Field Trip #33 Creating and Saving Fractals

Julia Set We consider a complex function, f(z) For each point on the complex plane (x,y), where z = x + iy, we consider the sequence of points f(z), f(f(z)), f(f(f(z))), f(f(f(f(z)))), etc. We define a point we call infinity For each point on the complex plane, we determine how many points in the sequence it takes until the magnitude of a point in the sequence is greater than infinity We then assign a different color based on how many points

Complex Numbers Recall that there is no real number, i, such that i*i = -1. We define the imaginary number i to be the square root of -1. We consider a plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. We define a complex number z to be x + iy

Iteration The sequence of points we generate is called an iteration An iteration involves taking the output of a function and putting that into the function to produce a result We sometimes call this recursion A recursion cannot go on indefinitely We define a maximum number of times we will perform the recursion

Loops To fill in the Julia Set, we will need to consider many points on the complex plane In order to do this, we will take advantage of the one of the pillars of structured programming: repeated execution or loops One form of a loop is the for loop A for loop uses a controlling variable that steps between two values and do something for each value of the controlling variable

Computing Color We create initially an array of values This array of values will contain the values we need to create colors The elements of the array determine what color is assigned to the number of iterations needed to exceed infinity If the iteration doesn't exceed infinity after the maximum number of iterations, then we assign the color black

Arrays The array is a structure that contains elements of the same type We use three parallel arrays The arrays contain the amount of red, green, and blue in the color We create an array with the syntax int[] red = new int[30]

Different Fractals There are some well known functions that produce interesting Julia Sets The quadratic function, f(z) = z^2 + c, where c is a complex number can produce different Julia Sets if we vary c The exponential function f(z) = ce^z also produces interesting results The sine function f(z) = csin(z) produces an interesting Julia Set

Quadratic Family, f(z) = z^

Exponential Family, f(z) = (1+i)e^z

Sine Family, f(z) = csin(z)

BufferedImage In the Java language, there is a package that deals with images One of the classes in the package is the BufferedImage After we create a BufferedImage, we can use the class javax.imageio.ImageIO to write the BufferedImage to a file where we specify the type of image file by using an appropriate setting: jpg, png, etc.

Creating the BufferedImage To create the BufferedImage, we first create an array that contain the color information For each Color, we specify how much red, green, and blue it contains. We specify this with integers between 0 and 255 The Color class contains a method called getRGB() that gives us a single integer We use an array of these single integers to create the BufferedImage

Saving an Image After the image has been created, we use the ImageIO class to write the BufferedImage to a file We use the method write which expects the BufferedImage, the type of file (“jpg”,”png”, etc.), and the location We specify the location to write the file by creating a new File object