Mandelbrot Set

Due Before Class on Tuesday, March 15

The Mandelbrot set is a set of complex numbers that remains bounded (does not increase to infinity) when iterated with a quadratic polynomial function. When the set is graphed on the complex plane, the set boundary has a complex pattern that does not simplify at any magnification and is therefore sometimes defined as a fractal.

Create a program that plots the Mandelbrot set for complex numbers in the range -2 > a, b < 2. To do this you will need to convert pixel x, y coordinates to complex a,b coordinates. For example, if the window is 200 by 200 pixels, the pixel (0, 0) would map to the complex coordinate (-2, -2) and the pixel (100, 100) to the complex coordinate (0, 0). If the complex number that cooresponds to a pixel is in the Mandelbrot set, draw the pixel as white, if it is not, draw it as a shade of gray depending on how many iterations of the quadratic polynomial function were required to determine that it is not in the set. For example, if the quadratic polynomial function is iterated 100 times before the absolute value becomes larger than 2, then draw the pixel with the RGB values (100, 100, 100). There is no `drawPixel`

or `drawPoint`

in the Graphics class. So instead, you will need to create a BufferedImage that is the size of the window, set a pixel's color using the `setRGB`

method of the BufferedImage class, and draw the image to the window. Note, the `setRGB`

method takes a single int parameter that represents all three RGB values to set. In order to convert three different ints that represent the RGB values, you can use the color class (or bit operations).

When the user clicks on the image of the Mandelbrot set the program should zoom in on the click location by changing the range of the complex numbers. For example, if the window is 200 by 200 and the complex range is halved on user clicks, a click on the pixel (100, 100) would set the complex range to -1 > a, b < 1. A click on the pixel (50, 50) would set the complex range to -2 > a, b < 0.

Lastly, add the ability to save the currently displayed image to a high-resolution file. Based on some user input (a key press, button press, menu selection, or modified mouse click) create a buffered image that is 2000 x 1600 pixels and render the portion of the Mandelbrot set that is currently being drawn to the window, to the large image. Note, creating an image of the Mandelbrot set this large may take a little while, so you might want to test it on a smaller image first. Lastly, the large image should be saved to a file. Include your favorite image with your code submission. Make it a good one, the best image (as selected by an on-line poll) will be framed and hung in Trexler.

**Submission**: Tar and submit your code on the course Inquire site.

**Color Map**

It is possible to create more visually pleasing images by using a color map to specify what color maps to which escape value. It is possible, for example, to have a color map between colors with RGB values (100, 255, 150) and (200, 50, 0) by interpolating between the two colors based on the escape value. It is possible to interpolate by using the equation from lab, but use it with each of the color channels. For example, if the escape value is 128, then the interpolated color of the above two colors would be r = (100 * (1 - i)) + (200 * i), g = (255 * (1 - i)) + (50 * i), b = (150 * (1 - i)) + (0 * i), where i = 128 / 255.

It is also possible to produce a color map with more than two colors by first determining which pair of colors an escape value is between, and then interpolating the colors. For example, if a color map consists of three colors and the escape value is 42, then because 42 / 255 is less than 0.5 the interpolated color would be produced by interpolating between the first and second colors with i = 42 / 127.

**Julia Set**

Julia sets are sets of complex numbers that are related to the Mandelbrot set and are also capable of producing interesting images. A complex number is determined to be in a Julia set in a similar fashion as for the Mandelbrot set. The difference is that the quadratic polynomial equation for a Julia set is:

c_{n+1} = c_{n}^{2} + c_{j}

Where c_{j} is any complex number. The more interesting Julia sets correspond to complex numbers that are near the boundary of the Mandelbrot set. Some interesting values for c_{j} you could try are 0.4+0.6i, 0.285+0.01i, and -0.835-0.2321i

**Smooth Gradient**

The colors of the fractal correspond to the distance that a particular point is from the boundary of the set. The colors produce bands of distinct regions because the distance is calculated using integers. In order to find distances that can be fractional, and therefore produce images with smooth gradient transitions, use the following equation to determine the distance, and therefore the color, of a complex number:

c = n + 1 - ln(ln(|c_{n}|) / ln(2)) / ln(2)

where *n* is the number of iterations of the quadratic polynomial function that were computed, *c _{n}* is the complex number that was computed from the n