# Curves

### Nephroid

- Parametric Cartesian equation:
- $x = a(3\cos(t) - \cos(3t)), y = a(3\sin(t) - \sin(3t))$

### Description

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 $a$ rolling externally on a fixed circle of radius $2a$.The nephroid has length $24a$ and area $12\pi a^{2}$.

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:**

Xah Lee