The name "nephroid" (from the Latin for "kidney-shaped") was introduced in 1878 by the English mathematician Richard Proctor in his book The Geometry of Cycloids. Prior to that it was known as a two-cusped epicycloid.
Specifically, the nephroid is the epicycloid formed by a circle of radius a rolling around the outside on a fixed circle of radius 2a. It has a length of 24a, an area of 12π2, and is given by the parametric equations:
x = a(3cos t - 3cos 3t)
y = a(3sin t - sin 3t).
The nephroid is the involute of Cayley's sextic and is also the envelope of circles with their centers on a given circle, touching a given diameter of that circle.
Freeth's nephroidThe nephroid has been described as the perfect shape for a multi-seat dining table. Not to be mistaken with the ordinary nephroid, just described, is Freeth's nephroid, named after the English mathematician T. J. Freeth (1819–1904) who first wrote about it in a paper published by the London Mathematical Society in 1879. Freeth's nephroid is the strophoid of a circle and has the polar equation r = a(1 + 2sinθ/2). Freeth's nephroid is also the name of group of mathematicians, mostly from Royal Holloway College, London, who gather weekly in a pub called the Beehive and compete in games of trivial pursuit.
Related category PLANE CURVES
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