Calculate a dynamic iteration value when zooming into a Mandelbrot

Question

I'm trying to figure out how to automatically adjust the maximum iteration value when moving around in the Mandelbrot fractal.

All examples I've found uses a constant of 1000 or less but that's not enough when zooming into the fractal set.

Is there a way to determine the number of max_iterations based on for example where you are in the Mandelbrot space (x_start,x_end,y_start,y_end)?

Answer 1

One method I tried was to repetitively pre-process a small area in the region of the Mset boundary with increasing iterations until the percentage change in status from one repetition to the next was small. The problem was, that would vary in different places on the current map, since the "depth" varies across it. How to find the right place to do it? By logging the "deepest" boundary area during the previous generation (that will still be within the next zoom area).

But my best strategy was to avoid iterating wherever possible:

Away from the boundary of the Mset, areas of equal depth can be "contoured" and then filled with that depth. It was not an easy algorithm. Basically I followed a raster scan but when I detected a boundary of iteration change (examining all the neighbours to ensure I wasn't close the the edge of the Mset), I would switch to a curve-stitching method to iterate around a contour back to where it started (obviously not recalculating spots I already did), and then make a second pass filling in the raster lines within the countour with the iteration level. It was fraught with leaks but eventually I cracked it.

Within the Mset, I followed the same approach, because the very last thing you want to do is to plough across vast areas and hit the iteration limit.

The difficult area is close the the boundary, where the iteration results can't be related to smooth contours with the neighbours. The contour stitching method won't work here, since there is only ever 1 pixel of a particular depth.

Using the contour method also will have faults to the lower or Mset sides of this region, but since this area looks chaotic until you zoom deeper, I lived with that.

So having said all that, I simply set the iteration depth as high as I can tolerate, but perhaps you can combine my first paragraph with the area-filling techniques.

BTW colouring the region adjacent to the Mset looks terrible when an animated smooth playback of the zoom is attempted. For that reason I coloured this area in a grey scale, by comparing with neighbours. If there was too much difference, I coloured to 0x808080 at first, then adapted that depending on the predominance of the neighbours' depth. All requiring fine tuning!