International Journal of Computer Applications (0975 – 8887) Volume 48– No.1, June 2012 1 Game-enabling the 3D-Mandelbulb Fractal by adding Velocity-induced Support Vectors Bulusu Rama Assistant Professor of CSE SLC Institute of Engg. And Technology Piglipur(V),Hayatnagar(M), Hyderabad – 501512(A.P.) India Jibitesh Mishra Head of Dept. of Information Technology College of Engg. & Tech. BPUT,Ghatikia. Bhubaneswar- 751030(Odisha) India ABSTRACT Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self- similarity, scale invariance, and dimensionality. Of the many different types of fractals that have come into limelight since their origin, the family of Mandelbrot Set fractals has eluded both mathematicians and computer scientists alike. And the „true‟ 3D realization of the Mandelbrot set has been a challenging centre piece of research with its limits extending only to that of the sky. An earlier paper co-authored by us in 2011 explained a method of realizing a „true‟ 3D simulation of the Mandelbrot set and the rendering of the same onto 3- dimensional space. This paper takes a step further in using this variant of the Mandelbulb as input and outlines a method of the game-enabling of the same Mandelbulb by using direction-oriented vectors that are analogous in function to that of Support Vectors in the Support Vector Graphics (SVG) domain. A real-world application of the same can translate to examples of understanding an entire coast-line set to motion in space by adding 3D-animation enabled elevation to the corresponding fractal image. General Terms Algorithms, Fractal Geometry Keywords Fractals, Mandelbrot Set, Mandelbulb, Three Dimensional Velocity, Rendering, Support Vectors 1. INTRODUCTION A fractal is a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced size copy of the whole or in other words, is self-similar when compared with respect to the original shape. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin word “fractus” meaning “broken” or “fractional”. The primary characteristic properties of fractals are self-similarity, scale invariance and general irregularity in shape due to which they tend to have a significant detail even after magnification-the more the magnification the more the detail. In most cases, a fractal can be generated by a repeating pattern constructed by a recursive or iterative process. Natural fractals possess statistical self- similarity whereas regular fractals such as Sierpinski Gasket, Cantor set or Koch curve contain exact self-similarity. A 3D rendering of the Mandelbrot set is popularly termed as the Mandelbulb. This paper takes a step further in game-enabling the Mandelbulb by adding variant input based velocity induction that sets the Mandelbulb in a dynamically controllable motion. It does this by using a dynamically set speed 2-tuple that acts as a Support Vector lever to kick-off the game-enablement. The source image of the 3D Mandelbulb is generated based on rotation of the Mandelbrot set away from the azimuthal or the z-axis, followed by an IFS- based repeated execution and the rendering of the same in 3D. The game-enabling program was implemented using the Python Visual Development Kit and the Python Game API and rendered using the VIDLE GUI based on the algorithms presented in the following sections. The displayed output of the same based on a typical set of inputs is presented. The experimental results are shown in section 4 and concluding remarks in section 5. 2. ABOUT MANDELBROT SET AND 3D MANDELBULB A brief description of Mandelbrot Set and the 3D Mandelbulb along with their properties followed by a description of game- enabling the Mandelbulb based on adding variant support vectors game-enabling on iterative function systems is presented in the paragraphs that follow. The Mandelbrot Set was invented by the French mathematician Benoit Mandelbrot in 1979, when he was working on the simple equation z = z 2 + c. In this equation, z and c denote complex numbers. In other words, the Mandelbrot set is the set of all such complex numbers c, that iterating z = z 2 +c does not diverge. Hence, it is a connected set of points, which is bounded. An Iterated Function System (IFS) based on the maximum number of iterations and an initially defined region denoting the lower and upper limits of the bounded space, is iterated as many times as the maximum number of iterations. The resulting set of points can be span an indeterminable amount of space that is a function of the number of iterations involved. Then the set of all points for which the line spacing from the origin to that point in each of the co-ordinate directions is zero constitutes the Mandelbrot Set. Applying the set of affine transformations iteratively on each point in the starting region set, the resulting fractal is a self-similar shaped image that resembles an approximation to the original image. This effect is best visualized when rendered in 3D. To generate the Mandelbrot set graphically, the computer screen becomes the complex plane. Each point on the plane is tested into the equation z = z2+c. If the iterated z stayed within a given boundary forever, i.e., is convergent, the point is inside the set and the point is plotted black. If the iteration went of control, i.e., is divergent, the point was plotted in a colour with respect to how quickly it escaped. When testing a point in a plane to see if it is part of the set, the initial value of z is always zero. This is so because zero is the critical point of the equation used to generate the set. The true canonical form