A four-pointed star traced by a small wheel rolling inside a big one. Two circles give the shape; the sharp points take a few more.
Measuring the shape…
x = cos³ t, y = sin³ t
The astroid is a hypocycloid: a circle a quarter the size rolls inside a larger one, and a point on its rim traces four cusps. In the formula it is exactly two circles, but here the pen moves at an even speed along the path, so the cusps take a handful more.
Each path of the astroid 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.