Lamé curve

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A plane algebraic curve whose equation in rectangular Cartesian coordinates has the form

$$\left(\frac xa\right)^m+\left(\frac yb\right)^m=1,$$

where $m=p/q$, $p$ and $q$ are coprime numbers, $a>0$ and $b>0$. The order of Lamé's curve is $pq$ if $m>0$ and $2pq$ if $m<0$. If $m=1$, Lamé's curve is a straight line, if $m=2$ it is an ellipse, and if $m=2/3$ and $a=b$ it is an astroid. The Lamé curves are named after G. Lamé, who considered them in 1818.

References

 [1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

References

 [a1] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962) [a2] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
How to Cite This Entry:
Lame curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lame_curve&oldid=23365