The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive).

The nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690 (MacTutor Archive). (Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)

Since the nephroid has cusps, , and the equation for in terms of the parameter is given by epicycloid equation

(1)

with ,

(2)

where

(3)

This can be written

(4)

The parametric equations are

			(5)
			(6)
			(7)

The Cartesian equation is

(8)

The nephroid has area and arc length,

			(9)
			(10)

The arc length, curvature, and tangential angle as a function of parameter are

			(11)
			(12)
			(13)

where the expressions for and are valid for .

The nephroid can be generated as the envelope of circles centered on a given circle and tangent to one of the circle's diameters (Wells 1991).

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 169-173, 1972.

Lockwood, E. H. "The Nephroid." Ch. 7 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 62-71, 1967.

MacTutor History of Mathematics Archive. "Nephroid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Nephroid.html.

Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 158, 1991.

Yates, R. C. "Nephroid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 152-154, 1952.

CITE THIS AS:

Weisstein, Eric W. "Nephroid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Nephroid.html