# Difference between revisions of "Lamé curve"

From Encyclopedia of Mathematics

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A plane algebraic curve whose equation in rectangular Cartesian coordinates has the form | 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 | + | 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|astroid]]. The Lamé curves are named after G. Lamé, who considered them in 1818. |

====References==== | ====References==== |

## Latest revision as of 14:51, 1 May 2014

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) |

#### Comments

#### 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:**

Lamé curve.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_curve&oldid=32042

This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article