二叉树递归分形，牛顿分形图案

1. 牛顿分形(Newton Fractal)
在复数域上使用牛顿迭代生成分形图像，函数公式F(z) = z^3 – 1在复数域上面有
三个根，一个是1，另外两个分别是复数-0.5+0.87i 与 -0.5 – 0.87i根据计算出来根
的值不同转换为RGB三种不同的颜色，根据迭代次数的多少设置颜色值的大小，

即颜色强度。

2. 曼德布罗特集合分形(Mandelbort Set Fractal) 使用复数函数公式F(z) = z^2 + c其中

c是一个复数

3. 递归分形树 (recursion tree)– 类似二叉树的递归生成树干，同时不断的缩小树干长
度，根据递归次数不同与角度不同可以得到不同的递归分形树，注意Java最大栈
深度是64，过度的归次数可能导致java栈溢出错误。递归次数建议不要超过32.

根据角度不同，可以生成不同的二叉递归树。

牛顿迭代与曼德尔波特分形算法需要复数范围内的加减乘除计算，请先google一下

然后就知道啦。本人实现的复数计算的类如下：

package com.gloomyfish.fractal;public class Complex{  private float real;  private float imaginary;  public Complex(float paramFloat1, float paramFloat2)  {    this.real = paramFloat1;    this.imaginary = paramFloat2;  }  public float real()  {    return this.real;  }  public float imaginary()  {    return this.imaginary;  }  public float modulus()  {    return (float)Math.sqrt(this.real * this.real + this.imaginary * this.imaginary);  }  public boolean equal(Complex paramComplex)  {    return ((this.real == paramComplex.real()) && (this.imaginary == paramComplex.imaginary()));  }  public Complex add(Complex paramComplex)  {    return new Complex(this.real + paramComplex.real(), this.imaginary + paramComplex.imaginary());  }  public Complex subtract(Complex paramComplex)  {    return new Complex(this.real - paramComplex.real(), this.imaginary - paramComplex.imaginary());  }  public Complex multiply(Complex paramComplex)  {    return new Complex(this.real * paramComplex.real() - (this.imaginary * paramComplex.imaginary()), this.real * paramComplex.imaginary() + this.imaginary * paramComplex.real());  }  public Complex divide(Complex paramComplex)  {    float f1 = paramComplex.real() * paramComplex.real() + paramComplex.imaginary() * paramComplex.imaginary();    float f2 = (this.real * paramComplex.real() + this.imaginary * paramComplex.imaginary()) / f1;    float f3 = (this.imaginary * paramComplex.real() - (this.real * paramComplex.imaginary())) / f1;    return new Complex(f2, f3);  }  public String toString()  {    String str = (this.imaginary >= 0.0F) ? "+" : "-";    return this.real + str + Math.abs(this.imaginary) + "i";  }}

牛顿分形的算法代码如下：

package com.gloomyfish.fractal;public class NewtonFractal extends Fractal {private static final Complex ONE = new Complex(1.0F, 0.0F);private static final Complex THREE = new Complex(3.0F, 0.0F);public NewtonFractal(int widthImage, int heightImage) {super(widthImage, heightImage);// default start point and end point// primary group, this.x1 = -1.0f;this.y1 = -1.0f;this.x2 = 1.0f;this.y2 = 1.0f;// second group//this.x1 = -3.0f;//this.y1 = -1.76f;//this.x2 = 3.0f;//this.y2 = 1.76f;// end comment}@Overridepublic void BuildFractal() {int[] inPixels = new int[getWidth()*getHeight()];        getRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels );        int index = 0;        float xDelta = ((x2 - x1) / (float)width);        float yDelta = ((y2 - y1) / (float)height);        for(int row=0; row<height; row++) {        int ta = 0, tr = 0, tg = 0, tb = 0;        for(int col=0; col<width; col++) {        Complex localComplex2;                float f1 = this.x1 + col * xDelta;                float f2 = this.y2 - (row * yDelta);                Complex localComplex1 = new Complex(f1, f2);                int k = 0;                do {                  Complex localComplex3 = localComplex1.multiply(localComplex1);                  Complex localComplex4 = localComplex3.multiply(localComplex1);                  localComplex2 = localComplex1;                  localComplex1 = localComplex1.subtract(localComplex4.subtract(ONE).divide(THREE.multiply(localComplex3)));                }                while ((++k < MAX_ITERS) && (!(localComplex1.equal(localComplex2))));                int l = 20 * k % 10; // keep value scope between 0 and 255                // if root is 1 then                if (localComplex1.real() > 0.0F)                {                tr = tg = l;                tb = 255;                }                                // if root is second complex = -0.5+0.87i                else if (localComplex1.imaginary() > 0.0F)                {                tr = tb = l;                tg = 255;                }                else                {                  tr = 255;                  tg = tb = l;                }                index = row * width + col;        ta = 255;        inPixels[index] = (ta << 24) | (tr << 16) | (tg << 8) | tb;        }        }        setRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels);}}

曼德尔波特分形算法如下：

package com.gloomyfish.fractal;public class MandelbrotSetFractal extends Fractal {private float delta = 0.01f;public MandelbrotSetFractal(int widthImage, int heightImage) {super(widthImage, heightImage);    this.delta = 0.01F;    this.x1 = (-(this.width / 2) * this.delta);    this.y1 = (-(this.height / 2) * this.delta);    this.x2 = (-this.x1);    this.y2 = (-this.y1);}@Overridepublic void BuildFractal() {int[] inPixels = new int[getWidth()*getHeight()];        getRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels );        int index = 0;        for(int row=0; row<height; row++) {        int ta = 0, tr = 0, tg = 0, tb = 0;        float f1 = y2 - (row * delta);        for(int col=0; col<width; col++) {        float f5;                int i1;                float f2 = x1 + col * delta;                Complex localComplex1 = new Complex(f2, f1);                Complex localComplex2 = new Complex(0.0F, 0.0F);                int k = 0;                int l = 0;                do                {                  localComplex2 = localComplex2.multiply(localComplex2).add(localComplex1);                  f5 = localComplex2.modulus();                  k = (f5 > 2.0F) ? 1 : 0; }                while ((++l < 32) && (k == 0));                index = row * width + col;                if (k != 0) {                  i1 = 255 - (255 * l / 32);                  i1 = Math.min(i1, 240);                  tr = tg = tb = i1;                }                else                {                  i1 = (int)(100.0F * f5) / 2 + 1;                  int i2 = 101 * i1 & 0xFF;                  int i3 = 149 * i1 & 0xFF;                  int i4 = 199 * i1 & 0xFF;                  tr = i2;                  tg = i3;                  tb = i4;                }                ta = 255;        inPixels[index] = (ta << 24) | (tr << 16) | (tg << 8) | tb;        }        }        setRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels);}}

递归分形树代码如下：

package com.gloomyfish.fractal;import java.awt.BorderLayout;import java.awt.Color;import java.awt.Dimension;import java.awt.Font;import java.awt.FontFormatException;import java.awt.Graphics;import java.awt.Graphics2D;import java.awt.RenderingHints;import java.io.IOException;import java.io.InputStream;import java.util.Date;import javax.swing.JComponent;import javax.swing.JFrame;public class FractalTree extends JComponent {/** *  */private static final long serialVersionUID = 8812325148970066491L;private int maxRecursions = 8; //never make this too big or it'll take foreverprivate double angle = 0.2 * Math.PI; //angle in radiansprivate double shrink = 1.8; //relative size of new branchespublic FractalTree() {super();}protected void paintComponent(Graphics g) {Graphics2D g2 = (Graphics2D) g;g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);g2.setPaint(Color.WHITE);g2.fillRect(0, 0, 400, 400);renderTree(g2);g2.setPaint(Color.RED);try {g2.setFont(loadFont());} catch (FontFormatException e) {// TODO Auto-generated catch blocke.printStackTrace();} catch (IOException e) {// TODO Auto-generated catch blocke.printStackTrace();}g2.drawString("Created by Gloomyfish " + new Date(System.currentTimeMillis()), 10, 320);}/** * create fractal tree using recursion * @param Graphics2D g2 */private void renderTree(Graphics2D g2) {g2.setPaint(new Color(128, 96, 64));recursion(400.0d / 2.0d, 400.0d -1.0d, 0.0d, -1.0d, 400.0d / 2.3d, 0, g2);}// http://www.cg.info.hiroshima-cu.ac.jp/~miyazaki/knowledge/teche31.htmlvoid recursion(double posX, double posY, double dirX, double dirY, double size, int n, Graphics2D g2){    int x1, x2, y1, y2;    x1 = (int)posX;    y1 = (int)posY;    x2 = (int)(posX + size * dirX);    y2 = (int)(posY + size * dirY);    g2.drawLine(x1, y1, x2, y2);    if(n >= maxRecursions) return;    double posX2, posY2, dirX2, dirY2, size2;    int n2;        // calculate the new start point coordinate    posX2 = posX + size * dirX;    posY2 = posY + size * dirY;    size2 = size / shrink; // make different length of line.    n2 = n + 1;        // rotate angle and get the new directX, directY    // http://www.jimloy.com/geometry/trigz.htm    // sin(theta + angle) = sin(theta) * cos(angle) + cos(theta) * sin(angle)    // cos(theta + angle) = -sin(angle) * sin(theta) + cos(theta) * cos(angle)    dirX2 = Math.cos(angle) * dirX + Math.sin(angle) * dirY;    dirY2 = -Math.sin(angle) * dirX + Math.cos(angle) * dirY;    recursion(posX2, posY2, dirX2, dirY2, size2, n2, g2);        dirX2 = Math.cos(-angle) * dirX + Math.sin(-angle) * dirY;    dirY2 = -Math.sin(-angle) * dirX + Math.cos(-angle) * dirY;    recursion(posX2, posY2, dirX2, dirY2, size2, n2, g2);}/** * http://en.wikipedia.org/wiki/Mandelbrot_set * http://www.urbanfonts.com/fonts/sans-serif-fonts.htm * @return * @throws FontFormatException * @throws IOException */public Font loadFont() throws FontFormatException, IOException{    String fontFileName = "AMERSN.ttf";    InputStream is = this.getClass().getResourceAsStream(fontFileName);    Font actionJson = Font.createFont(Font.TRUETYPE_FONT, is);    Font actionJsonBase = actionJson.deriveFont(Font.BOLD, 12);    return actionJsonBase;}public static void main(String[] args) {JFrame frame = new JFrame("Fractal Tree UI - GloomyFish");frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);frame.getContentPane().setLayout(new BorderLayout());// Display the window.frame.getContentPane().add(new FractalTree(), BorderLayout.CENTER);frame.setPreferredSize(new Dimension(450,400));frame.pack();frame.setVisible(true);}}