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 *(x*^{2} + y^{2}
- 4a^{2})^{3} = 108a^{4}y^{2}.

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