, is named as the ReadBajraktarevic equation.
The ReadBajraktarevic equation has the following form:
where are functions and is a closed interval in . We are looking for a function satisfying (1) with given functions and .
Following [8], [10] we recall the theorem:
Theorem
Let I be a closed interval in R. Assume that Further, assume that there exists a constant r, 0 < r < 1 such that for every x in I, is satisfied:
Then the operator Φ:
The operator Φ is the socalled Read Bajraktarevic operator. From the wellknown Hahn Banach Fixed Point Theorem it follows that there exists a unique fixed point (x)  the solution to the equation (1), that may be obtained as a limit of the following iterations:
starting from any function (x).
The sequence of iterations (2) is convergent to (x) in the following sense:
In the next section examples of ReadBajraktarevic equations will be presented together with their solutions. Observe that in considered examples all assumptions about functions v and b are satisfied.
2. Maple Program
Procedure for approximately solving the ReadBajraktarevic equation (n is a number of iterations)
Procedure for plotting approximate solution to the ReadBajraktarevic equation (n is a number of iterations)
3. Examples
Let us consider the following examples:
Example 3.1
Given functions:

(3.1.1) 

(3.1.2) 
Initial function

(3.1.3) 
Iterations
Analytic Form
Graphical Form
Animation
Example 3.2
Take functions b and v the same as in Example 3.1. But the starting function is now different.
Initial function

(3.2.1) 
Iterations
Analytic Form

(3.2.1.1) 
Graphical Form
Animation
Remark
We can see that both cases (Example 3.1 and 3.2) the limit function is the same independently on the initial function.
Example 3.3
Given functions

(3.3.1) 

(3.3.2) 

(3.3.3) 
Initial function

(3.3.4) 
Iterations
Analytic Form
Graphical Form
Animation
Example 3.4
Given function

(3.4.1) 

(3.4.2) 

(3.4.3) 
Iterations
Analytic Form
Graphical Form
Animation
4. Remarks
In this worksheet we presented approximations of solutions to the ReadBajraktarevic equation. It is easily seen that solutions to the ReadBajraktarevic equation are fractal like functions. Applying more general theorem to theorem 1 (see: [8], [10]) it is possible to prove that quadratic curve is the solution to some ReadBajraktarevic equation with functions:
Directly one can check that the function satisfies the following identity
Observe, that there also exists the second solution That is because the function does not map the interval [0,1] onto [0,1]. Summing up, any quadratic curve is selfsimilar. That means that quadratic curve can be generated in a fractal way. The same result can be obtained applying subdivision arguments presented by Goldman in [2], who gave the form of IFS for rendering of any B�zier curve. The authors applied Goldman's result and its generalizations to any subdivision to generate fractally many shapes presented in their earlier Maple worksheets [3][6]. It should be mentioned that the first author described a method of fractal modeling of shapes in [7].
5. References
[1] Barnsley M., Fractals Everywhere, Academic Press, New York, 1988.
[2] Goldman R., The Fractal Nature of B�zier Curves, Proceedings of the Geometric Modeling and Processing, April 1315, 2004, Beijing, China, 311.
[3] Kotarski W., Lisowska A., On Fractal Modeling of Contours, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651.
[4] Kotarski W., Lisowska A., Probabilistic Approach to Fractal Modeling of Shapes, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657 .
[5] Kotarski W., Lisowska A., Fractal Teapot from Utah, Maplesoft, 2006, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1956.
[6] Kotarski W., Lisowska A., Fractal Rendering of 3D Patches, Maplesoft, 2006, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1955.
[7] Kotarski W., Fractal modeling of Shapes, EXIT, Warsaw 2008, (in Polish).[8] McClure M., The ReadBajraktarevic Operator, Mathematica in Education and Research, Vol. 11, No. 3, 356362, 2006.
[9] Mandelbrot B.,The Fractal Geometry of Nature, Freeman and Company, San Francisco, 1983.
[10] Massopust P. R., Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, Inc., San Diego, CA, 1994.
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