I'm working on rendering the Mandelbrot set using K. I. Martin's method (http://www.superfractalthing.co.nf/sft_maths.pdf), and I am able to successfully use it to approximate the values of points in the set under iteration. However, now I have found myself faced with a small problem that I can't seem to get around.
Usually points not in the Mandelbrot set are colored based on the number of iterations it takes for them to escape, but since I only have access to the value of the points after a certain number of iterations and not the number of iterations it would have taken those points to escape, I have no way of creating the usual image of the Mandelbrot set.
I've tried using methods like taking one divided by the absolute value of the point (since larger points generally would have diverged faster), but methods like these leave me with pictures like this:
So, my question is, how can you color a point in the Mandelbrot set knowing only the value of that point after a certain number of iterations and not the number of iterations it took to diverge?