So I'm relatively new to fractals (I've just gotten into the Mandelbrot set in the past few months) and I'm trying to make a reasonably fast fractal program capable of deep zooms. I came across K. I. Martin's SuperFractalThing paper (http://www.superfractalthing.co.nf/sft_maths.pdf) and I decided this would be the best way to go. I've successfully implemented his algorithm, but in actually rendering the Mandelbrot set I've come across a small problem.

Usually, points not in the Mandelbrot set are colored based on the number of iterations it takes them to escape. However, with the method I am using, this is impossible because it does not calculate these points iteratively (except for the reference point). Since I only have access to the values of the points after a certain number of iterations, and not the iterations that it would take these points to escape, I have no way of coloring points outside of the set. Obviously I can just color all points not in the Mandelbrot set the same color and still get a good picture of the set itself, but I would like to be able to see the Julia sets and other interesting patterns that show up when the points outside the set are colored.

I know there's a way to do this, as I have seen K. I. Martin's algorithm implemented before (I have his SuperFractalThing program), but I can't seem to figure out to do it from the information I have access to, and I figured someone on here might be able to help me.

Thanks!

Linkback: https://fractalforums.org/index.php?topic=1050.0

Thanks! I’ll go check it out!

Thanks! That actually makes a lot of sense!

I just have a couple of questions about the implementation. First of all, how would I know how many iterations to skip before the picture becomes too inaccurate (it seems to me from some experimentation that the series approximation tends to diverge faster than using the normal method)? And separately, I’m assuming I calculate the reference orbit with high precision, so I don’t see how I would be able to gain any performance speedup if I’m comparing low precision deltas to the high precision orbit (wouldn’t that just be a high precision calculation?). I’m also kind of curious about how to choose a good reference orbit, but I think I can figure that out myself by experimenting/ looking at other forum posts.

Sorry for all the questions, I’m a noob to Mandelbrot rendering and I’m trying to understand this topic as best I can.