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!

Yilmaz4/MV2