Read the epicycloid writeup first. The nephroid is a kidney-shaped plane algebraic curve. It is an epicycloid such that the outer of the two circles in the definition of an epicycloid has radius twice that of the other, a, say.

An equation for the nephroid is (x2 + y2 - 4a2)3 = 108a4y2.

Like other epicycloids it finds applications in the study of gears.