DEM/M





DEM/M =Distance Estimation Method for Mandelbrot set = Milnor algorithm


Idea:
In typical algorithms coordinate of pixel ( integer value = screen coordinate) is matched with one value of world coordinate ( real number),
but between 2 real values there are infinitely many other real values.
If narrow part of M-set ( dendrit or filaments) is between 2 values it is a little chance that it would be drawn by such typical algorithm.
So we can :

Programers point of view:
This algorithm is based on escape time algorithm:
Each pixel ( c ) is iterated ( Z0:=0 ) to compute Zn, DZn ( first derivative with respect to c) and last iteration.
If point is not in madelbrot set ( LastIteration < iterationMax) then compute distance from that point to M-set.
Distance = F ( Zn, DZn)
if distance< distanceMax then draw a pixel ( = border of M-set).

Theory:
the approximate distance between the point c and the nearest point in the Mandelbrot set

distance = 2 * |Zn| * log|Zn| / |dZn|

where: Zn = F(n)(Z0) (complex number)
dZn is first derivative of the Mandelbrot map with respect to c ( complex number)
|Zn| = magnitude(Zn)


format of floating point numbers = extended
iterationMax:=1000;
excape_Radius:=2
distanceMax:= exp(-9)/100000

This image is made with delphi program, which is translation ( with modification) of Mandelbrot Dazibao program in QBasic
It is slow program, but works.
It draws only border of M-set.
It means points for which:

This image was drawn with MSExplorer

How to compute DistanceMax ?

This value is obtained empirically and is different than in Mandelbrot Dazibao program.
probably because of used type of floating point numbers( I use extended ).

DistanceMax is proportional to width of "border" of M-set;

distanceMax:= ePixelSize;
where ePixelSize:= (eCxMax-eCxMin)/(iXmax-iXmin);

This procedure is time consuming, because more computations in main loop ( derivatives).
What can we do to make it faster?
Is it worth trying ?
30 times faster !!!

There are a few versions/modifications of this algorithm:
Questions :

What is the relation between distance (DEM) and distance between 2 points ?

" For distance estimate it has been proved that the computed value differs from the true distance at most by a factor of 4. " (Wolf Jung)

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Author: Adam Majewski
adammaj1-at-o2-dot-pl   About

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