The Research On Distance Ratio Fractal And Complex Iterated Function System

Since the introduction of fractal by Mandelbrot, fractals have experiencedconsiderable success in quantifying the complex structure exhibited by many naturalpatterns and have captured the imaginations of both scientists and artists. Manyresearchers perform research on the generation method of fractal and draw manybeautiful images. The research of fractal has experienced a slow progressing phase. Itis needed to conclude current rendering methods and introduce new algorithm.The research of fractal needs the application of computer graphics method. One ofthe reasons why fractal concept can be accepted by people rapidly is the help of thecomputer graphics method. By giving a deep research on the application of computergraphics in fractal rendering, we can increase the comprehension of fractal and extendfractal application region. Meanwhile, the nature of complex structure of fractal needsthe fractal rendering method to be improved.The basic idea of current fractal rendering algorithm is to treat fractal as point sets,select different color for pixel during iteration process and render it on screen, such aschaos game method for rendering IFS and escape time algorithm for renderingcomplex mapping. These methods have following shortages:1. Current render method of complex mapping can only render the borderbetween the stable region and escape region, and cannot display the detail of stableregion and escape region and it is difficult to understand fractal image structure.2.The iteration fractal image of some complex mapping is difficult to rendering.It is needed to modify current method or develop new method to research the iterativeproperties of the mapping.3. There is no related work about making fractal image with some painting style,so it is possible to extend the application region of fractal image.From above discussion we can see there are many problems in fractal technology.This dissertation tries to innovate and break through on these problems, introduce newfractal rendering method to rendering process and improve fractal image rendering andutilize it to a new level. The innovations of this dissertation are listed as follows:1. Propose the distance ratio iteration method to construct fractal image.The distance ratio iteration method takes two points to perform iteration andutilizes its convergence speed to render fractal image of complex mapping. Thegeneralized Mandelbrot and Julia sets based on distance ratio have rich detail in stableregion. The distance ratio iteration method can render fractal image of some complexmapping which cannot be rendered by escape time algorithm. This dissertation sturdieson the iteration properties of distance ratio and proves its convergence theorem. Basedon these theorems, we introduce three methods for rendering distance ratio fractal,including convergence time method, inverse iteration layer method and mix method.These algorithms can render many fractal images of complex mapping, such aspolynomial mapping, exponent mapping, logarithm mapping and 3x+1 mapping.The escape time algorithm treat all convergence points as a same point, so itcannot render structure of stable region. However, distance ratio method useconvergence time to classify point set and can get complex structure.The fractal image rendered by distance ratio method is the two dimensionprojection of complete four dimension point sets because distance ratio takes twopoints to perform iteration. Distance ratio iteration method provide a general methodfor rendering 2d-4d fractal object.2.Research on the generalized Julia set based on distance ratio for f(z)=zα+c.Firstly we research the visual structure of DRJ for nonparameter mapping f(z)=z2and analysis its buds distribute rule, give formula of each level width. Secondly, weresearch the evolution rule of DRJ for mapping f(z)=zα(1<α<2) and rendering variousexponent mapping 's fractal image.We study the generalized Julia set based on distance ratio for f(z)=z2+c anddiscuss the image properties for mapping with attractive fixed point, two period orbitand p period orbit. We state the symmetry theorem of DRJ and prove the border ofDRJ is the mapping's Julia set. So the distance ratio iteration method can be a methodfor rendering the inner structure of Julia set. We state the reason of two phase structureexist in DRJ when mapping has two period orbit and give a new algorithm to renderDRJ when the mapping has p period orbit.This dissertation discusses the chooson of z2 for DRJ. Firstly, we research thesituation when z2 is a fixed complex number and discuss image structure for differentlocation of z2. Then we research the situation which z2=g(z1) and give the notation oftwo mapping compose distance ratio Julia set. We discuss some complex mappingcomposed distance ratio Julia set and state the extend ability of distance ratio iterationmethod.We research the generalized Julia set based on distance ratio for f(z)=zα+c andstate the shortage of escape time algorithm in this situation and the advantage ofdistance ratio iteration method. The distance ratio Julia set is a fine Julia set and it canrender the detail that not in Julia set.3.Research on generalized Mandelbrot set based on distance ratio for f(z)=zα+c.Taking complex mapping f(z)=zα+c as an example, we generates images of thegeneralized Mandelbrot set based on distance ratio and states conjectures on theirvisual characteristics. Comparing with escape time algorithm, it consists of abundantdetails in the convergent region and more complex structure can be generated.Therefore, distance ratio iteration method can be used as a new method for renderinginner structure of M-set when α>1.DRM for α<0 is composed of two parts: outer curve polygon and innerconstellation. The curve polygon has |α|+1 edges and is covered by petal structures inouter layer;Inner constellation is composed of a central planet and surroundingsatellites. The central planet has |α|+1 protuberances and tangent with curve edge ofoutside polygon in the middle points. The DRM is the complement of M-set incomplex plane.When 0<α<1, M-set generated by escape time algorithm is a kind of simplegeometry image, because the attractive region of the mapping changes.4.Research on the complex iterated function system f(z) =z2+ciThe dissertation researches the iteration properties of complex mapping setf(z)=z2+c by taking them as an complex iterated function system. Firstly we discuss thecondition of the complex mapping family to become an NIFS. Then the dissertationresearches some common problems in NIFS, such as the relationship between attractorand the fixed point of mapping, the selection of mapping parameter, and the bound ofattractor.Based on above discussion, we can make such conclusion: the complex mappingset is a iteration function system, only if |ci|≤0.25 and the initial iteration point z locatesin the intersection of the mappings of the Julia set or |z|<0.5. The attractor generated bythe complex IFS satisfying this condition converge to the cycle whose center is theorigin and radius is(1 ? 1 ? 4 R)/ 2. The equations for computing fixed points are listedin the former part.5.Propose a model to render dry brush effect of Chinese brush work.Some article states that there is fractal character in Chinese paint and calligraphy.So the fractal image can be used to Chinese painting and calligraphy. We propose acalligraphy model based on iteration function system construct by complex mappingfamily f(z)=z2+ci. The models construct iterated function system based on user's brushspeed and contrail renders the attractor by chaos game algorithm as final dry brushstroke. It is easy to combine this model to current brush model.In conclusion, this dissertation has academic significance and value of application,and it enriches the research of fractal geometry. It also provides constructive methodand techniques for research of fractal geometry.