A hypotrochoid, the looping flower a Spirograph toy draws. Six circles bring it within 1%.
Measuring the shape…
x = 2 cos t + 5 cos(2t/3), y = 2 sin t − 5 sin(2t/3), 0 ≤ t ≤ 6π
Spirograph drawings are hypotrochoids: a pen held inside a small gear that rolls inside a ring. That is already a chain of two circles, the same idea as epicycles, so the Fourier version needs very few.
Each path of the spirograph 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.