Darren New <dne### [at] sanrrcom> wrote:
> Figuring out what color to make one pixel near the end of the video doesn't 
> take more iterations than figuring out what color to make one pixel near the 
> start of the video?

  It does.

  The farther away you are from the border of the set, the fewer iterations
it takes for the complex numbers to bail out (ie. to exit the circle of
radius 2, which is usually what's used for this). The closer you get to
the border of the set, the more iterations it takes for it to bail out.

  If you do too few iterations, points very close to the set will look like
they belong to the set even though they don't, and thus get wrongly colored
like the interior of the set (in other words, all color bands which are
closer to the set than the iteration amount will get colored black).

  Zooming into the Mandelbrot set usually means zooming to some point which
is extremely close to the border of the set (any other place would be boring
because you would get a flat color after a little while). Zooming to a point
which is extremely close to the set means in practice that you are approaching
the border of the set, and hence you need larger iteration counts in order
to get the correct result. (If you don't increase the iteration count
appropriately, you end up having a wrongly-colored black image.)

  As I said in my earlier post, with the classical coloring each color
band represents one iteration (in other words, all the pixels of that
color took the same amount of iterations to bail out). The pixels of a
color band immediately inside another color band took 1 iteration more
to calculate. If you count the total number of distinct color bands,
you will get how many iterations it took for (each of the pixels of) the
innermost color band to calculate.

  In the video I estimate this count is in the millions.

> It sounds like you're saying that a 1 minute video takes fewer iterations 
> than a 10 minute video. Yes, that seems obvious.

  No, I'm talking about individual images. The amount of iterations needed
to get a proper result depends on the zoom level.

> But does drawing a 640x480 pixel section of the mandelbrot set (on the 
> border of course) that's 0.0001 on a size take fewer iterations than drawing 
> a 640x480 pixel section of the mandelbrot set that/s 0.000000000000001 on a 
> side?

  Yes, if you are looking at the border of the set (and you want a result
which is correct).

> >   It's more like you don't *need* so many iterations for a 2^30 zoom to
> > look good as you need for a 2^70 zoom. 

> I don't see how this works out, tho. If you have 256 colors, you (could) do 
> 256 iterations for each pixel, regardless of the mapping of the cartesian 
> plane to the screen, yes?

  The palette size doesn't affect how many iterations you need to perform
per pixel in order to get a correct image. It's simply that the iteration
count (before bailout) is mapped to the color palette, and you can do that
however you wish.

-- 
                                                          - Warp

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