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File:Erays.svg

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File:Erays.svg
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Summary

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 forc=−2{\displaystyle c=-2}.
বাংলা: জটিল গতিবিদ্যায় একক বৃত্ত
Français : Uniformisation du complémentaire du segment[−2,2]{\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.
Date4 November 2008 (original upload date)
SourceOwn work based on:Erays.png byAdam Majewski
AuthorVectorization:Alhadis
Other versions
  • português do Brasil
    português do Brasil
Source code
InfoField
Created usingMaxima.
R_max:5;R_min:1;dR:R_max-R_min;psi(w):=w+1/w;NmbrOfRays:10;iMax:100;/*numberofpointstodraw*/GiveCirclePoint(t):=R*%e^(%i*t*2*%pi);/*givespointofunitcircleforangletinturns*/GiveWRayPoint(R):=R*%e^(%i*tRay*2*%pi);/*givespointofexternalrayforradiusRandangletRayinturns*//*f_0plane=W-plane*//*Unitcircle*/R:1;circle_angles:makelist(i/(10*iMax),i,0,10*iMax-1);/*moreangles=morepoints*/CirclePoints:map(GiveCirclePoint,circle_angles);/*Externalcircles*/circle_radii:makelist(R_min+i,i,1,dR);WCirclesPoints:[];forRincircle_radiidoWCirclesPoints:append(WCirclesPoints,map(GiveCirclePoint,circle_angles));/*ExternalWrays*/ray_radii:makelist(R_min+dR*i/iMax,i,0,iMax);ray_angles:makelist(i/NmbrOfRays,i,0,NmbrOfRays-1);WRaysPoints:[];fortRayinray_anglesdoWRaysPoints:append(WRaysPoints,map(GiveWRayPoint,ray_radii));/*f_cplane=Zplane=dynamicplane*//*externalZrays*/ZRaysPoints:map(psi,WRaysPoints);/*Juliasetpoints*/JuliaPoints:map(psi,CirclePoints);Equipotentials:map(psi,WCirclesPoints);/*MarioRodríguezRiotorto(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
 
TheSVG code isvalid.
 
Thisvector image was created withAdobe 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 atext editor. Please donot upload edits saved or exported withInkscape or similarvector graphics editors, as well as with automated tools such asSVG Translate.
This filesupersedes the fileErays.png. It is recommended to use this file rather than the other one.

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minor quality

Long description

Here are two diagrams:

On left diagram one can see:

Right diagram is image of left diagram under functionΨc{\displaystyle \Psi _{c}\,} (theRiemann map) which maps thecomplement (exterior) of theclosed unit diskD¯{\displaystyle {\overline {\mathbb {D} }}} to the complement of thefilled Julia set Kc{\displaystyle \ Kc}

Ψc:C^∖D¯→C^∖Kc{\displaystyle \Psi _{c}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus Kc}

Forc=−2{\displaystyle c=-2\,}:

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

It is:

  • a simplest case for analysis,
  • only one case when formula for computingΨc{\displaystyle \Psi _{c}\,} is known (explicit Riemann mapping).

Ψ−2{\displaystyle \Psi _{-2}\,} maps[1]:

Licensing

w:en:Creative Commons
attributionshare alike
This file is licensed under theCreative CommonsAttribution-Share Alike 3.0 Unported license.
Attribution:Adam Majewski
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under thesame or compatible license as the original.
https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue
  1. ↑Peitgen, Heinz-Otto; Richter Peter (1986)The Beauty of Fractals,Heidelberg: Springer-VerlagISBN:0-387-15851-0.

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4 November 2008

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Date/TimeThumbnailDimensionsUserComment
current20:35, 16 February 2023Thumbnail for version as of 20:35, 16 February 20231,000 × 500(612 KB)AlhadisRecreated SVG using librsvg-compatible markup.
18:02, 16 February 2023Thumbnail for version as of 18:02, 16 February 20231,000 × 500(853 KB)Alhadis== {{int:filedesc}} == {{Information | Description = {{en|Polar coordinate system and mapping from the complement (exterior) of the closed unit disk to the complement of the filled Julia set for <math>c=-2</math>.}} {{pl|Układ współrzędnych biegunowych oraz funkcja odwzorowująca dopełnienie dysku jednostkowego na dopełnienie zbioru Julia.}} | Source = {{Own}} | Date = {{Original upload date|2008-11-04}} | Author = {{U|Soul windsurfer|Adam Majewski}} | Other fields = {{Created with code|+=Sour...

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