Wikimore

Summary

Description

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 (original upload date)

Source

Own work by uploader in Maxima and Gnuplot with help of many people (see references)

Author

Adam majewski

Other versions

File:Erays.svg is a vector version of this file. It should be used in place of this PNG file when not inferior.Category:Vector version available

File:Erays.png → File:Erays.svg

For more information, see Help:SVG.

In other languages

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Source code

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 = '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))
	) 
);

Long description

Here are two diagrams:

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

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 $\Psi _{c}\,$ (the Riemann map) which maps the complement (exterior) of the closed unit disk ${\overline {\mathbb {D} }}$ to the complement of the filled Julia set $\ Kc$

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

For $c=-2\,$ :

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

It is:

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

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

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

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:

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

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.

Category:CC-BY-SA-3.0#Erays.png

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.

Category:License migration redundant#Erays.png Category:GFDL#Erays.png

You may select the license of your choice.

Category:Self-published work

References

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

Category:Gnuplot graphics Category:Complex analysis Category:Concentric circles Category:Images with Maxima CAS source code Category:Contour plots Category:External rays

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

[1]