Mandelbrot Voyage 2

A fully interactive GPU-accelerated customizable fractal zoom program aimed at creating artistic and high quality images & videos.

Mandelbrot set is a set defined in the complex plane, and consists of all complex numbers which satisfy $|Z_n| < 2$ for all $n$ under iteration of $Z_{n+1}=Z_n^2+c$ where $c$ is any point in the complex plane and $Z_0=0$.

Points inside the set are colored black, while points outside the set are colored based on $n$ (how many iterations it took for the point to diverge).

Features

• Fully customizable equation in GLSL syntax
• Smooth coloring
• Normal mapping for shadows
• Super-sampling anti aliasing (SSAA)
• Customizable color palette with up to 16 colors
• Hold right-click to see the orbit and the corresponding Julia set for any point
• Zoom video rendering

2024_Jan_13_22_31_47_19.mp4

2024-05-22.00-45-10.mp4

Custom equations

Users can define custom equations to draw fractals in the complex plane. The following functions are available in addition to the standard functions in GLSL:

Custom functions reference

Double-precision transcendental functions

Function	Definition
double atan2(double, double)	$\tan^{-1}(x/y)$
double dsin(double)	$\sin(x)$
double dcos(double)	$\cos(x)$
double dlog(double)	$\ln(x)$
double dexp(double)	$e^x$
double dpow(double, double)	$x^y$

Complex-defined double-precision functions

Function	Definition
dvec2 cexp(dvec2)	$e^z $
dvec2 cconj(dvec2)	$\bar{z} $
double carg(dvec2)	$\arg{(z)}$
dvec2 cmultiply(dvec2, dvec2)	$z\cdot w$
dvec2 cdivide(dvec2, dvec2)	${z}/{w} $
dvec2 clog(dvec2)	$\ln{(z)}$
dvec2 cpow(dvec2, float)	$z^x, x \in \mathbb{R}$
dvec2 csin(dvec2)	$\sin(z)$
dvec2 ccos(dvec2)	$\cos(z)$

Furthermore, the following variables are also exposed to the user:

Local variables

Name	Description
dvec2 c	Corresponding point in the complex plane of the current pixel
dvec2 z	$Z_n$
dvec2 prevz	$Z_{n-1}$
int i	Number of iterations so far
dvec2 xsq	$\Re^2(Z_n)$, for optimization purposes
dvec2 ysq	$\Im^2(Z_n)$, for optimization purposes
float power	Uniform variable of type float, adjustable from the UI
int max_iters	Maximum number of iterations before point is considered inside the set
double zoom	Length of a single pixel in screen space in the complex plane
dvec2 center	Center point of the window in the complex plane
dvec2 mouseCoord	Point in the complex plane that the mouse cursor is on
dvec2 initialz	Initial value of $Z_n$

The first input (the main equation, must evaluate to a dvec2) is the new value of $Z_{n+1}$ after each iteration. The second input (bailout condition, must evaluate to a bool) is the condition which, when true, the current pixel will be considered inside the set. The third input (initial Z, must evaluate to a dvec2) is $Z_0$.

User-controlled variables can also be defined, which can then be used in the equation and adjusted in real time using the sliders below. "Power" is a default slider that cannot be deleted and corresponds to the power variable above.

Note

Due to limitations in GLSL, zoom is limited to $10^4$ for any non-integer or bigger than 4 power. See Limitations for more information. Click "Round" to round the power to the nearest integer to zoom further.

Examples

Burning ship fractal $Z_{n+1}=(|\Re(Z_n)| + i|\Im(Z_n)|)^2+c \quad Z_0=c \quad \text{Bailout: } |Z_n| > 100$

Nova fractal $Z_{n+1}=Z_n-\frac{Z_n^3-1}{3Z_n^2}+c \quad Z_0=1 \quad \text{Bailout: } |Z_n-Z_{n-1}| < 10^{-4}$

Magnet 1 fractal $Z_{n+1}=\bigg(\dfrac{Z_n^2+c-1}{2Z_n+c-2}\bigg)^2 \quad Z_0=0 \quad \text{Bailout: } |Z_n| > 100 \lor |Z_n-1| < 10^{-4}$

Building

Dependencies

• Git
• CMake version >= 3.21
• C++ build system (Make, Ninja, MSBuild, etc.)
• C++ compiler (GCC, Clang, MSVC, etc.)

Clone the repository with --recurse-submodules, then go into the directory

git clone --recurse-submodules https://github.com/Yilmaz4/MV2.git
cd MV2

Generate the build files with CMake and build

cmake -S . -B build
cmake --build build

You can then find the binary in the bin directory

This project has been tested on Windows and Linux.

Limitations

• Any custom equation utilizing dvec2 cpow(dvec2, float) where the second argument $\not\in [1,4] \cap \mathbb{N}$ will be limited to single-precision floating point, therefore limiting amount of zoom to $10^4$.
• Most of the double-precision transcendental functions are software emulated, which means performance will be severely impacted.
• Maximum zoom is $10^{14}$ due to finite precision.

Known issues

• Shader linkage takes too long on Intel iGPUs with Mesa drivers on Linux, causing the program to open only after several minutes, I have no idea why
• Taking a screenshot results in a blurry image regardless of how high SSAA is set

Contributing

Contributions are highly welcome, it could be anything from a typo correction to a completely new feature, feel free to create a pull request or raise an issue!