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" instead of iterating a point through a nice fractal-generating function until it exits the containing circle, I'm starting with the containing circle's function (2cos(t),2sin(t)) and iterating that circle function through the inverse of the fractal-generating function." Axis Angels[1]
If ER = 2 curves meet together ( have common point) c = −2. Thus they can't be equipotential lines.
If ER ≥ 2 they are outside of Mandelbrot set. They can also be drawn using Level Curves Method.
If ER >> 2 they aproximate equipotential lines ( level curves of real potential , see CPM/M ).
Maxima source code
/* based on the code by Jaime Villate */
load(implicit_plot); /* package by Andrej Vodopivec */
c: x+%i*y;
ER:2; /* Escape Radius = bailout value it should be >=2 */
f[n](c) := if n=1 then c else (f[n-1](c)^2 + c);
ip_grid:[100,100]; /* sets the grid for the first sampling in implicit plots. Default value: [50, 50] */
ip_grid_in:[15,15]; /* sets the grid for the second sampling in implicit plots. Default value: [5, 5] */
my_preamble: "set zeroaxis; set title 'Boundaries of level sets of escape time of Mandelbrot set'; set xlabel 'Re(c)'; set ylabel 'Im(c)'";
implicit_plot(makelist(abs(ev(f[n](c)))=ER,n,1,6), [x,-2.5,2.5],[y,-2.5,2.5],[gnuplot_preamble, my_preamble],
[gnuplot_term,"png size 1000,1000"],[gnuplot_out_file, "lemniscates6.png"]);
For curves 1-5 it works, but for curve number 6 this program fails( also Mathematica program[7]), because of floating point error.
One have to change the method of computing lemniscates . Here is the code and explanation by Andrej Vodopivec"
"You can trick implicit_plot to do computations in higher precision. Implicit_draw will draw the boundary of the region where the function
has negative value. You can define a function f6 which computes the sign of f[6] using bigfloats and then plot f6."
/* based on the code by Jaime Villate and Andrej Vodopivec*/
c: x+%i*y;
ER:2;
f[n](c) := if n=1 then c else (f[n-1](c)^2 + c);
F(x,y):=block([x:bfloat(x), y:bfloat(y)],if abs((f[6](c)))>ER then 1 else -1);
fpprec:32;
load(implicit_plot); /* package by Andrej Vodopivec */
ip_grid:[100,100];
ip_grid_in:[15,15];
implicit_plot(append(makelist(abs(ev(f[n](c)))=ER,n,1,5), ['(F(x,y))]),[x,-2.5,2.5],[y,-2.5,2.5]);
Questions
What is mathemathical description of these curves ?
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