Question on Distance estimator algorithms from Science of Fractal Images

[Duncan C]

Greetings all.

A while back I posted a question on how the black and white Mandelbrot and Julia images from "The Beauty of Fractals" were generated.

I've since found documentation in "The Science of Fractal Images" that states that these images were generated using the Distance Estimator Method (or DEM) as documented on pages 196 - 199.

The algorithm involves calculating the derivative of the iterated function

   Zn+1 = Zn+ C

That section includes separate mathematical definitions of DEM algorithms for Mandelbrot and Julia sets. The text also includes pseudo-code for the Mandelbrot version of the DEM algorithm, but no pseudo-code for the Julia version

It's been a LONG time since I took calculus, and the notation used in these texts is rather hard to follow, at least for me.

As far as I can tell, the only differences between the Mandelbrot and Julia versions of the DEM functions is that the Mandelbrot function has a "+1" term in the iterated derivative function, while the Julia version does not, and the derivative of the first Z0 iteration, Z'0, is 0 for the Mandelbrot version of the equation, and Z'0 is 1 for the Julia version of the equation.

The formula for the distance estimate of a Mandelbrot, where Zk is the result of the K iteration of the iterated function, is as follows:

   Z'k+1 = 2  Zk  Z'k +1, Z'0 = 0, k = 0, 1, ..., n-1

For Julia set DEM, the formula is

   Z'k+1 = 2  Zk  Z'k, Z'0 = 1, k = 0, 1, ..., n-1

Can you folks look over the algorithms (DEM M on page 196 and DEM/J on page 199) and see if there are any differences I'm missing?

Have any of you used these DEM algorithms to color plots, or as height maps for 3D plots? As far as I can tell, most of the 3D plots in "The Beauty of Fractals" and "The Science of Fractal Images" use the Continuous Potential Method (which is based on the log-log of the result of the result of the last iteration.)

The only example of a 3D plot that uses the DEM algorithm (map 30, on page 121 of "The Science of Fractal Images") looks like it only uses 2 height values, "far from the set" and "near to the set."




Duncan C