A loop inside a loop, from one short formula. Eight circles bring it within 1%.
Measuring the shape…
r = 1 + 2 sin θ
Étienne Pascal, Blaise Pascal's father, studied it, so it is also called Pascal's limaçon (French for snail). When the second number is bigger than the first, the curve passes back through the centre and makes the small inner loop. With equal numbers it becomes a cardioid.
Each path of the limaçon is traced by a chain of circles, each one spinning on the tip of the one before. A Fourier transform measures the path and works out every circle's size, speed and starting angle. The circles are sorted biggest first, so the first few give the rough shape and the rest add detail.
"Within 1%" means the drawing is, on average, less than 1% of its width away from the real outline. The bars show the circles of the main outline: height is size, position is how many times it turns per loop and in which direction. Click a bar to remove that circle and see what it was doing. Learn how it works.