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Plane algebraic curves of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581301.png" /> such that the product of the distances of each point of the curve from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581302.png" /> given points (foci) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581303.png" /> is equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581304.png" />-th power of a given number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581305.png" /> (the radius of the lemniscate). The equation of a lemniscate in rectangular Cartesian coordinates is
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Plane algebraic curves of order $2n$ such that the product of the distances of each point of the curve from $n$ given points (foci) $F_1,\dotsc,F_n$ is equal to the $n$-th power of a given number $r$ (the radius of the lemniscate). The equation of a lemniscate in rectangular Cartesian coordinates is
  
<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/l058/l058130/l0581306.png" /></td> </tr></table>
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$$|(z-z_1)\dotsm(z-z_n)|=r^n,\quad r>0,\quad z=x+iy.$$
  
 
A circle is a lemniscate with one focus, and a [[Cassini oval|Cassini oval]] is a lemniscate with two foci. See also [[Bernoulli lemniscate|Bernoulli lemniscate]] and [[Booth lemniscate|Booth lemniscate]].
 
A circle is a lemniscate with one focus, and a [[Cassini oval|Cassini oval]] is a lemniscate with two foci. See also [[Bernoulli lemniscate|Bernoulli lemniscate]] and [[Booth lemniscate|Booth lemniscate]].
  
 
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A lemniscate is a level curve of a polynomial. If all the foci $F_k$: $z_k=x_k+iy_k$, $k=1,\dotsc,n$, are distinct and the radius of the lemniscate is sufficiently small, then the lemniscate consists of $n$ continua that have pairwise no common points. For a sufficiently large radius a lemniscate consists of one connected component. As D. Hilbert showed in 1897, the boundary $\Gamma$ of an arbitrary simply-connected finite domain can be arbitrarily closely approximated by a lemniscate, that is, for any $\epsilon>0$ one can find a lemniscate $\Lambda$ such that in the $\epsilon$-neighbourhood of each point of $\Gamma$ there are points of $\Lambda$ and every point of $\Lambda$ is in the $\epsilon$-neighbourhood of an appropriate point of $\Gamma$.
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>
 
 
 
A lemniscate is a level curve of a polynomial. If all the foci <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581307.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l0581309.png" />, are distinct and the radius of the lemniscate is sufficiently small, then the lemniscate consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813010.png" /> continua that have pairwise no common points. For a sufficiently large radius a lemniscate consists of one connected component. As D. Hilbert showed in 1897, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813011.png" /> of an arbitrary simply-connected finite domain can be arbitrarily closely approximated by a lemniscate, that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813012.png" /> one can find a lemniscate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813013.png" /> such that in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813014.png" />-neighbourhood of each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813015.png" /> there are points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813016.png" /> and every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813017.png" /> is in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813018.png" />-neighbourhood of an appropriate point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058130/l05813019.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Walsh,  "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1965)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Walsh,  "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1965)</TD></TR>
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</table>
  
 
''E.D. Solomentsev''
 
''E.D. Solomentsev''

Latest revision as of 06:49, 7 April 2023

Plane algebraic curves of order $2n$ such that the product of the distances of each point of the curve from $n$ given points (foci) $F_1,\dotsc,F_n$ is equal to the $n$-th power of a given number $r$ (the radius of the lemniscate). The equation of a lemniscate in rectangular Cartesian coordinates is

$$|(z-z_1)\dotsm(z-z_n)|=r^n,\quad r>0,\quad z=x+iy.$$

A circle is a lemniscate with one focus, and a Cassini oval is a lemniscate with two foci. See also Bernoulli lemniscate and Booth lemniscate.

A lemniscate is a level curve of a polynomial. If all the foci $F_k$: $z_k=x_k+iy_k$, $k=1,\dotsc,n$, are distinct and the radius of the lemniscate is sufficiently small, then the lemniscate consists of $n$ continua that have pairwise no common points. For a sufficiently large radius a lemniscate consists of one connected component. As D. Hilbert showed in 1897, the boundary $\Gamma$ of an arbitrary simply-connected finite domain can be arbitrarily closely approximated by a lemniscate, that is, for any $\epsilon>0$ one can find a lemniscate $\Lambda$ such that in the $\epsilon$-neighbourhood of each point of $\Gamma$ there are points of $\Lambda$ and every point of $\Lambda$ is in the $\epsilon$-neighbourhood of an appropriate point of $\Gamma$.

References

[a1] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2] J.L. Walsh, "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc. (1965)

E.D. Solomentsev

How to Cite This Entry:
Lemniscates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lemniscates&oldid=11912
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article