Happy Valentines Day!

Anonymous, 14th February 2012

Happy Valentine's Day to all Chebfun users!

t = chebfun('t',[-pi pi]);
x = 16*sin(t).^3;
y = 13*cos(t)-5*cos(2*t)-2*cos(3*t)-1*cos(4*t);
fill(x,y,'.-r'); axis equal
hold on
plot(12.5*scribble('Happy Valentine''s Day!'),'k')
hold off
axis equal off

This 'heart curve' can be found on Wolfram's MathWorld site [1], where they compute its area. We do the same, and compare the result:

z = x+1i*y;
A = abs(sum(x.*diff(y)))
err = A - 180*pi

A =
     5.654866776461628e+02
err =
     0

Here are some more heart curves that can be found in [1]:

Heart eqn 1

t = chebfun('t',[-pi pi]);
r = 1-sin(t);
[x1 y1] = pol2cart(t,r);
plot(x1,y1,'.-r')
axis equal off
A1 = abs(sum(x1.*diff(y1)))
err = A1-3*pi/2 % Note, this is A2 on [1]!

A1 =
   4.712388980384689
err =
    -8.881784197001252e-16

Heart eqn 5

t = chebfun('t',[-pi pi]);
r = 2 - 2*sin(t)+sin(t).*sqrt(abs(cos(t))+.1)./(sin(t)+1.4);
[x5 y5] = pol2cart(t,r);
plot(x5,y5,'.-r')
axis equal off
A5 = abs(sum(x5.*diff(y5))) % Note, different from [1] was we include .1 in sqrt

A5 =
  11.645555308722983

References [1] http://mathworld.wolfram.com/HeartCurve.html