Summary

DescriptionCircle2heart.png	English: Circle to non-cardioid heart curve mapping.
Date	9 December 2008
Source	Own work by uploader in Maxima and Gnuplot with help of many people ( see references )  This plot was created with gnuplot by n. Category:PNG created with Gnuplot#Circle2heart.png
Author	Adam majewski
Other versions	This math image could be re-created using vector graphics as an SVG file. This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is available, please upload it and afterwards replace this template with {{vector version available|new image name}}. It is recommended to name the SVG file “Circle2heart.svg”—then the template Vector version available (or Vva) does not need the new image name parameter. Category:Math images that should use vector graphics#%20Circle2heart.pngCategory:PNG that should use vector graphics

Long description

"It's not complex-analytic, but

${\displaystyle (x,y)\mapsto (x,{\sqrt {|x|}}-y)}$

makes nice-looking heart shapes out of circles centered on the origin (different for different radii)" ^[1]

so in Maxima notation :

• point of circle is w :

w:x+y*%i;

• point of heart curve is

c:x+f(w)*%i;

where

f(w):=sqrt(abs(realpart(w)))-imagpart(w);

Compare it with

• Image showing non-cardioid heart curve as a shadow of ring
• similar image, but different position of ring
• Conformal mapping from circle to cardioid
• Image GJL-fft-herz.png
• File:Simple 4-quadrant heart curve.svg
• Image made by Monguin61 using MATLAB

Maxima source code

/* 
batch file for maxima
mapping from :
the  circle with center=0 and radius=r
given by equation : abs(2*z)=1
where : z:x+y*%i=r*cos(t)+%i*r*sin(t) = 
onto 
non-cardioid heart curve 
is given by equation:
c:x+(sqrt(abs(x))- y)*%i;
*/ 
/*  */
f(w):=sqrt(abs(realpart(w)))-imagpart(w);
iMax:1000; /* number of points to draw */
dt:1/iMax;
r:4; /* radius of circle */
/* 
circle D={w:abs(2w)=1 } where w=l(t) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
*/
l(t):=r*%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)); /* "exponential form prevents allroots from working", code by Robert P. Munafo */ 
circleList:makelist (w, j, 1, 1); 
for i:1 thru iMax step 1 do
block
(t:t+dt,
w:rectform(ev(l(t), numer)), /* "exponential form prevents allroots from working", code by Robert P. Munafo */
circleList:cons(w,circleList)
);
/* convert one list into 2 lists */
xxCircle:map(realpart, circleList);
yyCircle:map(imagpart, circleList);
/* convert one list into 2 lists and maps */
xxHeart:map(realpart, circleList);
yyHeart:map(f, circleList);
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",
    terminal  = 'png,	 pic_width  = 1000,
  pic_height = 500,
columns  = 2,
gr2d(title = " circle D={w:abs(2*w)=4 } ",
points_joined =true,
color         = red,
point_type = 0,
points(xxCircle,yyCircle)),
gr2d(title      = "non-cardioid heart shape curve ",
points_joined =true,
color         = blue,
point_type = 0,
points(xxHeart,yyHeart)) 
);

References

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