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，
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 } @Override public 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); } @Override public 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 forever private double angle = 0.2 * Math.PI; //angle in radians private double shrink = 1.8; //relative size of new branches public 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 block e.printStackTrace(); } catch (IOException e) { // TODO Auto-generated catch block e.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.html void 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); } }
