File:Circle2cardioid.png

Summary[edit]

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.
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 ∙ +/−

 

Long description[edit]

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]:

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

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

Maxima source code[edit]

/* 
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)) 
);

References[edit]

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

Licensing[edit]

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