Bernoulli's figure eight, the shape behind the infinity sign. Seven circles bring it within 1%.
Measuring the shape…
x = cos t / (1 + sin² t), y = sin t cos t / (1 + sin² t)
Jacob Bernoulli described it in 1694: every point on it has distances to two fixed points that multiply to the same number. Its circles come in equal pairs spinning opposite ways, because the curve is symmetric left to right.
Each path of the lemniscate 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.