A kidney shape with two cusps, traced by a wheel rolling round a circle twice its size. Seven circles bring it within 1%.
Measuring the shape…
x = 3 cos t − cos 3t, y = 3 sin t − sin 3t
The nephroid is an epicycloid: a point on a circle rolling around the outside of one twice as big. Shine light into a cup or a ring and the bright curve on the bottom is a nephroid. Christiaan Huygens and Tschirnhaus studied it in the 1670s while working on reflected light.
Each path of the nephroid 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.