This kidney formed curve 1) is the epicycloid with two cusps. The rolling circle is half the size as the rolled circle. The English mathematician R.A. Proctor (1837-1888) gave the curve its name, in 1878 in his book 'The geometry of cycloids'.

Some other properties of the curve are:

  • the curve is the catacaustic of the circle, with the source at infinity
    Christiaan Huygens, who was the first to describe the curve in 1678, found this (also in 1678), and he published this findings in 1690 in his essay 'Traité de la lumière'. One had to wait for 1838 that Airy proved the fact theoretically.
  • the curve is the catacaustic of the cardioid, with the cusp as source.
  • It was Jakob Bernoulli who showed this in 1692.
  • the nephroid is the evolute of Cayley's sextic
  • the nephroid is the envelope of a set of circles with centers on a circle and tangent to a diameter

The nephroid has a length of 24, and its area is 12π.
The curve is a sextic curve2).
The nephroid is said to be the perfect multi-seater dining-table.

The curve can be linearly stretched to an elongated nephroid.


1) Nephros (Greek) = kidney

2) With Cartesian equation: (x2 + y2 - 4)3 = 108 y 2