File:Rabbit Julia set with spine.svg
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Summary
DescriptionRabbit Julia set with spine.svg |
English: Rabbit Julia set with spine |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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Maxima CAS src code
/*
batch file for Maxima CAS
maxima
batch("r.mac")
*/
start:elapsed_run_time ();
kill(all);
remvalue(all);
/* --------------------------definitions of functions ------------------------------*/
f(z,c):=z*z+c; /* Complex quadratic map */
finverseplus(z,c):=sqrt(z-c)$
finverseminus(z,c):=-sqrt(z-c)$
/* */
fn(p, z, c) :=
if p=0 then z
elseif p=1 then f(z,c)
else f(fn(p-1, z, c),c)$
/*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */
F(p, z, c) := fn(p, z, c) - z $
/* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/
G[p,z,c]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(F(p,z,c)),
for i in f do
t:t*G[i,z,c],
g: F(p,z,c)/t,
return(ratsimp(g))
)$
GiveRoots(g):=
block(
[cc],
cc:bfallroots(expand(%i*g)=0),
cc:map(rhs,cc),/* remove string "c=" */
cc:map('float,cc),
return(cc)
)$
/* endcons the complex point to list in the format for draw package */
endconsD(point,list):=endcons([realpart(point),imagpart(point)],list)$
consD(point,list):=cons([realpart(point),imagpart(point)],list)$
GiveForwardOrbit(z0,c,iMax):=
/*
computes (without escape test)
forward orbit of point z0
and saves it to the list for draw package */
block(
[z,orbit,temp],
z:z0, /* first point = critical point z:0+0*%i */
orbit:[[realpart(z),imagpart(z)]],
for i:1 thru iMax step 1 do
( z:expand(f(z,c)),
orbit:endcons([realpart(z),imagpart(z)],orbit)),
return(orbit)
)$
/* Gives points of backward orbit of z=repellor */
GiveBackwardOrbit(c,repellor,zxMin,zxMax,zyMin,zyMax,iXmax,iYmax):=
block(
hit_limit:4, /* proportional to number of details and time of drawing */
PixelWidth:(zxMax-zxMin)/iXmax,
PixelHeight:(zyMax-zyMin)/iYmax,
/* 2D array of hits pixels . Hit > 0 means that point was in orbit */
array(Hits,fixnum,iXmax,iYmax), /* no hits for beginning */
/* choose repeller z=repellor as a starting point */
stack:[repellor], /*save repellor in stack */
/* save first point to list of pixels */
x_y:[repellor],
/* reversed iteration of repellor */
loop,
/* pop = take one point from the stack */
z:last(stack),
stack:delete(z,stack),
/*inverse iteration - first preimage (root) */
z:finverseplus(z,c),
/* translate from world to screen coordinate */
iX:fix((realpart(z)-zxMin)/PixelWidth),
iY:fix((imagpart(z)-zyMin)/PixelHeight),
hit:Hits[iX,iY],
if hit<hit_limit
then
(
Hits[iX,iY]:hit+1,
stack:endcons(z,stack), /* push = add z at the end of list stack */
if hit=0 then x_y:endcons( z,x_y)
),
/*inverse iteration - second preimage (root) */
z:-z,
/* translate from world to screen coordinate, coversion to integer */
iX:fix((realpart(z)-zxMin)/PixelWidth),
iY:fix((imagpart(z)-zyMin)/PixelHeight),
hit:Hits[iX,iY],
if hit<hit_limit
then
(
Hits[iX,iY]:hit+1,
stack:endcons(z,stack), /* push = add z at the end of list stack to continue iteration */
if hit=0 then x_y:endcons( z,x_y)
),
if is(not emptyp(stack)) then go(loop),
return(x_y) /* list of pixels in the form [z1,z2] */
)$
/*-----------------------------------*/
Psi_n(r,t,z_last, Max_R):=
/* */
block(
[iMax:200,
iMax2:0],
/* ----- forward iteration of 2 points : z_last and w --------------*/
array(forward,iMax-1), /* forward orbit of z_last for comparison */
forward[0]:z_last,
i:0,
while cabs(forward[i])<Max_R and i< ( iMax-2) do
(
/* forward iteration of z in fc plane & save it to forward array */
forward[i+1]:forward[i]*forward[i] + c, /* z*z+c */
/* forward iteration of w in f0 plane : w(n+1):=wn^2 */
r:r*2, /* square radius = R^2=2^(2*r) because R=2^r */
t:mod(2*t,1),
/* */
iMax2:iMax2+1,
i:i+1
),
/* compute last w point ; it is equal to z-point */
R:2^r,
/* w:R*exp(2*%pi*%i*t), z:w, */
array(backward,iMax-1),
backward[iMax2]:rectform(ev(R*exp(2*%pi*%i*t))), /* use last w as a starting point for backward iteration to new z */
/* ----- backward iteration point z=w in fc plane --------------*/
for i:iMax2 step -1 thru 1 do
(
temp:float(rectform(sqrt(backward[i]-c))), /* sqrt(z-c) */
scalar_product:realpart(temp)*realpart(forward[i-1])+imagpart(temp)*imagpart(forward[i-1]),
if (0>scalar_product) then temp:-temp, /* choose preimage */
backward[i-1]:temp
),
return(backward[0])
)$
GiveRay(t,c):=
block(
[r],
/* range for drawing R=2^r ; as r tends to 0 R tends to 1 */
rMin:1E-10, /* 1E-4; rMin > 0 ; if rMin=0 then program has infinity loop !!!!! */
rMax:2,
caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */
/* upper limit for iteration */
R_max:300,
/* */
zz:[], /* array for z points of ray in fc plane */
/* some w-points of external ray in f0 plane */
r:rMax,
while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */
R:2^r,
w:rectform(ev(R*exp(2*%pi*%i*t))),
z:w, /* near infinity z=w */
zz:cons(z,zz),
unless r<rMin do
( /* new smaller R */
r:r*caution,
R:2^r,
/* */
w:rectform(ev(R*exp(2*%pi*%i*t))),
/* */
last_z:z,
z:Psi_n(r,t,last_z,R_max), /* z=Psi_n(w) */
zz:cons(z,zz)
),
return(zz)
)$
/*
converts complex number z = x*y*%i
to the list in a draw format:
[x,y]
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$
ToPoints(myList):= points(map(d,myList))$
/* give Draw List from one point*/
ToPoint(z):=points([d(z)])$
GiveSpine(zc, Alfa,c, iMax):=block(
[s, center, cut],
/* first center of component, only one !!! */
center: zc,
s:[center],
/* first pair of cut points */
s: cons( Alfa,s),
cut : -Alfa,
s:endcons( cut,s),
for i: 1 thru iMax step 1 do (
/* pair of component's centers */
center: finverseminus(center,c),
s: cons(center, s),
center : - center,
s: endcons (center, s),
/* pair of cut points */
cut : finverseminus(cut,c),
s: cons( cut,s),
cut : - cut,
s:endcons( cut,s)
),
/* convert to draw format and return list */
s: ToPoints(s)
)$
compile(all)$
/* ----------------------- main ----------------------------------------------------*/
period:3$
/* external angle in turns */
/* resolution is proportional to number of details and time of drawing */
iX_max:1000$
iY_max:1000$
/* define z-plane ( dynamical ) */
ZxMin:-2.0$
ZxMax:2.0$
ZyMin:-2.0$
ZyMax:2.0$
/* give c a value */
c:0.74486176661974*%i-0.12256116687665$ /* center of period 3 component */
/* compute fixed points */
Beta:float(rectform((1+sqrt(1-4*c))/2))$ /* compute repelling fixed point beta */
alfa:float(rectform((1-sqrt(1-4*c))/2))$ /* other fixed point */
/* compute backward orbit of repelling fixed point */
xy: GiveBackwardOrbit(c,Beta,ZxMin,ZxMax,ZyMin,ZyMax,iX_max,iY_max)$
/* compute ray points & save to zz list */
eRayZero:GiveRay(0/1,c)$
eRay1o2:GiveRay(1/2,c)$
spine: GiveSpine(0, alfa,c, 4)$
/* time of computations */
time:fix(elapsed_run_time ()-start)$
/* draw it using draw package by */
load(draw)$
path:"~/maxima/batch/julia/spine/rabbit/"$ /* if empty then file is in a home dir */
/* if graphic file is empty (= 0 bytes) then run draw2d command again */
draw2d(
terminal = 'svg,
file_name = sconcat(path,"rabbitSpine_"),
user_preamble="set size square;set key top right",
title= concat("Dynamical plane for fc(z)=z*z+",string(c)),
dimensions = [iX_max, iY_max],
yrange = [ZyMin,ZyMax],
xrange = [ZxMin,ZyMax],
xlabel = "Z.re ",
ylabel = "Z.im",
point_type = filled_circle,
points_joined =true,
point_size = 0.2,
color = red,
points_joined =false,
color = black,
key = "backward orbit of z=beta",
points(map(realpart,xy),map(imagpart,xy)),
points_joined =true,
point_size = 0.2,
color = red,
key = "external ray 0",
ToPoints(eRayZero),
key = "external ray 1/2",
color = magenta,
ToPoints(eRay1o2),
points_joined =true,
point_size = 0.8,
color = gray,
key = "spine",
spine,
points_joined =false,
color = black,
point_size = 1.4,
key = "critical point z = 0.0",
ToPoint(0.0),
color = red,
point_size = 1.4,
key = "repelling fixed point z= beta",
ToPoint(Beta),
color = magenta,
key = "minus beta",
ToPoint(-Beta),
color = yellow,
key = "attracting fixed point z= alfa",
ToPoint(alfa)
)$
Items portrayed in this file
depicts
some value
6 January 2019
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 10:38, 6 January 2019 | 1,000 × 1,000 (1.45 MB) | Soul windsurfer | User created page with UploadWizard |
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Short title | Gnuplot |
---|---|
Image title | Produced by GNUPLOT 5.3 patchlevel 0 |
Width | 1000 |
Height | 1000 |
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