International Journal of Computer Applications (0975 – 8887) Volume 43– No.6, April 2012 A Relative Superior Julia Set and Relative Superior Tricorn and Multicorns of Fractals Priti Dimri Shashank Lingwal Ashish Negi Head Dept. of Computer Science and Associate Professor Dept. of Computer Science and Engineering Dept. of Computer Science and Engineering G.B.Pant Engineering College Engineering G.B.Pant Engineering College Ghurdauri, Pauri G.B.Pant Engineering College Ghurdauri, Pauri Ghurdauri, Pauri ABSTRACT 2. ELABORATION OF CONCEPT In this paper we investigate the new Julia set and a new INVOLVED Tricorn and Multicorns of fractals. The beautiful and useful fractal images are generated using Ishikawa iteration to study 2.1 Mandelbrot Set many of their properties. The paper mainly emphasizes on Definition 1. [17] The Mandelbrot set M for the quadratic 2 is defined as the collection of all reviewing the detailed study and generation of Relative QzC ( ) = z + c Superior Tricorn and Multicorns along with Relative Superior cC for which the orbit of point 0 is bounded, that is, Julia Set. n M{ c C :{ Qc (0)}; n 0,1,2,3... is bounded} Keywords Complex dynamics, relative superior Julia set, Ishikawa An equivalent formulation is n iteration, Relative superior Tricorn, Relative superior M{ c C :{ Qc (0) does not tends to as n }} Multicorns. We choose the initial point 0, as 0 is the only critical point of 1. INTRODUCTION Qc. The term „Fractal‟ was coined by Benoit B Mandelbrot, in 1975 to denote his generalisation of complex shapes. Fractal 2.2 Julia Set Definition 2. [17] The set of points K whose orbits are is derived from Latin word „fractus‟ which describes the bounded under the iteration function of Qc(z) is called the appearance of broken stone : irregular and fragmented. The Julia set. We choose the initial point 0, as 0 is the only critical object Mandelbrot set, given by Mandelbrot 1979 and its point of Qc(z) . relative object Julia set due to their beauty and complexity of their nature have become elite area of research nowadays. The 2.3 Tricorn and Multicorns fractal graphics are generally obtained by “colouring” the Please The study of connectedness locus for antiholomorphic 2 escape speed of the seed points within the certain regions of polynomials zc defined as Tricorn, coined by Milnor complex plane that give rise to the unbounded orbits. [14], plays intermediate role between quadratic and cubic polynomials. Many authors have presented the papers on several “orbit Definition 3. [5] The Multicorns Ac , for the quadratic n is defined as the collection of all traps” rendering methods to create the artistic fractal images. Ac ()' z z c cC An orbit trap is a bounded area in complex plane into which for which the orbit of the point 0 is bounded, that is, an orbiting point may fall. Motivated by this idea of “orbit A{ c C : A (0) is bounded}. An traps”, Chauhan [4] introduces the different type of orbit traps c c n0,1,2,3... for Ishikawa iteration procedure. They have considered Julia equivalent formulation is 2 A{ c C : A (0) not tends to as n } . sets of z az c , as orbit traps to explore their cc nn1 The Tricorn are special Multicorns when n=2. As quoted by relevant fractal images, since, it is connected and bounded for Devany [7][8], iteration of function A z'2 c , using the a and c. c Escape Time Algorithm, results in many strange and Recently Shizuo [15] has quoted the Multicorns as the surprising structures. Devany [7][8] has named it Tricorns and generalized Tricorns or the Tricorns of higher order and observed that f(z‟), the conjugate function of f(z), is presented various properties of them. Tricorn are being used antipolynomial. Further, its second iterates is a polynomial of for commercial purpose, e.g. Tricorn Mugs and Tricorn T degree 4. The function zc'2 is conjugate of zd'2 , shirts. Multicorns are symmetrical objects. Their symmetry where 2i /3 , which shows that the Tricorn is has been studied by Lau and Schleicher [11]. Negi [16] have de introduced a new class of Relative Superior Multicorns using symmetric under rotations through angle 2 / 3 . The critical Ishikawa iterates and also studied their corresponding point for A is 0, since has only one preimage c cA c (0) Relative Superior Julia sets. whereas any other wC , has two preimages. 29 International Journal of Computer Applications (0975 – 8887) Volume 43– No.6, April 2012 2.4 Picard’s Orbit function z zn c,2 n and z() zn c 1 for Definition 4. Let X be a nonempty set and f: X X . generating fractal images with respect to Ishikawa iterates, For any point , the Picard‟s orbit is defined as the where z and c are the complex quantities and n is a real xX0 number. Each of these fractal images is constructed as two- set of iterates of a point x0 , that is; dimensional array of pixel. Each pixel is represented by a pair . of (x,y) coordinates. The complex quantities z and c can be O( f , x ) { xn ; x n f ( x n1 ), n 1,2,3...} represented as: In functional dynamics, we have existence of two different z zxy iz types of points. Points that leave the interval after a finite c cxy ic number are in stable set of infinity. Points that never leave the interval after any number of iterations have bounded orbits. where i ( 1) and zx , cx are the real parts and zy , cy are So, an orbit is bounded if there exists a positive real number, the imaginary parts of z and c respectively. The pixel such that the modulus of every point in the orbit is less than coordinates (x,y) may be associated with (cx,cy) or (zx,zy) . this number. The collection of points that are bounded, i.e. n there exists M, such that Q() z M , for all n, is called Based on this concept, the fractal images can be classified as as a prisoner set while the collection of points that are in the follows: stable set of infinity is called the escape set. Hence the boundary of a prisoner set is simultaneously the boundary of (a) z-Plane fractals, wherein (x,y) is a function of escape set and that is Julia set for Q. (zx,zy). 2.5 Ishikawa Iteration (b) c-Plane fractals, wherein (x,y) is a function of Definition 5. Ishikawa Iterates [17]: Let X be a subset of real (cx,cy). or complex number and f: X X for all xX , we 0 In the literature, the fractals for n=2 in z plane are termed as have the sequence {xn} and {yn} in X in the following the Mandelbrot set while the fractals for n=2 in c plane are manner: known as Julia sets [4, 5] yn S' n f ( x n ) (1 S ' n ) x n xn1 S n f( y n ) (1 S n ) x n 4. ESCAPE CRITERIA FOR RELATIVE where and & are both 0S 'n 1, 01Sn S 'n Sn SUPERIOR MANDELBROT AND JULIA convergent to non-zero number. SET 2.6 Relative Superior Mandelbrot Set 4.1 Escape Criterion for Quadratics Now we define Mandelbrot set for the function with respect to [13] Suppose that z max{ c ,2 / s ,2 / s '} , then Ishikawa iterates. We call them as Relative Superior zz(1 )n and z as n . So, Mandelbrot sets. n Definition 6. [17] Relative Superior Mandelbrot set RSM for zc and zs 2/ as well as zs 2 / ' shows the the function of the form n , where n = Qc () z z c escape criteria for quadratics. 1,2,3,… is defined as the collection of cC for which the orbit of 0 is bounded i.e. 4.2 Escape Criterion for Cubics k is bounded. [13] Suppose zmax{ b ,( a 2/) s1/2 ,( a 2/ s ') 1/2 } RSM{ c C : Qc (0): k 0,1,2,3...} then as n . This gives the escape criterion In functional dynamics, we have existence of two different zn types of points. Points that leave the interval after a finite for cubic polynomials. number are in stable set of infinity. Points that never leave the interval after any number of iterations have bounded orbits. 4.3 General Escape Criterion So, an orbit is bounded if there exist a positive real number. [13] Consider z max{ c ,(2/ s )1/2 ,(2/ s ') 1/2 } then as n is the escape criterion. 2.7 Relative Superior Julia Set zn Definition 7. [17] The set of points RSK whose orbits are bounded under relative superior iteration of function Q(z) is Note that the initial value z0 should be infinity, since infinity called Relative Superior Julia sets. Relative Superior Julia set is the critical point of z() zn c 1 . However, instead of Q is boundary of Julia set RSK. of starting with z0 = infinity, it is simpler to start with z1 = c, which yields the same result. A critical point of 3. GENERATING PROCESS z F() z c is a point where Fz'( ) 0 . The basic principle of generating fractals employs the iterative formula: z f() z where z0 = the initial valueof z, nn1 th and zi = the value of complex quantity z at the i iteration [17]. For example, the Mandelbrot‟s self-squared function for generating fractal is: f(z) = z2+c , where z and c are both complex quantities. We propose the use of transformation 30 International Journal of Computer Applications (0975 – 8887) Volume 43– No.6, April 2012 5.