The Iterated Function System Based On Polynomial Transformation

Fractal, as a complex geometric figure, involves mathematics, physics, material science, biology and medicine and computer graphics, etc. So the research on fractal has important theoretical significance and extensive applied value. In recent years, IFS has become a very important method of natural scenes with research in the field of fractal in-depth, then many scholars are concerned about fractal images as a part of computer graphics now, while the traditional IFS is confined to mapping approach, then the construction of natural scenes is too monotone and the IFS code is too cumbersome when the complicated set is described. Besides, the produced fractal images exists shortcomings in the term of diversity and vitality. The IFS based on polynomial transformation in this paper is introduced to solve these problems.Firstly, the basic theories of fractal and some typical algorithms of generating fractal graphics is reviewed in this paper, such as the L-system, escape-time algorithms, the DLA and iterated function system. On the basis of IFS constructed by mapping transformation, Collage theorem and attractor theorem are given, then the random NIFS and the aggregate NIFS are described.The IFS is based on the control of points coordinate. Several forms of the IFS are expressed by means of geometric approach after analyzing the method of IFS, such as secondary transformation with single parameter, double parameters, three parameters, the transformation of separated cubic spline, the non-uniform and weak transformation of cubic spline and the transformation of cubic spline based on the minimum acceleration, etc. In addition, the IFS is given by interpolation method and is discussed in the respect of uniqueness and continuity of attractors.A kind of method generating fractal images with polynomial transformation is put forward for the realization of the IFS, which a triangle is mapped into curve triangle with polynomial transformation and the algorithm can be extended to the IFS with hyperbolic triangle, curve triangle, curve quadrilateral and even curve polygon. The experiment shows that the method can not only produce more elaboratedly and beautiful images, and provide a more powerful description for the production of fractal graphics and models, but also has a momentous significance meaning for designing and creating IFS for interactive and practical fractal images.