File:Erays.svg
Original file (SVG file, nominally 1,000 × 500 pixels, file size: 612 KB)
Captions
Summary
[edit]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 .
বাংলা: জটিল গতিবিদ্যায় একক বৃত্ত
Français : Uniformisation du complémentaire du segment .
Bahasa Indonesia: Lingkaran satuan dalam dinamika kompleks.
Polski: Układ współrzędnych biegunowych oraz funkcja odwzorowująca dopełnienie dysku jednostkowego na dopełnienie zbioru Julia. |
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Date | 4 November 2008 (original upload date) | ||
Source | Own work based on: Erays.png by Adam Majewski | ||
Author | Vectorization: Alhadis | ||
SVG development InfoField |
|
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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))
)
);
|
This file supersedes the file Erays.png. It is recommended to use this file rather than the other one.
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Long description
[edit]Here are two diagrams:
- on the left is dynamical plane for
- on the right is dynamical plane for
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 (the Riemann map) which maps the complement (exterior) of the closed unit disk to the complement of the filled Julia set
For :
It is:
- a simplest case for analysis,
- only one case when formula for computing is known (explicit Riemann mapping).
maps [1]:
- red unit circle to blue line segment (Julia sets)
- concentric circles to ellipses (equipotential lines)
- rays of unit circle to hyperbolas (external rays)
Licensing
[edit]- 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 the same or compatible license as the original.
- ↑ Peitgen, Heinz-Otto; Richter Peter (1986) The Beauty of Fractals, Heidelberg: Springer-Verlag ISBN: 0-387-15851-0.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:35, 16 February 2023 | 1,000 × 500 (612 KB) | Alhadis (talk | contribs) | Recreated SVG using librsvg-compatible markup. | |
18:02, 16 February 2023 | 1,000 × 500 (853 KB) | Alhadis (talk | contribs) | == {{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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Width | 1000 |
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Height | 500 |