Difference between revisions of "Lamé curve"
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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]]. The Lamé curves are named after G. Lamé, who considered them in 1818. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> F. Gomes Teixeira, "Traité des courbes" , '''1–3''' , Chelsea, reprint (1971)</TD></TR> | |
| − | + | </table> | |
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Latest revision as of 18:29, 4 May 2023
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) |
| [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=22701
Lamé curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_curve&oldid=22701
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article