The boundary of the Mandelbrot Set is the limit of the Nth lemniscate

Lemniscate = contour line = border of level set = level curve

From the Mandelbrot Set Glossary and Encyclopedia, İ 1987-2004 Robert P. Munafo.

z:0;

for i:1 thru 6 do

(

z:z^2+c,

print(expand(z)),

print(factor(z))

);

Miguel Romera gives an algorithm for drawing contour lines of M-set.

According to Miguel Romera definitions contour lines may be :

- escape lines ( for escape radius = 2 ); lines meet at point (-2,0) so thay can't be equipotential lines because each point shuld have only one value of potential
- "equipotential" lines ( for escape radius >2 ); lines don't meet

thera are 2 tricks

- ( 2 coats of paint). For description see 2D curves page
- mirror symmetry around x axis ( M Romera program uses it only in case y:=0 is in the middle of image)

- using Pixels property - slow but easy to read: Delphi unit and whole program
- using ScanLine method without zoom Delphi unit and whole program
- using ScanLine method with zoom program
- version 2006.02.11 not finished, only contour lines method, only one dialog box
- version 2006.03.07 3 dialog boxes
- version 2006.03.08 new procedures
- version 2006.03.13 potential ( not finished)
- for new versions see Mandelbrot set explorer

and here is image made with M Romera procedure:

As Inigo Quilez says:

This "method is basically equivalent to drawing a normal escape time picture of Mandelbrot set,

and then detectingthe edges with sobel or lapalacian filter".

Also one can create " these images with the open source fractal tool Xaos, using the mandelbrot formula and the edge detection filter".

- MathWorld
- Mandelbrot lemniscate by jan wassenaar
- M. Romera et al : Graphic tool to analyse one-dimensional quadratic maps. Comput & graphics v. 20 No 2, pp 333-339
- Mu-ency
- Coefficents of Lemniscates from Mu-ency
- Integer sequence
- Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
- Fractal Plotter by Gonzalo Álvarez Maran~ón
- Drawing equipotential lines - c procedure: mandel.c by Stanley A. Wileman, Jr.
- Drawing M-set by contour lines method -discussion on sci.fractals
- Mandelbrot Set Lemniscates on sci.math

Autor: Adam Majewski

adammaj1-at-o2-dot-pl

Feel free to e-mail me. (:-))

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