Summary

Licensing

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

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

Category:CC-BY-SA-3.0#Circle%20to%20cardioid.svg

Long description

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

where :

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

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

• 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

 /* 
 batch file for Maxima CAS
 http://maxima.sourceforge.net/

 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

 Adam Majewski
 */ 

 kill(all);
 remvalue(all);

 /* ----------- functions ---------------------*/
 /* conformal map */
 f(w):=w/2-w*w/4;
 
 /* 
 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);

compile(all)$

/* ---- const ----*/

 iMax:100; /* number of points to draw */
 dt:1/iMax;
 

 /* 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); 
 while (t<=1) do
 (  t:t+dt,
    w:rectform(ev(l(t), numer)), 
    circleList:cons(w,circleList)
 );
 
 /* conformal mapping from  circle to cardioid */
 cardioidList: map(f, circleList);	

/* convert one list into 2 lists */
 xxCircle:map(realpart, circleList);
 yyCircle:map(imagpart, circleList); 

 /* convert one list into 2 lists */
 xxCardioid:map(realpart, cardioidList);
 yyCardioid:map(imagpart, cardioidList);

 /* ------------ draw ----------------------------- */
 path:"~/maxima/batch/mandelbrot/circe2cardioid/"$ /* pwd  */
 FileName:"a5"$

 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(
  file_name = concat(path,FileName),
  dimensions=[1000,500],
  terminal  = 'svg,
  columns  = 2,
  gr2d(title = " circle D=\\{w:abs(w)=1 \\} ",
   points_joined =true,
   color         = red,
   point_type = 0,
   points(xxCircle,yyCircle)),
  gr2d(title      = "cardioid \\{c: c = w/2-w*w/4 \\} ",
   points_joined =true,
   color         = blue,
   point_type = 0,
   points(xxCardioid,yyCardioid)) 
 );

References

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

</syntaxhighlight>