with(plots):
animate(1+(2-4*abs(x))*cos(theta),theta=0..2*Pi,
   x=-1..1,coords=polar,thickness=2,numpoints=100,
   scaling=constrained,tickmarks=[0,0],frames=50,
   labels=[``,``],color=blue);

1. The family of limaçons is varied by making a range from -2 to 2, and then back to -2 again.  This is done by the function , with .
2. An animation editor was used to slow down the animation.  Each frame is displayed for 0.1 seconds, except the end frames, which were given longer display times.