First : thx for Inigo Quilez for help and many informations. (:-))

G

where:

M is a Mandelbrot set

G

C is a complex plane

C-M is a complement of M to C

c is a complex number = point of complex plane C

R is a

G

where n --> infinity

If c is a member of M

then G

else G

g

Take a

Take a scalar function of point position

z=F(x,y)

Then one have

If field has no rotational component then F(x,y) is

And field is irrotational = conservative vector field <==> curl of that vector field is zero

There are an infinite number of potential functions that lead to the same field,

Usually F(x,y) is 0 at infinity but for Mandelbrot set it is 0 in the interior of that set.

G

Arnaud Chéritat uses negative vallues for potential of M-set in his program DH_Drawer .

Field lines are normal to equipotential lines.

Field lines of potential are external rays

This images are drawn using simplified definition :

Potential ( c, iter) := 0.5 * log2(hypot(Zn))/IntPower(2, iter);

where Zn:= Fc ( Z0) and Z0 := 0

g:=MSetPotential(Zx,Zy,Iteration);

- first image: bG:= round(255 - 255*g );
- second image: bG:= round(255*g );

begin

B := bG;

G := bG;

R := bG;

//A := 0;

end ;

One can apply

theory:

leves set is a set of points of exterior of M-set for which potential is greater then 2

and smaller or equel to 2

L

how to compute it in delphi:

r:= log2(abs(potential));

k:=ceil(r);

The images of Level Set Method for potential ( pLSM )

are similar to LevelSetMethod for escape time ( eLSM).

All above images are made with my program Mandelbrot set explorer.

- CPM/J
- Hubbard Douady Potential by Inigo Quilez
- Hubbard Douady Potential, Field Lines by Inigo Quilez
- Douady-Hubbard Potential by Linas Vepstas
- Continuous Potential in Fractint
- Periodic Orbits, Externals Rays And The Mandelbrot Set: An Expository Account by John Milnor
- CPM/M by ???
- Douady-Hubbard Potential - discussion on sci.fractals
- Drawing M-set by contour lines method - sci.fractals discussion
- Polarizability of 2D monster and light scattering M.V. Entin and G.M. Entin
- John H. Ewing and Glenn Schober
On the coefficients of the mapping to the exterior of the Mandelbrot set. Michigan Math. J. 37, iss. 2 (1990), 315–320

- Mandelbrot Set via Continuous Potential Method with detection of Periods 1 & 2, January 30th 1994 by Fausto Barbuto. Here is powl definition
- THE MANDELBROT SET Math by O. Knill

- Programs that can draw equipotential lines:
- DH_Drawer by Arnaud Chéritat. Use negative values of potential in the complement of M-set

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