Three cusps from a wheel rolling inside a circle three times its size. Within 1% with seven circles.
Measuring the shape…
x = 2 cos t + cos 2t, y = 2 sin t − sin 2t
Also called Steiner's hypocycloid, after Jakob Steiner. A circle rolls inside one three times as big. It turns up in the Kakeya needle problem: a needle can turn all the way round inside a deltoid.
Each path of the deltoid 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.