DescriptionErays.svg	English: Polar coordinate system and mapping from the complement (exterior) of the closed unit disk to the complement of the filled Julia set for ${\displaystyle c=-2}$. বাংলা: জটিল গতিবিদ্যায় একক বৃত্ত Français : Uniformisation du complémentaire du segment ${\displaystyle [-2,2]}$. Bahasa Indonesia: Lingkaran satuan dalam dinamika kompleks. 日本語: リーマン写像による単位円の像としての単連結ジュリア集合。 Polski: Układ współrzędnych biegunowych oraz funkcja odwzorowująca dopełnienie dysku jednostkowego na dopełnienie zbioru Julia.
Date	4 November 2008 (original upload date)
Source	Own work based on: Erays.png by Adam Majewski
Author	Vectorization: Alhadis
Other versions	português do Brasil
Source code InfoField	Created using Maxima. 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); /* Mario Rodríguez Riotorto (http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html) */ load(draw); draw( file_name = "erays", pic_width = 1000, pic_height = 500, terminal = 'svg, 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)) ) );
SVG development InfoField	The SVG code is valid.   This vector image was created with Adobe Illustrator, and then manually edited. This file is saved in human-editable plain text format. Any editing of the image or creation of any derivative work should be performed using a text editor. Please do not upload edits saved or exported with Inkscape or similar vector graphics editors, as well as with automated tools such as SVG Translate.

This file supersedes the file Erays.png. It is recommended to use this file rather than the other one. Deutsch ∙ English ∙ español ∙ français ∙ magyar ∙ Bahasa Indonesia ∙ italiano ∙ 日本語 ∙ македонски ∙ മലയാളം ∙ Nederlands ∙ polski ∙ prūsiskan ∙ português do Brasil ∙ русский ∙ slovenščina ∙ svenska ∙ 中文（简体） ∙ 中文（繁體） ∙ +/−

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 (explicit Riemann mapping).

${\displaystyle \Psi _{-2}\,}$ 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 ellipses (equipotential lines)
• rays of unit circle to hyperbolas (external rays)

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Attribution: Adam Majewski

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• to remix – to adapt the work

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• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

1. ↑ Peitgen, Heinz-Otto; Richter Peter (1986) The Beauty of Fractals, Heidelberg: Springer-Verlag ISBN: 0-387-15851-0.