Your task in this assignment is to compute and visualize the Mandelbrot set. This set is the result of studying the convergence of the iterative function xt+1=xt2+c with x0=0+i0=0. This function is defined in the complex plane, i.e. xj for j=0,1,... and c are complex numbers. A point c is said to belong to the Mandelbrot set if the value of xt as t goes to infinity is bounded. The following figure shows this set:
Each of the black points represents a value of c that belongs to the Mandelbrot set. In the previous figure, the real part of each of the points is placed on the x-axis, while the imaginary part is placed on the y-axis. In spite of its simple definition, the structure of this set is strikingly complex.
In practice, we can explore this set in the complex plane by evaluating the values of xt after some number of iterations (e.g. t=30) and comparing |xt| with 2 (it is possible to prove that if |xt| exceeds 2, then the sequence of xt for t=0,1,... grows to infinity). A particularly attractive way of plotting this set is to color each point in the complex plane according to the number of iterations that it takes to determine whether the point belongs to the set or not (i.e. value of t the first time xt > 2). The following figures show two sample visualizations:
In this assignment, you have to write a Java program that computes and visualizes the Mandelbrot set. The visualization should use color as in the two figure above, but you are free to choose the color scheme, as long as it remains informative. The program should be invoked with:
java Mandelbrot a b c d
After this, your program should create a window with a visualization of the set in the subplane with (a+ib) as its upper left corner and (c+id) as its lower right corner. The resolution should be at least 200 by 200 pixels (i.e. evaluate the function for at least 40,000 points.)
Remember that a complex number consists of the pair (a+ib), where a and b are real numbers. a is known as the real part, and ib as the imaginary part. The absolute value of a complex number is given by
|(a+ib)|=(a2+b2)1/2
and the following rules are used to add and multiply complex numbers:
(a+ib)+(c+id)=(a+c)+i(b+d)
(a+ib)*(c+id)=(ac-bd)+i(ad+bc)
You can find numerous tutorials on how to draw and create images in Java. For instance, the official Java Tutorial gives a good introduction to the topic in
http://java.sun.com/docs/books/tutorial/uiswing/painting/index.html
If you need help working with frames, check
http://java.sun.com/docs/books/tutorial/uiswing/mini/index.html
Understanding Your first Swing program section is enough for the basic solution of this assignment.
You can extend your solution in several ways:
This assignment is due on Jan 26 (during the first ten minutes of class). Late assignments get 25% off each day they are late (e.g. Jan 26 after 12:40 PM, -25%; Jan 30, -50%). Your submission will be electronic. Your electronic submission will be an e-mail sent to wluebke@cs.unc.edu with the subject
COMP 144 Assignment 1 Submission
and a single .zip file attached to it. This file should have the following content:
I will compile the program using Sun's SDK v1.3.1_02 (see http://java.sun.com/j2se/1.3/) only. You can develop you program with any Java programming environment, but you must make sure it compiles and runs under SDK 1.3.
It is very easy to install this package, since it comes as a Windows installer. You can find very detailed information on how to compile and run Java programs with SDK at
http://java.sun.com/docs/books/tutorial/getStarted/cupojava/win32.html
In addition, you will probably want to change your DOS PATH variable, so you do not have to type the directory where the SDK executable is located each time you compile or run a Java program. See
http://java.sun.com/j2se/1.3/install-windows.html#Environment
for more details.
I will grade both the quality of the source code (design and documentation) and the output. I will not grade programs that do not compile. Collaboration is encouraged, but you cannot share source code. Honor Code applies to this program. Please, read http://www.cs.unc.edu/Admin/Courses/HonorCode.html for more details.