The flat shadow of the simplest knot. Four circles get it within 1%.
Measuring the shape…
x = sin t + 2 sin 2t, y = cos t − 2 cos 2t
This is the trefoil knot seen from above. The formula is already two circles, one turning once and one twice as fast in the other direction, so the Fourier drawing finds them almost straight away.
Each path of the trefoil 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.