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Nova fractals

In the previous post we looked at creating Newton fractals. We can generalize the method even further, which is then known as a Nova fractal.

The "Mandelbrot" version of the Nova fractal, with $f(z) = z^3 - 1$, and $z=1$.

The Nova fractal is created by using Newton's method, and adding a parameter $c$ to the end: $$ z_{n+1} = z - \frac{f(z)}{f'(z)} + c. $$ We can get a "Julia" version of the fractal if we have fixed value for $c$, and use $z$ for the pixel position. Likewise, we can also get a "Mandelbrot" version by setting $z$ to a fixed point, and using $c$ for the pixel position. Note that the value of $z$, should be a stable point, e.g. a root.

To get the image that is displayed above, the function $f(z) = z^3 - 1$ is used, which has a stable point at $z = 1$. This gives the function: $$ z_{n+1} = z - \frac{z^3-1}{3z^2} + c, $$ or in more general terms, where $p$ indicates the power: $$ z_{n+1} = z - \frac{z^p-1}{pz^{p-1}} + c. $$ To color this, we will keep track of how many iterations it takes before $z$ reaches a root, which is when $f(z) = 0$, or when $z_{n+1}-z_n = 0$, meaning that we haven't moved this iteration.

The live demo below is available on ShaderToy. It will display the "Mandelbrot" version of the Nova fractal, and if the mouse is used the "Julia" version is displayed.

Another variation can be found if we set $z_0$ to $c$. This produces a more regular version, which can be seen in the image below.

Nova fractal for $f : z \rightarrow z^4 - 1$, and $z_0 = c$.

Now all that is missing is a method to get a smooth iteration count. I will come back to this post once I found it, and update the images.

References

Most of the references are links for possible solutions to a smooth iteration count I'm still researching.
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