Fract'ol

This project is about creating graphically beautiful fractals.

---

Table o'Contents

• About 📌
  • Mandatory Features
  • Bonus Features
• Implementation 📜
  • t_display Structure
  • ft_args.c : Argument Parsing Functions
    • ft_no_args()
    • ft_args()
      • ft_select_fractal()
      • ft_set_args()
  • ft_init.c : Initialization Functions
    • ft_init_display()
    • ft_init_events()
      • ft_kill_handle();
      • ft_handle_keys();
      • ft_handle_mouse();
    • ft_init_data()
    • ft_usage()
    • ft_render()
    • mlx_loop()
• Usage 🏁
• Testing 🧪
• Appendixes
  • MinilibX 🪟
    • X-Window System
    • X client-server Architecture
  • Complex Numbers
  • Complex Arithmetic
    • Addition
    • Subtraction
    • Multiplication
      • Complex * Real
      • Complex * Complex
    • Expanding a Complex Number
  • Complex Plane
  • Fractals
    • Julia Set
    • Mandelbrot Set
    • Burning Ship Set
    • Tricorn Set
• Footnotes

About 📌

Fract'ol is the first computer graphics project of the Common Core curriculum.

It is a simple graphics program using minilibx, an opportunity to learn how to use the mathematical notion of complex numbers, have a first contact with the concept of optimization in computer graphics, and event handling.

---

Mandatory Features

• General

  • The program must take the type of the fractal to be displayed as a parameter and any other relevant option.
  • The program must display the fractal in the window powered by minilibx.
  • The project must contain a Makefile that compiles all sources. It must not relink.
  • Global variables are forbidden.

• Rendering

  • The program must offer the Julia and Mandelbrot sets.
  • The mouse wheel zooms in and out almost infinitely, within the limits of the computer.
  • A different Julia set must be rendered if the program is passed the appropriate parameters.
  • A parameter passed on startup must be the type of the fractal to be rendered.
    • Adding more parameters is optional. If no parameter is provided, or the parameters are invalid, it displays the help page and exits cleanly.
  • A few different color schemes must be implemented.

• Graphic Management

  • The program has to display the image in a window.
  • The management of the window must remain smooth.
  • Pressing ESC must close the window and exit the program in a clean way.
  • Clicking on the cross on the top window frame must have the same effect.
  • It is mandatory to use images from minilibx.

---

Bonus Features

• One extra fractal.
• The zoom follows the mouse position.
• Moving the view by pressing the arrow keys.
• Make the color range shift.

---

Implementation 📜

Before anything else, the main() function declares a t_display variable named display that stores all the necessary data, conveniently packed to be passed around the program.

---

t_display Structure

typedef struct s_display
{
	// mlx Data
	void        *mlx_conn;   // Stores pointer to mlx connection
	void        *mlx_win;    // Stores pointer to mlx window
	t_img       img;         // Stores the image data
	int         width;       // Stores the width of the window
	int         height;      // Stores the height of the window
	t_range     win_size;    // Stores the size of the window
	double      x_offset;    // Stores how much to shift when moving the view 
	double      y_offset;    // Stores how much to shift when moving the view
	double      zoom;        // Stores the zoom factor
	// Fractal Data
	char        *name;       // Stores the name of the fractal
	int         set;         // Stores the type of fractal
	long        iter;        // Stores the number of iterations
	t_complex   z;           // Stores z for Mandelbrot/Julia/Tricorn/Burning Ship
	t_complex   c;           // Stores c for Mandelbrot/Tricorn/Burning Ship
	t_complex   c_julia;     // Stores c for Julia
	t_complex   z_newton;    // Stores z for Newton
	t_range     frac_range;  // Stores the range of the complex plane
	double      escape;      // Stores the complex plane escape value
	double      newton_esc;  // Stores escape value for Newton
	t_range     color_iter;  // Stores a range of 0 to n iterations 
	t_range     color_range; // Stores a range from black to white
	int         color;       // Stores a color for the Newton fractal
}               t_display;

---

ft_args.c : Argument Parsing Functions

The main logic for argument parsing can be found inside the ft_args.c file.

ft_no_args() and ft_args() are used to parse the input arguments and ensure that if there is something wrong the program exits correctly (without memory leaks).

if (argc < 2)
	return (ft_no_args());
else if (!ft_args(&display, argc, argv))
	exit(EXIT_FAILURE);

---

ft_no_args()

If the program is passed no arguments:

• It prints an error to stderr;
• Displays the help page and exits cleanly.

---

ft_args()

Checks if the arguments passed are valid.

int	ft_args(t_display *d, int argc, char **argv)
{
	if (!ft_select_fractal(d, argc, argv))
		return (ft_invalid_args(argv[1]));
	if (!ft_set_args(d, argc, argv))
		return (0);
	return (1);
}

• First checks if the fractal type selected is valid.
• Then attempts to set the input arguments:

---

ft_select_fractal()

This function checks if the fractal type is valid.

• It first converts the fractal type (first argument) to lowercase.
• If it is valid, ft_set_fractal() is called and the function outputs 1.
• If it is NOT valid it outputs 0.

---

ft_set_args()

Here we make sure we got the right number of arguments and check if they are the right type before the program initializes anything.

• First checks the iterations argument:

  • If the 2nd argument is a valid input for the number of iterations, we set it to d->iter. In case it is a negative value a default value is set instead.
  • Otherwise the program prints an error to stderr and exits.

• Then we check for the Julia case in which we get a complex number as the third and fourth arguments.

  • If the input arguments are a valid doubles we set them to d->c_julia.r and d->c_julia.i.
  • Otherwise the program prints an error to stderr and exits.

---

ft_init.c : Initialization Functions

After all validation tests are passed, the program calls ft_init_display().

---

ft_init_display()

ft_init_display(&display, argv);

It initializes:

• the mlx connection into d->mlx_conn by calling mlx_init();
• the mlx window into d->mlx_win by calling mlx_new_window();
• the image pointer into d->img.img by calling mlx_new_image();
• the image pixels into d->img.pix by calling mlx_get_data_addr();

All these calls are properly protected by calls to cleanup functions in case a initialization error arises.

After everything is properly allocated we proceed to initialize the event handling functionality.

---

ft_init_events()

This function initializes three event handlers to be triggered when certain events are received:

---

ft_kill_handle();

• Listens for DestroyNotify event;
  • Destroys the image data;
  • Destroys the mlx window;
  • Destroys the mlx connection;
  • Frees the t_display pointer to the mlx_conn;

---

ft_handle_keys();

• Listens for KeyPress events;
  • If Escape is received, it exits by calling ft_kill_handle();
  • If the arrow keys are pressed, ft_handle_offsets() is called;
  • If PageUp or PageDown are pressed, the d->iter is increased or decreased by 1 respectively;
  • If Space, 1, 2, 3, 4, 5 are pressed, ft_swith_set() is called.
  • If Left-Shift, Right-Shift, r, g or b are pressed, ft_switch_color() is invoked.
  • Else if the key press received is not being handled, a message with the keysym value is printed to stdout.
  • If an event was successfully caught ft_render() is called causing a re-render of the window.

---

ft_handle_mouse();

• Listens for ButtonPress events;
  • If the left mouse button is pressed inside the window when on the Mandelbrot set the fractal settings are changed and a re-render is triggered with a Julia set with its c set to the current mouse position;
  • Else if the right button is pressed the window re-renders the Mandelbrot set.
  • Else if the mouse wheel is scrolled up or down ft_handle_zoom() is called.

Note

Understanding ft_handle_zoom() :

Centering & Scaling

The keys to zooming in computer graphics are :

• Adjusting the view's center, by changing the d->x_offset and d->y_offset;
• Adjusting the view's scale, by changing the d->zoom factor;

Mouse Position & Zoom Center

The x and y coordinates of the mouse are used to determine the zoom center;

• This is done by mapping the mouse position to the range of the complex plane;

Zoom Factor & Scaling

• The zoom factor (SCALE_FACTOR) determines how much the view is scaled with each zoom operation.
• Increasing the zoom level, divides d->zoom value by the SCALE_FACTOR, enlarging the view;
• Decreasing the zoom level, multiplies d->zoom value by the SCALE_FACTOR, shrinking the view.
• fabs() is used to ensure that the scale factor is always positive, regardless of the current zoom level.

Offset Adjustment

• The offset adjustment (0.13 * fabs(d->zoom)) is a scaling factor that controls how much the view is moved in response to zooming.
• This factor is multiplied by the mapped mouse position to ensure that the zoom center is adjusted proportionally to the zoom level, providing a smoother and more controlled zooming.

Now that we got the X connection, the window and event handling up and running all there is left to do it the data initialization.

---

ft_init_data()

In this function we initialize the data inside the t_display structure to be passed and used by the program.

ft_init_display(&display, argv);

Check out ft_init.c and fractol.h for a closer look at what is being initialized and to what values.

---

ft_usage()

The ft_usage() function prints the usage of the program and all available commands to stdout.

ft_usage();

---

ft_render()

This is where the pixel-by-pixel drawing of the window takes place.

ft_render(&display);

• It iterates over each pixel in the window;
• Selects the rendering function based on the chosen fractal type;
• For each pixel it evaluates the function describing the selected set;

while (++y <= HEIGHT)
{
	x = -1;
	while (++x < WIDTH)
		ft_select_set(d, x, y);
	ft_printf("\r%sRendering:%s [%d%%]", YEL, NC, ((y * 100) / d->height));
}
ft_printf("\t%sComplete!%s\n", MAG, NC);

• Once the calculations are done mlx_put_image_to_window() is called to render the image to the window.

mlx_put_image_to_window(d->mlx_conn, d->mlx_win, d->img.img, 0, 0);

• Then ft_render_ui() is called to print a simple UI to the window.

ft_render_ui(d);

Note

This is a function that can produce memory leaks if the usage of ft_itoa() and ft_strjoin() are not handled correctly. Take a look for yourself at ft_ui.c for details.

---

mlx_loop()

Finally, the program enters an infinite loop, keeping the window open listening for user events.

mlx_loop(d->mlx_conn);

---

Usage 🏁

First, clone the contents of this repository over SSH:

git clone git@github.com:PedroZappa/42_fractol.git

Then, make sure that the program is compiled with all its dependencies using make:

make

One way to find out all available startup options and keybindings, is to run the program without arguments:

./fractol

---

Testing 🧪

If you want to test the program with valgrind, you can use the following make rule:

make valgrind

There is also a convenient make rule to run a Norminette check:

make norm

---

Appendixes

---

MinilibX 🪟

MinilibX is a small library, a simplified version of XLib (X11R6) written in C , designed to introduce students to the X-Window System. ¹

---

X-Window System

The X-Window System is an architecture independent windowing system for bitmap displays that provides a basic framework for creating graphical user interfaces. ² It enables users to draw and move windows on a display using the mouse and keyboard.

Note

In computing, a bitmap (also known as bit array or bitmap index) is a mapping from a given domain (for instance, a range of integers) to bits. ³

---

X client-server Architecture

X is based on a client-server model:

• one X server connects to multiple X client programs.

flowchart TB
	Display[Display]
	Keys[Keyboard]
	Mouse[Mouse]

	Keys[Keyboard] --->|input| Xserv[X Server]
	Mouse[Mouse] --->|input| Xserv
	Display[Display] <---|output| Xserv
	subgraph W[User Workstation]
		Xserv[X Server]
		Xserv --> X-client[X client1]
		Xserv --> X-client2[X client2]
	end
	subgraph Remote Machine
		Xserv -->|Network Conn| X-client3[X client3]
	end

Loading

The X Server receives requests to output graphics on the display (through windows) and sends back user input (from a keyboard, mouse, etc).

Note

There are many implementations of the X Window System (Xlib), minilibx being just one among many following the X Consortium standard; ⁴

• Xlib : X Consortium Standard

---

Complex Numbers

Complex numbers are numbers in the form (a + bi) where:

• a is the real part:
• b is the imaginary part;
• i is the imaginary unit, defined by the equation $i^2 = -1$.

Note

$i = \sqrt-1$

---

Complex Arithmetic

Like with real numbers, we can perform arithmetic on complex numbers.

---

Addition

$(a + bi) + (c + di) = (a + c) + (b + d)i$

Example of how to add two complex numbers:

$((3 - 4i) + (2 + 5i)) =$

$((3 + 2) + (-4 + 5)i) =$

$(5 + i)$

---

Subtraction

$(a + bi) - (c + di) = (a - c) + (b - d)i$

---

Multiplication

Multiplication is similar to multiplying binomials but with complex numbers we work with the real and imaginary parts separately.

Complex * Real

$c(a + bi) = (c * a) + (c * b)i$

Example:

$3(6 + 2i) =$

$(3 * 6) + (3 * 2i) =$ # Distribute

$(18 * 6i)$ # Simplify

---

Complex * Complex

$(a + bi)(c + di) = ac + adi + bci + bdi^2$

• Because $i^2 = -1$, we can simplify the expression to:

$(a + bi)(c + di) = ac + adi + bci - bd$

• Simplifying, we combine the real parts, and then the imaginary parts:

$(a + bi)(c + di) =$

$(ac - bd) + (ad + bc)i$

Example:

$(4 + 3i)(2 - 5i) =$

$(4 * 2) + (4 * (-5i)) + (3i * 2) + (3i * (-5i)) =$

$8 - 20i + 6i - 15i^2 =$

$8 + 15 - 20i + 6i =$

$(23 - 14i)$

---

Expanding a Complex Number

Here is an example on how to expand a squared complex number:

$(a + bi)^2 =$

$(a * a) + (a * bi) + (a * bi) - (bi * bi)$

$(a^2 - bi^2) + 2(a * bi))$

• The real part is $(a^2 - b^2)$;
• The imaginary part is $2(a * bi)$;

---

Complex Plane

We can take complex numbers and plot them in a plane known as the Complex Plane.

This plane is formed by the mapping of the real and imaginary parts of a complex number to a Cartesian coordinate system. The real part mapped to the x-axis and the imaginary part to the y-axis.

---

Fractals

Fractals are infinitely complex self-similar patterns across multiple scales.

Generated by:

• Initializing a complex number $z = (x + yi)$ where: $i^2 = -1$
• x and y are image pixel coordinates mapped to a range between -2 to 2.
• A formula is iterated until the value of |z| becomes greater than 2.
  • If the point never escapes the range it IS considered to be part of the set.
  • If the point escapes the range it means it is NOT part of the set.
  • The color of each pixel is determined by the number of iterations it took to escape the set.

---

Julia Set

Formula : $f(z_{n+1}) = z_n^2 + c$

There are infinitely many Julia sets. To generate them, we use the same complex number c for all pixels.

• For each pixel in the image:
  • z is initially set to 0.
  • z is updated repeatedly following the formula $z_{n+1} = z^2 + c$.
  • c is a complex number that seeds a specific Julia set.

---

Mandelbrot Set

Formula : $f(z_{n+1}) = z_n^2 + c$

For the Mandelbrot set, we use different complex numbers for each pixel. It is the one map to all Julia sets.

• For each pixel in the image:
  • z is initially set to 0.
  • z is updated repeatedly following the formula $z_{n+1} = z^2 + c$.
  • c is a complex constant defined as: $c = (x + yi)$ where: $i^2 = -1$

---

Burning Ship Set

$f(z_{n+1}) = (|{Re}(z_n)| + |{Im}(z_n)|i)^2 + c$

The Burning Ship Set is generated by the equation above where:

• $z_n$ is the current complex number;
• c is a complex constant (just like in the Julia Set formula);
• $z_{n+1}$ is the next complex number in the sequence;
• The real and imaginary components are set to their absolute values before squaring at each iteration.

This modification results in the distinctive "burning ship" appearance of the fractal.

---

Tricorn Set

Formula : $f(z_{n+1}) = \overline{z_n}^2 + c$

The Tricorn fractal is a variant of the Mandelbrot set and is characterized by its triangular shape. It is generated by using a slightly different formula where:

• The complex conjugate of z is squared instead of z itself.
• The complex conjugate of z is represented by $\overline{z_n}$
• c is a complex constant that varies for each pixel in the image.

Note

To get the complex conjugate of a complex number $z_n = (a + bi)$, we simply invert the sign of the imaginary part like so: $\overline{z_n} = (a - bi)$

For example: The conjugate of (4 + 7i) is (4 - 7i).

---

Footnotes

Footnotes

1. The Fractal Geometry of Nature - Benoit B. Mandelbrot - Google Livros ↩

2. Are Fractals or Fractal Curves Differentiable? ↩

3. How to Draw Fractals by Hand: A Beginner's Guide ↩

4. Complete List of Books by Benoit Mandelbrot ↩