Nephroid

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

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

with $n=2$,

$$r^2 = {a^2\over 2^2} [(2^2+2\cdot 2+2)-2(2+1)\cos(2\phi)]$$

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

where

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

This can be written

$$\left({r\over 2a}\right)^{2/3}=[\sin({\textstyle{1\over 2}}\theta)]^{2/3}+[\cos({\textstyle{1\over 2}}\theta)]^{2/3}.$$ (4)

The parametric equations are

$$x = a[3\cos t-\cos(3t)]$$ (5)

$$y = a[3\sin t-\sin(3t)].$$ (6)

The Cartesian equation is

$$(x^2+y^2-4a^2)^3=108 a^4 y^2.$$ (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

References

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

Lee, X. ``Nephroid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html.

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.

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