File:Scalar field, potential of Mandelbrot set.svg
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Summary
| Description |
English: Scalar field: potential of Mandelbrot set |
| Date | |
| Source | Own work |
| Author | Adam majewski |
| Other versions |
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.svg)
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Maxima CAS src code
/*
Batch file for Maxima CAS
save as a o.mac
run maxima :
maxima
and then :
batch("quiver_plot.mac");
Potential(0.5624000299307079*%i+0.3023655896056563 );
=
0.5624000299307079*%i+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079*%i
+(0.5624000299307079
*%i
*/
kill(all);
remvalue(all);
ratprint:false; /* It doesn't change the computing, just the warnings. */
display2d:false;
numer: true;
/* functions */
/*
converts complex number z = x*y*%i
to the list in a draw format:
[x,y]
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$
log2(x) := float(log(x) / log(2))$
/*
point of the unit circle D={w:abs(w)=1 } where w=l(t)
t is angle in turns
1 turn = 360 degree = 2*Pi radians
*/
give_unit_circle_point(t):= float(rectform(%e^(%i*t*2*%pi)))$
/* circle points */
give_circle_point(center, Radius, t) := float(rectform(center + Radius*give_unit_circle_point(t)))$
/*
(scalar) potential
https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/MandelbrotSetExterior#Complex_potential
real potential
potential = log(modulus)/2^iterations
complex quadratic polynomial
https://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
Potential(c):= block(
[i, iMax, z, ER, n,t],
z:0,
i:0,
iMax : 1000,
n:1,
ER:1000,
t: cabs(z),
while i< iMax and t< ER
do
( /* print("i = ", i, "z = ", z), */
z : z*z+c, /* complex quadratic polynomial */
z :float(rectform(z)), /* Maxima encountered a Lisp error: Condition in MACSYMA-TOP-LEVEL [or a callee]: INTERNAL-SIMPLE-ERROR: Bind stack overflow. */
t : cabs(z),
n : n*2,
i : i+1
),
/* print("from Potential t = ", t), */
if floatnump(t) and t>0 and i<iMax
then t : log2(t)/n /* exterior = escaping */
else t : 0, /* interior and non escaping */
return (t)
)$
/*
P(x+y*%i):=Potential(x+y*%i)$
*/
/*
numerical aproximation of the potential ( scalar funtion) gradient
on the parameter plane ( c-plane)
https://en.wikipedia.org/wiki/Complex_quadratic_polynomial#Parameter_plane
https://commons.wikimedia.org/wiki/File:Gradient_of_potential.svg
input:
* n = number of points (on the circle) to check
Gradient vector can be descibed by the
* target point cMax. When the origin of the vector is known then target point describes the gradient vector
output is a complex point cMax
* on the circle with center = Center and radius = Radius
*
--------------
GradientPoint(0,0.1,2);
Unrecoverable error: bind stack overflow.
Przerwane
solved : if pCenter=0 then return (Center),
---------------------------------
*/
GradientPoint(Center, Radius, n) := block(
[
pCenter,
p,
dp , /* finite difference of potential between center and circle point */
dpMax, /* max dp */
c, /* point on the circle */
cMax, /* c : dp = dpMax */
t, /* angle in turns */
tMax,
dt /* angle step */
],
/* */
pCenter : Potential(Center),
/* print("pCenter = ", pCenter),*/
if is(pCenter=0)
then (
/* print("pCenter=0"),*/
return (Center) /* magnitude of gradient vector is 0 inside M set. It removes stack error */
),
dpMax : 0,
dt : 1/n,
cMax : Center, /* in case when Center is in the interior */
t : 0, /* start with t=0 ; it can be modified to start with previous direction ??? */
while (t < 1) do ( /* compute values (of c and dp) for all points on the circle, it can be modified to search in increasing direction and stop when decreasing */
c : give_circle_point(Center, Radius,t),
/* print ("c = ", c), */
p : Potential(c),
/* print("p= ", p), */
dp : p - pCenter, /* https://en.wikipedia.org/wiki/Finite_difference#Relation_with_derivatives */
if (dp > dpMax) then (
dpMax : dp,
cMax : c,
tMax : t
),
t : t + dt
),
/* knowing good direction one can check 2 more points around tMax: */
/* first point : tMax + dt/2 */
c : give_circle_point(Center, Radius,tMax + dt/2),
p : Potential(c),
dp : p - pCenter, /* https://en.wikipedia.org/wiki/Finite_difference#Relation_with_derivatives */
if (dp > dpMax) then (
dpMax : dp,
cMax : c,
tMax : t
),
/* second point tMax -dt/2 */
c : give_circle_point(Center, Radius,tMax - dt/2),
p : Potential(c),
dp : p - pCenter, /* https://en.wikipedia.org/wiki/Finite_difference#Relation_with_derivatives */
if (dp > dpMax) then (
dpMax : dp,
cMax : c,
tMax : t
),
return(cMax)
)$
/*
gives vector from c1 to c2
width = dp
using:
from draw package :
vector([x, y], [dx,dy])
http://maxima.sourceforge.net/docs/manual/maxima_52.html#Item_003a-vector
plots vector
* with width [dx, dy]
* with origin in [x, y].
*/
give_vector(c1, c2 ):=block(
[x,y,dx,dy ],
x: realpart(c1),
y: imagpart(c1),
/*
length = cabs(c2-c1)
angle is direction from c1 to c2
*/
dx : realpart(c2) - x,
dy : imagpart(c2) - y,
s : vector([x, y], [dx,dy])
)$
/*
return list of complex points z
*/
GiveGridList(xMin, xMax, ixMax, yMin, yMax, iyMax):=block(
[xx, dx, x_step,
yy, dy, y_step,
GridList, z],
dx : xMax - xMin,
x_step : dx/ixMax,
dy : yMax - yMin,
y_step : dy/iyMax,
GridList : [],
xx : makelist(xMin+i*x_step, i, 0, ixMax ), /* list length = iMax+1 */
yy : makelist(yMin+i*y_step, i, 0, iyMax ), /* list length = iMax+1 */
for iy : 1 thru length(yy) step 1 do
for ix : 1 thru length(xx) step 1 do
(
z : xx[ix]+yy[iy]*%i,
GridList : endcons(z, GridList)
),
return(GridList)
)$
/*
input :
* cc = list of complex points
* Radius =
* n =
output : list of vectors for draw package
*/
GiveVectors(cc, Radius, n):= block (
[g, v, vv],
vv:[],
for c in cc do (
/* print("c = ", c),*/
g : GradientPoint(c, Radius, n),
/*print("g = ", g),*/
if g#c
then ( v : give_vector(c, g),
vv : cons(v, vv)
)
),
return (vv)
)$
/* ==================================== ===================================*/
NumberOfSteps : 42;
/* world coordinate, note that dx = dy */
xMin : -2.5 $
xMax : 0.5 $
dx: xMax - xMin$
yMin : -1.5 $
yMax : 1.5 $
dy : yMax - yMin$
/* for screen coordinate see dimensions */
NumberOfVectors : 100$
Radius : (xMax - xMin)/(2*NumberOfSteps)$
cc : GiveGridList(xMin, xMax,NumberOfSteps, yMin, yMax, NumberOfSteps)$
/* */
potential_list : map(Potential, cc)$
potential_array : make_array(flonum, NumberOfSteps+1, NumberOfSteps+1)$ /* 2d array */
potential_array : fillarray (potential_array, potential_list )$
arrayinfo(potential_array);
potential_matrix: genmatrix(potential_array,NumberOfSteps,NumberOfSteps)$
dd : map(d,cc)$ /* convert grid points to draw format */
vv : GiveVectors(cc, Radius, NumberOfVectors )$
/* strings */
path:"~/c/mandel/p_e_angle/trace_last/test5/q/q2/"$ /* if empty then file is in a home dir , path should end with "/" */
FileName: string(NumberOfSteps)$
load(draw)$
/* http://riotorto.users.sourceforge.net/Maxima/gnuplot/index.html */
draw2d(
terminal = svg,
file_name = sconcat(path, FileName, "_a"),
title = "grid points of rectangular mesh",
dimensions = [1000, 1000],
xlabel = "cx ",
ylabel = "cy",
point_type = filled_circle,
point_size = 0.5,
color = black,
key = "",
points(dd)
)$
draw2d(
terminal = svg,
file_name = sconcat(path, FileName,"_b"),
title = "Scalar field: potential of Mandelbrot set",
dimensions = [1000, 1000],
xlabel = "cx ",
ylabel = "cy",
image(potential_matrix,xMin,yMin, dx, dy)
)$
draw2d(
terminal = svg,
file_name = sconcat(path, FileName,"_c"),
title = "Vetor field : gradient of Mandelbrot set potential",
dimensions = [1000, 1000],
xlabel = "cx ",
ylabel = "cy",
nticks = 200,
key = "",
/* key="main cardioid", */
parametric( cos(t)/2-cos(2*t)/4, sin(t)/2-sin(2*t)/4, t,-%pi,%pi),
/* key="period 2 component", */
parametric( cos(t)/4-1, sin(t)/4, t,-%pi,%pi),
head_length = Radius/10,
color = black,
vv
)$
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 21:03, 18 August 2018 | | 1,000 × 1,000 (30 KB) | Soul windsurfer | User created page with UploadWizard |
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