JavaAlgorithms/Others/Mandelbrot.java

package Others;

import java.awt.*;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import javax.imageio.ImageIO;

/**
 * The Mandelbrot set is the set of complex numbers "c" for which the series "z_(n+1) = z_n * z_n +
 * c" does not diverge, i.e. remains bounded. Thus, a complex number "c" is a member of the
 * Mandelbrot set if, when starting with "z_0 = 0" and applying the iteration repeatedly, the
 * absolute value of "z_n" remains bounded for all "n > 0". Complex numbers can be written as "a +
 * b*i": "a" is the real component, usually drawn on the x-axis, and "b*i" is the imaginary
 * component, usually drawn on the y-axis. Most visualizations of the Mandelbrot set use a
 * color-coding to indicate after how many steps in the series the numbers outside the set cross the
 * divergence threshold. Images of the Mandelbrot set exhibit an elaborate and infinitely
 * complicated boundary that reveals progressively ever-finer recursive detail at increasing
 * magnifications, making the boundary of the Mandelbrot set a fractal curve. (description adapted
 * from https://en.wikipedia.org/wiki/Mandelbrot_set ) (see also
 * https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set )
 */
public class Mandelbrot {

  public static void main(String[] args) {
    // Test black and white
    BufferedImage blackAndWhiteImage = getImage(800, 600, -0.6, 0, 3.2, 50, false);

    // Pixel outside the Mandelbrot set should be white.
    assert blackAndWhiteImage.getRGB(0, 0) == new Color(255, 255, 255).getRGB();

    // Pixel inside the Mandelbrot set should be black.
    assert blackAndWhiteImage.getRGB(400, 300) == new Color(0, 0, 0).getRGB();

    // Test color-coding
    BufferedImage coloredImage = getImage(800, 600, -0.6, 0, 3.2, 50, true);

    // Pixel distant to the Mandelbrot set should be red.
    assert coloredImage.getRGB(0, 0) == new Color(255, 0, 0).getRGB();

    // Pixel inside the Mandelbrot set should be black.
    assert coloredImage.getRGB(400, 300) == new Color(0, 0, 0).getRGB();

    // Save image
    try {
      ImageIO.write(coloredImage, "png", new File("Mandelbrot.png"));
    } catch (IOException e) {
      e.printStackTrace();
    }
  }

  /**
   * Method to generate the image of the Mandelbrot set. Two types of coordinates are used:
   * image-coordinates that refer to the pixels and figure-coordinates that refer to the complex
   * numbers inside and outside the Mandelbrot set. The figure-coordinates in the arguments of this
   * method determine which section of the Mandelbrot set is viewed. The main area of the Mandelbrot
   * set is roughly between "-1.5 < x < 0.5" and "-1 < y < 1" in the figure-coordinates.
   *
   * @param imageWidth The width of the rendered image.
   * @param imageHeight The height of the rendered image.
   * @param figureCenterX The x-coordinate of the center of the figure.
   * @param figureCenterY The y-coordinate of the center of the figure.
   * @param figureWidth The width of the figure.
   * @param maxStep Maximum number of steps to check for divergent behavior.
   * @param useDistanceColorCoding Render in color or black and white.
   * @return The image of the rendered Mandelbrot set.
   */
  public static BufferedImage getImage(
      int imageWidth,
      int imageHeight,
      double figureCenterX,
      double figureCenterY,
      double figureWidth,
      int maxStep,
      boolean useDistanceColorCoding) {
    if (imageWidth <= 0) {
      throw new IllegalArgumentException("imageWidth should be greater than zero");
    }

    if (imageHeight <= 0) {
      throw new IllegalArgumentException("imageHeight should be greater than zero");
    }

    if (maxStep <= 0) {
      throw new IllegalArgumentException("maxStep should be greater than zero");
    }

    BufferedImage image = new BufferedImage(imageWidth, imageHeight, BufferedImage.TYPE_INT_RGB);
    double figureHeight = figureWidth / imageWidth * imageHeight;

    // loop through the image-coordinates
    for (int imageX = 0; imageX < imageWidth; imageX++) {
      for (int imageY = 0; imageY < imageHeight; imageY++) {
        // determine the figure-coordinates based on the image-coordinates
        double figureX = figureCenterX + ((double) imageX / imageWidth - 0.5) * figureWidth;
        double figureY = figureCenterY + ((double) imageY / imageHeight - 0.5) * figureHeight;

        double distance = getDistance(figureX, figureY, maxStep);

        // color the corresponding pixel based on the selected coloring-function
        image.setRGB(
            imageX,
            imageY,
            useDistanceColorCoding
                ? colorCodedColorMap(distance).getRGB()
                : blackAndWhiteColorMap(distance).getRGB());
      }
    }

    return image;
  }

  /**
   * Black and white color-coding that ignores the relative distance. The Mandelbrot set is black,
   * everything else is white.
   *
   * @param distance Distance until divergence threshold
   * @return The color corresponding to the distance.
   */
  private static Color blackAndWhiteColorMap(double distance) {
    return distance >= 1 ? new Color(0, 0, 0) : new Color(255, 255, 255);
  }

  /**
   * Color-coding taking the relative distance into account. The Mandelbrot set is black.
   *
   * @param distance Distance until divergence threshold.
   * @return The color corresponding to the distance.
   */
  private static Color colorCodedColorMap(double distance) {
    if (distance >= 1) {
      return new Color(0, 0, 0);
    } else {
      // simplified transformation of HSV to RGB
      // distance determines hue
      double hue = 360 * distance;
      double saturation = 1;
      double val = 255;
      int hi = (int) (Math.floor(hue / 60)) % 6;
      double f = hue / 60 - Math.floor(hue / 60);

      int v = (int) val;
      int p = 0;
      int q = (int) (val * (1 - f * saturation));
      int t = (int) (val * (1 - (1 - f) * saturation));

      switch (hi) {
        case 0:
          return new Color(v, t, p);
        case 1:
          return new Color(q, v, p);
        case 2:
          return new Color(p, v, t);
        case 3:
          return new Color(p, q, v);
        case 4:
          return new Color(t, p, v);
        default:
          return new Color(v, p, q);
      }
    }
  }

  /**
   * Return the relative distance (ratio of steps taken to maxStep) after which the complex number
   * constituted by this x-y-pair diverges. Members of the Mandelbrot set do not diverge so their
   * distance is 1.
   *
   * @param figureX The x-coordinate within the figure.
   * @param figureX The y-coordinate within the figure.
   * @param maxStep Maximum number of steps to check for divergent behavior.
   * @return The relative distance as the ratio of steps taken to maxStep.
   */
  private static double getDistance(double figureX, double figureY, int maxStep) {
    double a = figureX;
    double b = figureY;
    int currentStep = 0;
    for (int step = 0; step < maxStep; step++) {
      currentStep = step;
      double aNew = a * a - b * b + figureX;
      b = 2 * a * b + figureY;
      a = aNew;

      // divergence happens for all complex number with an absolute value
      // greater than 4 (= divergence threshold)
      if (a * a + b * b > 4) {
        break;
      }
    }
    return (double) currentStep / (maxStep - 1);
  }
}