The Normal Mandelbrot

The Mandeblort figure ist computed by the following series, wheres the real part of c is the x and the imaginary part the y axis of the plot. This series can either converge or diverge. It can be shown, that if a intermediate value is grater then 2, the series will diverge. The colour indicates, how many steps are required, until divergence can be detected.

\[z_n=z_{n-1}^2 + c\]

Manipulating z0

For most mandelbrot figures, the start value $z_0$ is set to 0. But its not clear, if this has a non trivial impact (scaling, shift, rotate) to the overall image. So i wrote a mandelbrot program based on OpenCL and added some keymappings to manipulate $z_0$. The program will interpolate between two set of parameters and render the image as ppm images, which can then be converted to a video.

Sweeping the real part of $z_0$ from -1.6 to 1.6

Sweeping the imaginary part of $z_0$ from -1.2 to 1.2

Rotating

Lets call $c.i$ the imaginary and $c.r$ the real part of $c$. A Point in the 4-D Mandelbrot space including the $z_0$ starting value can then be written as $(c.r, c.i, z_0.r, z_0.i)$. A normal mandelbrot will use the direction vector (1,0,0,0) for the X axis and (0,1,0,0) for the Y axis. Instead of viewing the mandelbrot in the $c.r$ $c.i$ plane, we can also try to rotate this plane. In this case starting from a fixed point, I used $X=(cos(\phi),0,sin(\phi),0)$ and $Y=(0,cos(\phi),0,sin(\phi))$ to drwa the mandelbrot image.

Sweeping $\phi$ from 0 to $\pi$ with $(1,0,0,0)$ as start:

Implementation

The Implementation is using OpenCL for computing the mandelbrot values and GLUT for drawing the image (source) The following key mappings are used to controll the Mandelbrot:

Key		Meaning
Arrows		Move in X/Y Plane
1/q		Modify c.r
2/w		Modify c.i
3/e		Modify z0.r
4/r		Modify z0.i
5/t		Modify $\phi$
7/u		Modify Max Steps
+/-		Zoom
k		Store Parameters
0		Reset to last stored position
.		Render From last stored position
,		Render From last stored position and store images in /tmp/out
;		Screenshot