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Comment by Wolf Jung : "Do you know that this is related to elephants (demo 2 page 10) ? There I am using just a rotation and rescaling on a small region instead of the non-linear transformation." See program Mandel demo 2 page 10^[1]

Maxima CAS src code

/*

 b batch file for maxima

http://mathr.co.uk/blog/2013-12-16_stretching_cusps.html
Claude Heiland-Allen

Figure 4.22 on pages 204-205 of The Science Of Fractal Images is presented with this description:

    A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. ... Our blow-up factor is chosen accordingly to the result that all disks in Figure 4.22 have the same size.

transformation from line, thru half of the circle, to half of the cardioid 

There are 3 complex planes :
* z-plane ( where are lines) = k-plane in Claude Heiland-Allen notation
* w-plane ( where are circles)
* c-plane ( for cardioid )

2 steps :
* from z plane go to w plane using w=fi(z)
* from w plane go to the  c plane using  c = gi(w)

so c = gi( fi ( z)) 
*/

kill(all);
remvalue(all);

/*

*/

z1:0;
z2:1;
/* z3:infinity;  z3 must be infinity */
zMax: 5; /* zMax << z3=infinity */

/* width between 2 rays */
z_width:%i/3; /*  gfi(%i/3)= -2  and z= i gives error  in fi function  */

/* 3 points on the unit hemicircle */
w1:-1;
w2:%i;
w3:1;

iMax:100; /* number of points to draw */
dz:abs(zMax-z1)/iMax; /* z step */

/* =================== functions ============ */

/* 
ray { z : z<=0 and imagpart(z)=0 }  to unit hemicircle where 0<= t <= pi
z= i gives error 
*/
fi(z):=(-%i-z)/(%i-z);

 /* conformal map from  circle to cardioid  */
 gi(w):=w/2-w*w/4; /* (1-(w+1)^2)/4;  */

/* check */
gfi(z):=gi(fi(z));

/* converts complex number into list for draw package  */
 draw_format(z):= [realpart(z),imagpart(z)];

compile(all);

/* ============== compute ===============  */

/* point to point method of drawing 
 compute first point of curve, create list and save point to this list */
  
 z:z1;    
 LineList:[z]; 
 for i:1 thru iMax step 1 do
  (  z:z+dz,
     LineList:cons(z,LineList)
 );

 z:z1+z_width;
 LineList2:[z]; 
 for i:1 thru iMax step 1 do
  (  z:z+dz,
     LineList2:cons(z,LineList2)
 );

 z_points:[z1,z2];
 z_points2:[z1+z_width,z2+z_width]; /* only 32 points, third is infinity */

/* mapping from  line to hemi circle */
 CircleList: map(fi, LineList) $
 CircleList2: map(fi, LineList2) $
 w_points:[w1,w2,w3]; 
 w_points2: map(fi, z_points2)$

/* conformal mapping from  circle to cardioid */
 CardioidList: map(gi, CircleList) $
 CardioidList2: map(gi, CircleList2) $ 
 c_points: map(gi, w_points) $
 c_points2: map(gi, w_points2) $

/* convert list to draw format */

 LineList:map(draw_format,LineList)$ 
 z_points: map(draw_format, z_points) $
 

  LineList2:map(draw_format,LineList2)$ 
  z_points2: map(draw_format, z_points2) $

 CircleList:map(draw_format,CircleList)$ 
 w_points: map(draw_format, w_points) $

 CircleList2:map(draw_format,CircleList2)$  
 w_points2: map(draw_format, w_points2) $

 CardioidList:map(draw_format, CardioidList)$
 c_points: map(draw_format, c_points) $

 CardioidList2:map(draw_format, CardioidList2)$ 
 c_points2: map(draw_format, c_points2) $

/* =================  draw ======================================*/

path:"~/maxima/batch/mandelbrot/circe2cardioid/maximus/w5/"$ /* pwd  */
 FileName:"a6"$ /* without extension which is the terminal name */

 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(
  terminal  = 'svg,
  file_name = concat(path,FileName),
  columns  = 3,
  dimensions=[1000,333], /*  x = y*columns  */
  
  gr2d(title = " rays ",
   /* xrange = [0,3],  */
   points_joined =true,
   color         = green,
   point_size    = 0.5,
   point_type = filled_circle,
   points(LineList),
   points(LineList2), 
   points_joined =false,
   point_type    = filled_circle,
   color         = black,
   point_size    = 1.0,
   points(z_points),
   points(z_points2) 
   ),

  gr2d(title = " circles = image(rays) ",
   points_joined =true,
   color         = red,
   point_size    = 0.5,
   point_type = filled_circle,
   points(CircleList),
   points(CircleList2), 
   points_joined =false,
   point_type    = filled_circle,
   color         = black,
   point_size    = 1.0,
   points(w_points),
   points(w_points2)),

  gr2d(title      = "cardioids = image(circles) ",
   points_joined =true,
   color         = blue,
   point_size    = 0.5,
   point_type = filled_circle,
   points(CardioidList),
   points(CardioidList2),
   points_joined =false,
   point_type    = filled_circle,
   color         = black,
   point_size    = 1.0,
   points(c_points),
   points(c_points))

 );

References

1. ↑ program Mandel by Wolf Jung