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\begin{figure}\BoxedEPSF{nephroid.epsf scaled 400}\end{figure}

The 2-Cusped Epicycloid is called a nephroid. Since $n=2$, $a=b/2$, and the equation for $r^2$ in terms of the parameter $\phi$ is given by Epicycloid equation

r^2= {a^2\over n^2}\left[{(n^2+2n+2)-2(n+1)\cos(n\phi)}\right]
\end{displaymath} (1)

with $n=2$,
$\displaystyle r^2$ $\textstyle =$ $\displaystyle {a^2\over 2^2} [(2^2+2\cdot 2+2)-2(2+1)\cos(2\phi)]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}a^2[10-6\cos(2\phi)]={\textstyle{1\over 2}}a^2[5-3\cos(2\phi)],$ (2)

\tan\theta={3\sin\phi-\sin(3\phi)\over 3\cos\phi-\cos(3\phi)}.
\end{displaymath} (3)

This can be written
\left({r\over 2a}\right)^{2/3}=[\sin({\textstyle{1\over 2}}\theta)]^{2/3}+[\cos({\textstyle{1\over 2}}\theta)]^{2/3}.
\end{displaymath} (4)

The parametric equations are
$\displaystyle x$ $\textstyle =$ $\displaystyle a[3\cos t-\cos(3t)]$ (5)
$\displaystyle y$ $\textstyle =$ $\displaystyle a[3\sin t-\sin(3t)].$ (6)

The Cartesian equation is
(x^2+y^2-4a^2)^3=108 a^4 y^2.
\end{displaymath} (7)

The name nephroid means ``kidney shaped'' and was first used for the two-cusped Epicycloid by Proctor in 1878 (MacTutor Archive). The nephroid has Arc Length $24a$ and Area $12\pi^2a^2$. The Catacaustic for rays originating at the Cusp of a Cardioid and reflected by it is a nephroid. Huygens showed in 1678 that the nephroid is the Catacaustic of a Circle when the light source is at infinity. He published this fact in Traité de la luminère in 1690 (MacTutor Archive).

See also Astroid, Deltoid, Freeth's Nephroid


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

Lee, X. ``Nephroid.''

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.''

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

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© 1996-9 Eric W. Weisstein