DeskripsiErays.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.
Tanggal	4 November 2008 (original upload date)
Sumber	Karya sendiri berdasarkan: Erays.png by Adam Majewski
Pembuat	Vektorisasi: Alhadis
Versi lainnya	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 genesis 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.

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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 (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)

Berkas ini dilisensikan dengan lisensi Creative Commons Atribusi-BerbagiSerupa 3.0 Tanpa Adaptasi

Atribusi:

Adam Majewski

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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.