Summary

DescriptionIray.png	English: Internal ray for angle 1/3 of main cardioid of Mandelbrot set as an image of internal ray of unit circle under conformal map ${\displaystyle \gamma _{1}\,}$ which has the explicit equation Polski: Obraz koła jednostkowego i jego wewnętrznego promienia o kącie 1/3 po zasosowaniu funkcji konforemnej ${\displaystyle \gamma _{1}\,}$
Date	4 October 2008 (upload date)
Source	Own work by uploader in Maxima and Gnuplot with help of many people ( see references )
Author	Adam majewski

Long description

compare with

• 
  Mandelbrot set - multiplier map, internal rays , internal coordinate

Definitions

• Coordinates of point ${\displaystyle w\,}$ in exponential form are :

${\displaystyle w=r*e^{2\pi i\theta }\,}$

• Unit circle ${\displaystyle \partial D\,}$ is a boundary of unit disk

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

• Internal ray of angle ${\displaystyle \theta \,}$ of unit circle is :

${\displaystyle {\mathcal {R}}_{\theta }=\{\left(w=r*e^{2\pi i\theta }\right):\ 0\leq r<1\}}$

note that ${\displaystyle \theta \,}$ is measured in turns modulo 1

1 turn = 360 degrees = 2 * Pi radians

Maxima source code

 /*  batch file for maxima */ 
 NmbrOfPnts:100;
 /* ------------------------ definitions -------------------------*/

 gamma1(w):=w/2-w*w/4; /* conformal map from circle to cardioid  */

 l(t,r):=r*%e^(%i*t*2*%pi); /* exponential for of complex number with angle in turns */

 GiveIntRayOfCircleL(angle,NmbrOfPnts):=
 makelist(rectform(ev(l(angle,j/NmbrOfPnts), numer)),j, 0,NmbrOfPnts);

 GiveUnitCircleL(NmbrOfPnts):=
 makelist(rectform(ev(l(j/NmbrOfPnts,1), numer)),j, 0,NmbrOfPnts);

 /* ------------------  computes points of circle and ray ------------------*/
 circleList:GiveUnitCircleL(100);
 rayCircleList:GiveIntRayOfCircleL(1/3,100);

 /* ------------------- conformal mapping from  circle to cardioid ----------------*/
 cardioidList: map(gamma1, circleList);
 rayCardioidList:map(gamma1, rayCircleList);

 /* ---------- ------------ drawing -------------------------*/
 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(file_name = "iray",
  pic_width=1000, 
  pic_height= 500,
  terminal  = 'png,
  columns  = 2,
  gr2d(title = " unit circle D={w:abs(w)=1 } with internal ray for angle 1/3",
   points_joined =true,
   color         = red,
   point_type = 0,
   points(map(realpart,circleList),map(imagpart,circleList)),
   color         = green,
   points(map(realpart,rayCircleList),map(imagpart,rayCircleList))  ),
  gr2d(title      = "cardioid with internal_ray(1/3) as an image of circle under map gamma1(w):=w/2-w*w/4;",
   points_joined =true,
   color         = blue,
   point_type = 0,
   points(map(realpart,cardioidList),map(imagpart,cardioidList)),
   color         = green,
   points(map(realpart,rayCardioidList),map(imagpart,rayCardioidList)))  );

Licensing

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