Mandelbrot Voyage 2

A fully interactive open-source GPU-based fully 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 the particular point in the Mandelbrot set and $Z_0=0$.

Points inside the set are colored black, and points outside the set are colored based on $n$.

Features

• Smooth coloring with $n^{\prime}=n-\log_P\left(\log|Z_n|\right)$ where $n$ is the first iteration number after $|Z_n| \geq 2$
• Fully customizable equation in GLSL syntax, supporting 10 different complex defined functions
• Normal vector calculation for Lambert lighting
• 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 sequence creation

2024_Jan_13_22_31_47_19.mp4

2024-05-22.00-45-10.mp4

Custom equations

The expression in the inputs are directly substituted into the GLSL shader code. Because double-precision bivectors are used, most of the built-in GLSL functions are unavailable; and because vector arithmetic such as multiplication or division are component-wise, the following list of custom implemented functions have to be used instead:

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)	$\cos(z)$

Local variables

You can use these variables in the custom equation however you want

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 degree	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

The first input (dvec2) is the new value of $Z_{n+1}$ in each next iteration. The second input (bool) is the condition which when true the current pixel will be considered inside the set. The third input (dvec2) is $Z_0$.

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

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.

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!