**Parametric Cartesian equation: **

*x* = *a*(3cos(*t*) - cos(3*t*)), *y* = *a*(3sin(*t*) - sin(3*t*))

**Click below to see one of the Associated curves.**

If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

The name nephroid (meaning 'kidney shaped') was used for the two-cusped epicycloid by Proctor in 1878. The nephroid is the epicycloid formed by a circle of radius

The nephroid has length 24*a* and area 12π^{2}.

Huygens, in 1678, showed that the nephroid is the catacaustic of a circle when the light source is at infinity. He published this in *Traité de la lumière*in 1690. An explanation of why this should be was not discovered until the wave theory of light was used. Airy produced the theoretical proof in 1838.

R A Proctor was an English mathematician. He was born in 1837 and died in 1888. In 1878 he published *The geometry of cycloids*in London.

The involute of the nephroid is Cayley's sextic or another nephroid since they are parallel curves. To see the nephroid as an involute of itself see Involute 2 above constructing the involute through the point where the nephroid cuts the *y*-axis.

**Other Web site:**

JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Nephroid.html