File:Erays.png

DescriptionErays.png	English: polar coordinate system and mapping from the complement (exterior) of the closed unit disk to the complement of the filled Julia set for c=-2 Polski: Układ współrzędnych biegunowych oraz funkcja odwzorowująca dopełnienie dysku jednostkowego na dopełnienie zbioru Julia.
Date	4 November 2008
Source	Own work by uploader in Maxima and Gnuplot with help of many people ( see references )
Author	Adam majewski

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Here are two diagrams :

• on the left is dynamical plane for ${\displaystyle c=0\,}$
• on the right is dynamical plane for ${\displaystyle c=-2\,}$

On left diagram one can see :

• Julia set ( unit circle ) in red
• concentric circles outside unit circle
• external rays ( radial lines outside unit circle)

Right diagram is image of left diagram under function ${\displaystyle \Psi _{c}\,}$ (the Riemann map) which maps the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} }}}$ to the complement of the filled Julia set ${\displaystyle \ Kc}$

${\displaystyle \Psi _{c}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus Kc}$

For ${\displaystyle c=-2\,}$ :

${\displaystyle \Psi _{-2}(w)=w+{\frac {1}{w}}\,}$

It is :

• a simplest case for analysis,
• only one case when formula for computing ${\displaystyle \Psi _{c}\,}$ is known.

${\displaystyle \Psi _{-2}\,}$ maps ^[1]:

• red unit circle ${\displaystyle {w:abs(w)=1}\,}$ to blue line segment ${\displaystyle {z:z\in [-2,2]}\,}$ ( Julia sets )
• concentric circles to elipses ( equipotential lines )
• rays of unit circle to hyperbolas ( external rays )

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R_max:5;
R_min:1;
dR:R_max-R_min;
psi(w):=w+1/w;
NmbrOfRays:10;
iMax:100; /* number of points to draw */
GiveCirclePoint(t):=R*%e^(%i*t*2*%pi); /* gives point of unit circle for angle t in turns */
GiveWRayPoint(R):=R*%e^(%i*tRay*2*%pi); /* gives point of external ray for radius R and angle tRay in turns */ 
/* f_0 plane = w-plane */
/* unit circle */
R:1;
circle_angles:makelist(i/(10*iMax),i,0,10*iMax-1); /* more angles = more points */
CirclePoints:map(GiveCirclePoint,circle_angles);
/* external circles */
circle_radii:makelist(R_min+i,i,1,dR);
WCirclesPoints:[];
for R in circle_radii do 
  WCirclesPoints:append(WCirclesPoints,map(GiveCirclePoint,circle_angles));
/* external w rays */
ray_radii:makelist(R_min+dR*i/iMax,i,0,iMax);
ray_angles:makelist(i/NmbrOfRays,i,0,NmbrOfRays-1);
WRaysPoints:[];
for tRay in ray_angles do 
  WRaysPoints:append(WRaysPoints,map(GiveWRayPoint,ray_radii));
/* f_c plane = z plane = dynamic plane */
/* external z rays */
ZRaysPoints:map(psi,WRaysPoints);
/* Julia set points */
JuliaPoints:map(psi,CirclePoints);
Equipotentials:map(psi,WCirclesPoints);
load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
draw(file_name = "erays",
  pic_width=1000, 
  pic_height= 500,
  terminal  = 'png,
  columns  = 2,
  gr2d(title = " unit circle with external rays & circles ",
   point_type = filled_circle,
   points_joined =true,
   point_size    =  0.34,
   color         = red,
   points(map(realpart, CirclePoints),map(imagpart, CirclePoints)),
   points_joined =false,
   color         = black,
   points(map(realpart, WRaysPoints),map(imagpart, WRaysPoints)),
   points(map(realpart, WCirclesPoints),map(imagpart, WCirclesPoints))
   ),
  gr2d(title      = "Image under psi(w):=w+1/w; ",
   points_joined =true,
   point_type = filled_circle,
   point_size    =0.34,
   color         = blue,
   points(map(realpart, JuliaPoints),map(imagpart, JuliaPoints)),
   points_joined =false,
   color         = black,
   points(map(realpart, ZRaysPoints),map(imagpart, ZRaysPoints)),
   points(map(realpart, Equipotentials),map(imagpart, Equipotentials))
   ) 
 );