DescriptionCircle2cardioid.png	English: Conformal mapping from circle to cardioid Polski: Przekształcenie koła w kardioidę
Date	4 October 2008
Source	Own work by uploader ( using Maxima and Gnuplot ) with help of many people   This plot was created with Gnuplot by n.
Author	Adam majewski

File:Circle to cardioid.svg is a vector version of this file. It should be used in place of this PNG file when not inferior. File:Circle2cardioid.png → File:Circle to cardioid.svg For more information, see Help:SVG. In other languages Alemannisch ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

 

Cardioid ${\displaystyle \partial H_{1}\,}$ is an image of circle ${\displaystyle \partial D_{\frac {1}{2}}\,}$ under complex map ${\displaystyle g_{1}\,}$ ^[1]. It can be seen in nature^[2]

where :

• ${\displaystyle \partial D_{\frac {1}{2}}}$ is a circle with center at the origin and radius=1/2

${\displaystyle \partial D_{\frac {1}{2}}=\left\{w:abs(2w)=1\right\}}$ 

• Cardioid ${\displaystyle \partial H_{1}\,}$ is a boundary or period=1 hyperbolic component of Mandelbrot set

• ${\displaystyle g_{1}\,}$ is a complex map from circle to cardioid

${\displaystyle g_{1}:\partial D_{\frac {1}{2}}\rightarrow \partial H_{1}\,}$

or in other words :

${\displaystyle c=g_{1}(w)\,}$

Explicit equation of this map is :

${\displaystyle g_{1}(w)=w-w^{2}\,}$

Compare ${\displaystyle g_{1}\,}$ with inverse multiplier map ${\displaystyle \gamma _{1}\,}$ ^[3]:

${\displaystyle \gamma _{1}:\partial D_{1}\rightarrow \partial H_{1}\,}$

${\displaystyle \gamma _{1}(w)={\frac {w}{2}}-{\frac {w^{2}}{4}}\,}$

/* 
batch file for maxima
conformal mapping from :
the  circle with center=0 and radius=1/2
given by equation : abs(2*z)=1
where : z:x+y*%i=r*cos(t)+%i*r*sin(t) = 
onto 
cardioid ( boundary of main hyperbolic component of Mandelbrot set
is given by equation:
c:w-w*w;
based on :Conformal Mappings And The Area Of The Mandelbrot Set by David Allingham page 18
http://www.eng.warwick.ac.uk/staff/doa/reports/allingham-thesis1995.pdf archive copy at the Wayback Machine
*/ 
/* conformal map */
f(w):=w-w*w;
iMax:100; /* number of points to draw */
dt:1/iMax;
/* 
circle D={w:abs(2w)=1 } where w=l(t) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
*/
l(t):=%e^(%i*t*2*%pi)/2;
/* point to point method of drawing */
t:0; /* angle in turns */ 
/* compute first point of curve, create list and save point to this list */
/* point of unit circle   w:l(t); */
w:rectform(ev(l(t), numer)); 
circleList:makelist (w, j, 1, 1); 
for i:1 thru iMax step 1 do
block
(t:t+dt,
w:rectform(ev(l(t), numer)), 
circleList:cons(w,circleList)
);
/* convert one list into 2 lists */
xxCircle:map(realpart, circleList);
yyCircle:map(imagpart, circleList); 
/* conformal mapping from  circle to cardioid */
cardioidList: map(f, circleList);	
/* convert one list into 2 lists */
xxCardioid:map(realpart, cardioidList);
yyCardioid:map(imagpart, cardioidList);
load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html archive copy at the Wayback Machine */
draw(file_name = "a",
 pic_width=1000, 
 pic_height= 500,
 terminal  = 'png,
 columns  = 2,
 gr2d(title = " circle D={w:abs(2*w)=1 } ",
  points_joined =true,
  color         = red,
  point_type = 0,
  points(xxCircle,yyCircle)),
 gr2d(title      = "cardioid {c: c = w - w*w } ",
  points_joined =true,
  color         = blue,
  point_type = 0,
  points(xxCardioid,yyCardioid)) 
);

1. ↑ 3D-XplorMath \ Conformal Maps \ a*z^b+b*z
2. ↑ Cardioid as a shadow of ring. Foto by Sebastian Tkacz
3. ↑ Boundary of components of Mandelbrot set computed using boundary equations

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.

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

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.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

You may select the license of your choice.