Lamé Curve

A curve with Cartesian equation

$$\left({x\over a}\right)^n+\left({y\over b}\right)^n=c$$

first discussed in 1818 by Lamé. If $n$ is a rational, then the curve is algebraic. However, for irrational $n$, the curve is transcendental. For Even Integers $n$, the curve becomes closer to a rectangle as $n$ increases. For Odd Integer values of $n$, the curve looks like the Even case in the Positive quadrant but goes to infinity in both the second and fourth quadrants (MacTutor Archive). The Evolute of an Ellipse,

$$(ax)^{2/3}+(by)^{2/3}=(a^2-b^2)^{2/3}.$$

$n$	Curve
${\textstyle{2\over 3}}$	Astroid
${\textstyle{5\over 2}}$	Superellipse
3	Witch of Agnesi

See also Astroid, Superellipse, Witch of Agnesi

References

MacTutor History of Mathematics Archive. ``Lamé Curves.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lame.html.