Mandelbrot Set II

2017-01-14, post № 157

Java, mathematics, programming, #compiled, #filled Julia set, #generalized filled Julia set, #generalized Mandelbrot set, #Julia, #speed increase, #viewer

Over a year ago, I published my first Mandelbrot Set viewer, a Python program using Pygame. Since then, I have published a rather short program highlighting errors that can occur when calculating the set (Mandelbrot Set Miscalculations).
Since my first viewer was written in Python, which is an interpreted programming language, and I wanted to make my viewer faster, I decided to write one in Java. I was hoping for a speed increase since Java is compiled and thus should run at higher speeds. I was not disappointed. The new Java-based viewer runs noticeably faster and additionally I added a lot of new features, all listed below.

To use this application, you can either trust me and download the .jar-file or view the source code listed below, verify it and compile the program yourself.
The program will start in fullscreen mode, to change to windowed mode, just press ‘F11’ (as listed above).

New Year

2016-12-31, post № 156

art, haiku, poetry, #2017, #Silvester

Christmas

2016-12-24, post № 155

art, haiku, poetry, programming, Python, #ascii, #ascii art, #star tree, #tree, #xmas

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       *****       

Advent IV

2016-12-18, post № 154

art, haiku, poetry, #nature, #sleep

Orange

2016-12-17, post № 153

art, haiku, poetry, #color, #falling, #falling up, #fruit, #gif, #screw gravity

Advent III

2016-12-11, post № 152

art, haiku, poetry, #Christmas, #fir, #tree

Advent II

2016-12-04, post № 151

art, haiku, poetry, #ice

Mandelbrot set miscalculations

2016-12-03, post № 150

Java, mathematics, programming, Python, #assignment, #complex numbers

While developing a Java program to create an image of the Mandelbrot set, I stumbled upon a small error which completely changes the set’s look. To fix this bug, you need to swap two lines of code.

The bug arises when trying to convert convenient Python features to Java.
To iteratively apply the function , you update your complex number 𝑧 a certain amount of times. When you are not using a complex number class, but instead you use two floating point numbers (in Java doubles to gain precision) a and b to define the real and imaginary part (), logically both numbers need to be updated.
In Python you may write the following, when 𝑐 is defined as being a complex number with parts c and d ().

a, b = a**2 - b**2 + c, 2 * a * b + d

Which seems to be very similar to those two lines.

a = a**2 - b**2 + c
b = 2 * a * b + d

But notice that in the first code snippet you define a tuple consisting of the real and imaginary part and then assign it to the variables. The first snippet really looks like this.

t = (a**2 - b**2 + c, 2 * a * b + d)
a, b = t

Using this assignment of variables, which corresponds to , you get an image of the correct Mandelbrot set.

In contrary, the second code snippet assigns the old a its new value, then uses this new a to define the value of new b, thus does not calculate , which is equivalent to , but rather .

In Java it would look like this.

a = a*a - b*b + c;
b = 2 * a * b + d;

Which results in this rather unusual depiction of the famous fractal.

You can easily avoid this bug when using two sets of variables to define old 𝑧 and new 𝑧, as shown in the following.

_a = a*a - b*b + c;
_b = 2 * a * b + d;
a = _a;
b = _b;

Or you can define variables and and swap the assignment. Using variables for the squares of the parts of 𝑧 also helps to improve performance^[1].

b = 2 * a * b + d;
a = asqr - bsqr + c;
asqr = a*a;
bsqr = b*b;