INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 6, JUNE 2014 ISSN 2277-8616 1 IJSTR©2014 www.ijstr.org Geometric Modelling Of Complex Objects Using Iterated Function System Ankit Garg, Ashish Negi, Akshat Agrawal, Bhupendra Latwal Abstract: In the field of computer graphics construction of complex objects is difficult process. Objects in nature are complex such as tree, plants, mountains and clouds. Traditional geometry is not adequate to describe these objects. Researchers are investigating different techniques to model such types of complex objects. Algorithms presented in this paper are deterministic algorithm and random iteration algorithm which comes under iterated function system. The fundamental property of any IFS is that image generated by it is also a fractal which is called attractor. Any set of affine transformation and associated set of probabilities determines an Iterated function system (IFS). This paper presents the role of iterated function system in geometric modeling of 2D and 3D fractal objects. Key words: CMT, IFS ———————————————————— 1 INTRODUCTION Binoit Mandelbrot invented the word fractal. Latin adjective - fractus verb – frangere means ‗to break‗to create irregular fragments [12]. Fractals generated by dynamical systems are called Algebraic fractals, Ex: Mandelbrot & Julia set. In the field of computer graphics researchers are always try to find out new ways to construct geometric model of objects. Computer graphics provides various ways to construct man- made objects e.g. building, plants etc. Well developed mathematical polynomials are available to model such type of objects. These well defined mathematical polynomials can generate smooth geometry. As fractals are non smooth and highly irregular traditional polynomial methods requires more specification information. The concept of fractal was described by IBM mathematician Benoit Mandelbrot. He found that traditional geometry was inadequate to describe the structure of natural objects which are complex such as mountain, cloud, coastlines and tree. The non-Euclidean geometry or fractal geometry deals with irregular and fragmented patterns. Fractals are complex objects which has property of self similarity- A small section of fractal object is similar to whole object, hence fractal are the repetition of the same structural form. There are two main groups of fractals: linear and nonlinear [2]. The latter are typified by the popular Mandelbrot set and Julia sets, which are fractals of the complex plane [2]. Fractal may have condensation sets. Fractal with condensation set are not quite self similar. In general to create any fractal three things are required: a set of transformations (IFS), a base from which iteration starts, and a condensation set (possibly the empty set). IFS provide a very compact representation, efficient computation, and a very small amount of user specifications [1]. An IFS is a set of contraction mappings acting on a space X. The set of contraction mapping has a set of probabilities. Construction of fractal image with IFS starts with original image and some successive transformation are applied over the image. The result of IFS is called attractor which is a fix point. This fix point after contraction mapping is nothing but an image. An IFS maps the corresponding fractal onto itself as a collection of smaller self similar copies. Seemingly a photocopy machine has been designed by mean of which coefficients of map are computed [3]. The concept of photo copy machine has also been extended to the case of gray scale images [3]. IFS can be used in fractal image compression. Barsnley has derived a special form of the Contractive Mapping Transform (CMT) applied to IFS‘s called the College Theorem. College theorem suggests that the hausdorff distance between two images should be as minimum as possible, means the attractor of IFS should be close approximation of original image. The distance between original image and attractor is known as college error and it should be as small as possible. Due to self similar property of fractal IFS is applied on whole image in fractal image compression to find out the redundancy in image. There may be some images in which only part of image is similar to the other part of image, not whole image. In this case instead of applying affine transformation on whole image, contractive affine transformations are applied on parts of image, and the union of affine transformation is the final image. This can be achieved by applying PIFS over image. Some object or classical geometry can be generated as attractor of IFS. Objects of classical geometry generated through IFS are somewhat unsatisfactory only close approximation is possible. 2 MATHEMATICAL FOUNDATION OF IFS Definition A (hyperbolic) iterated function system consists of a complete space (X, d) together with a finite set of contraction mappings w n : X  X, with respective contractivity factors s n , for n = 1, 2, ..., N. The notation for this IFS is {X ; w n , n = 1, 2, , N} and its contractivity factor is s = max{s n : n = 1, 2, , N}. Definition An iterated function system with probabilities [8] consists of IFS {X; w 1 , w 2, , w N } _________________________  Ankit Garg, pursuing PHD from Uttarakhand Technical University, Dehradun. He is working as assistant professor in Amity University, Haryana.  Ashish Negi is Associate professor in GBP Engineering College, Pauri Garhwal.  Akshat Agrawal, Pursuing PHD from Amity University, Haryana. He is working as assistant professor in Amity University, Haryana.)  Bhupendra latwal, pursuing PHD, and working as assistant professor