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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573901.png" /></td> </tr></table>
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$$\left(\frac xa\right)^m+\left(\frac yb\right)^m=1,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573904.png" /> are coprime numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573906.png" />. The order of Lamé's curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573907.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l0573909.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l05739010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l05739011.png" />, Lamé's curve is a straight line, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l05739012.png" /> it is an ellipse, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l05739013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057390/l05739014.png" /> it is an [[Astroid|astroid]]. The Lamé curves are named after G. Lamé, who considered them in 1818.
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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 16:51, 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://www.encyclopediaofmath.org/index.php?title=Lam%C3%A9_curve&oldid=11566
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article