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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,


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

See also Astroid, Superellipse, Witch of Agnesi


MacTutor History of Mathematics Archive. ``Lamé Curves.''

© 1996-9 Eric W. Weisstein