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3x+1推广函数引起分形的性质与应用研究/自上世纪七十年代B.B.Mandelbrot明确地提出分形思想和概念以来，分形的理论研究和实际应用取得了巨大进展。分形的研究离不开计算机，如果不是计算机图形图像处理功能的增强，不能想象如何直观的表示出分形的精细结构。反之，分形的理论与方法又极大的丰富了计算机图形学的内容。分形研究中不断有新类型的美妙图像被发现，3x+1猜想引起的分形就是一类，3x+1推广函数引起的分形性质与艺术分形图像的绘制及应用是本文的主要工作。下面我们从三个方面具体阐述本文的主要内容。 1．3x+1推广函数引出分形的一些性质整数3n+1猜想是一个尚未解决的著名数学难题。猜想的内容为：对任意正整数，为奇数时乘3加1，为偶数时除以2，如此继续，则所得序列必到达整数1后进入循环(1,2)。文中采用的3x+1推广函数为T(x)和T(z)： #(x是实数) #(z是复数) 为基础展开研究。较详细的讨论了T(x)在实轴上周期点的一些性质、T(z)在复平面上的不动点，得出T(z)在复平面上的不动点存在的结论。绘制了3x+1推广函数的分形图像。用数学推理与函数图像相结合的方法证明了T(x)在区间(-1,0)上无n周期点；在区间(0,1)上，除不动点0.31582外，无其他周期点；在区间(-2,-1)上，除不动点-1.15387外，无其他周期点。将理论分析与函数图像相结合得出以下猜想：在实轴区间(8k,8k+1)(k不等于0)上T(x)必有3周期点；在区间( 2~n.k-1,2~n.k) (k是不为零的整数，n为自然数，且n=1时，k不等于1)上存在n周期点；在区间(1,2)上，除不动点1.51555外，无其他周期点；由以上结论和猜想进一步得出在区间(k,k+1)(|k|为大于等于2的整数)上有n周期点。在前人工作的基础上计算出了T(x)在实轴上的3周期吸引点及其吸引的周期轨道。 T(z)在复平面上的不动点是存在的，并指出其不动点是以实轴对称分布的，猜想这些不动点是斥性的。文中用基本逃逸时间算法和周期分类法绘制了T(z)在复平面的分形图像。 2．3x+1推广函数构造广义的M－J集由T(z)出发，我们又提出了一类新的广义3x+1映射： #(z，c均为复数) 利用这个复映射构造其广义M－J集，并在文中绘制了M－J的分形图像、分析并阐述了它们的性质、广义M集周期芽苞分布以及广义M－J集的对应关系。在这一部分提出了另外两种3x+1推广函数：F(z)和G(z) #(z为复数) #(z为整数) 研究了它们丰富的分形图像的构成、简单性质及特点。 3．3x+1推广函数引出的艺术分形分形图像需要应用于科学探索，因此不是简单的科学绘画，可以看作是艺术作品，又具有科学内涵，表现出科学与艺术的和谐与统一。艺术分形图像能够在绘画与动画，影视艺术，科学计算可视化及信息可视化等较多领域得到应用，有希望取得良好的社会效益和经济效益。文中对3x+1推广函数引出的分形在科学和艺术相结合的某些具体方法方面上进行了探索。通过对分形图像的绘制和利用全面系统的计算机实验，用艺术的风格揭示分形及与之相关形体的精细结构，生成了具有独特风格新一类艺术作品。本文采用两种方法绘制3x+1推广函数引起的艺术分形图像且生成的图像具有某种程度拟3维艺术效果。一是在基本逃逸时间算法的基础上，对此算法进行改进，利用调色板技术和距离划分法相结合绘制3x+1推广函数T(z)和F(z)的分形图的收敛区图像，利用调色板技术和轨迹井跟踪技术相结合绘制分形图发散区图像。并且根据3x+1推广函数引起的艺术分形图像的特点，提出了条形带状轨迹井、带状M集轨迹井、相切圆轨迹井和环状M集轨迹井。二是利用牛顿迭代法，绘制出了3x+1推广函数T(z)的美妙分形图像。把牛顿迭代法稍加改进，让其与调色板技术和轨迹井跟踪技术相结合应用于T(z)，能绘制出美丽的花形图案，这里采用了带状轨迹井、圆环轨迹井和相切圆轨迹井。 The theories and applications of the fractal have got a great improvement since B.B.Mandelbrot specifically put forward the idea and concept of the fractal in the 1970s. The fractal’s research can not separate from the computer. Without the powerful functions of the computer’s graph and image, we can not imagine how to directly show the subtle structures of the fractal. On the contrary, the fractal’s theories and methods also enrich the contents of the computer graphics so much. Many new and beautiful images are found continually during the studies of the fractal. The fractal generated by 3x+1 conjecture is one of them. The main contents of the paper conclude: the fractal attributes, applications and artistic fractal images producted by the generalized 3x+1 function. We begin to expound the contents of the paper from three respects as follows: 1. Some attributes of the fractal producted by the 3x+1 generalized function. The integer 3x+1 conjecture is a famous mathematical puzzle which is unsolved. The conjecture results in 3x+1 for odd positive integers x and half of x for even positive integer x, and the iterations of the function on positive integers eventually lead to the value 1, then enter the circle (1,2). We studied the generalized 3x+1 functions---T(x) and T(z) in the paper: #(x is the real number) #(z is the complex number) We detailedly discussed T(x)’s some attributes of the periodic points in the real axis, the fixed points of T(z) in the complex plane, then we got the conclusion: the fixed points of T(z) exist. We also drew the fractal images of the generalized 3x+1 function. We combined the mathematic method with the function images to prove that T(x) has no n-period points in the area (-1,0); T(x) only has the fixed point 0.31582 but no other n-period points in the area (0,1); T(x) has no n-period points except for the fixed point -1.15387 in the area (-2,-1). We combined theoretical analysis with the function images to get these conjectures: T(x) must have 3-period points in the area (8k,8k+1)(k is not 0) of the real axis; T(x) exists n-period point in the area (2~n.k-1,2~n.k)( k is not 0 and k is a integer, n is a natural number, and when n=1, k is not 1); T(x) has on other n-period points but the fixed point 1.51555; finally we get the conjecture that T(x) has n-period points in the area (k,k+1)(|k|is greater than 2 and k is a integer) according to the former conclusions and conjectures. We calculated the attracted 3-period points and the attracted periodic orbits of the function T(x) in the real axis based on the job of the previous scholars. T(z) has the fixed points in the complex plane, and these fixed points are symmetrical by the real axis. We supposed these fixed points are all repellent. We used the escaping time algorithm and periodic classifying algorithm to draw the fractal images of the function T(z) in the paper. 2. The generalized M-J set producted by the generalized 3x+1 function. We put forward a new type of generalized 3x+1 function according to the function T(z): #(z and c are all complex numbers) We used the complex mapping to product the generalized M-J set, and we drew the fractal images, analyzed and expounded the attributes of them and the periodic buds of the generalized M set. Furthermore, we studied the relations between the M set and the J set. We also brought forward another two kinds of generalized 3x+1 functions---F(z) and G(z) in this part: #(z is the complex number) #(z is the iteger) We studied the structures of the rich fractal images of the F(z) and G(z), and analyzed the attributes and features of them simply. 3. Artistic fractals producted by the generalized 3x+1 function. Fractal images need to be applied into the science research. Therefore they are not the simple scientific drawing. They can be treated not only as the artistic works but also as the scientific connotation. So the fractal images express the harmony and unification between the science and art. The artistic fractal images can be applied into many fields, such as the drawing, cartoon, the film art, the visualizing of the scientific calculation and the visualizing of the information and so on. It is possible to get the social benefits and economic benefits from the fractal images. We studied the fractal producted by the generalized 3x+1 function in the way of combination between the science and art in the paper. A new special kind of artistic works are producted by using the drawing of the fractal images and the computer experiments totally and systemically. We use the artistic style to express the related subtle structures. Two methods were taken to draw the artistic fractal images producted by the generalized 3x+1 function in the paper. Meanwhile, some images have the pseudo-3D effect to some extent. The first method is to improve the basic escaping time algorithm. We used the method which combined the color map with the orbit trap tracing approach to paint the escape areas, used the method which combined the color map with the distance divided approach to paint the non-escape areas of the function T(z) and F(z). What’s more, we put forward strip-band orbit trap, band M set orbit trap, tangent circles orbit trap, ring M set orbit trap according to the character of the artistic images producted by the generalized 3x+1 function. The second one is the Newton-Raphson method. We drew the wonderful fractal images of the function of T(z). Then we improved the Newton method, let it combine with the color map and the orbit trap tracing technology. By using the new algorithm, we got the beautiful fractal images like flowers of the function T(z). In this sector we also took the strip-band orbit trap, ring orbit trip and the tangent circles orbit trip
本文来自: 聚合吧(http://www.juhe8.com/) 详细出处参考：http://www.juhe8.com/lunwen/qita/2008-01-15/90530.html
liner and blur:尝试用liner和blur构图
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保形变换
一个复变函数w=f(z)(z∈C),从几何观点来看,可以解释为从z平面到w平面之间的一个变换.本章将讨论解析函数所构成的变换(简称解析变换),特别是线性变换的某些重要特性.我们将看到,这种变换在导数不为零的点处具有一种特殊的保角的特性,它在数学本身及在解决流体力学、弹性力学、电学等学科的某些实际问题中，都是一种重要的工具.
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