I'm sure this has been done, and there are probably other ways to do it also, but I've never found any info on how to do it.
Points that are inside the mandelbrot set are usually colored black. But that's boring! Here is an easy way to get a value for ALL points.
Just keep track of the distance that z moves while iterating z = z^2+c
Then use the total distance each point traveled to look up a color.
The minibrots that once all looked the same have a little more character
There are two other differences with using distance instead of iteration count.
With the usual iteration count, you will see bands of a certain value, so you end up with a solid color in that area. But with distance the values you get are a little more unique.
The other advantage is with points in the set that would escape to infinity but don't because your bail out value wasn't high enough.
Normally these would be solid colored like points that are a part of the set, but when using distance you get some pretty interesting detail.
This picture will demonstrate both. On the left is iteration based coloring and on the right is distance based. The bail out is 500 iterations for both.
The only problem I ran into is that the distances found for points inside the set vary greatly from the points outside the set. This would make it hard to do a straightforward lookup into a color table.
There are a few easy ways to fix this.
You can multiply the distances found of points in the set by one number and points outside the set by another number. Something like ~5 for outside points and 100+ for inside points.
Another thing you can do is to only apply the distance based coloring to points that reach the bail out, then color the rest of them like normal. This can actually be better for some images. Because the distance based method gives more unique values, you can sometimes have too much contrast and a chaotic look in areas where the distances change quickly.
Here is a hybrid using distance and iteration based coloring
This way shows less detail but has smoother transitions between different areas.
One last picture. Purely distance based coloring.
Feel free to post some pics if you have ever made fractals like this, or know other ways to give value to the points inside the set. I'd like to see how it looks with other types of color palettes.