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A plane algebraic curve of order four whose equation in orthogonal Cartesian coordinates is
 
A plane algebraic curve of order four whose equation in orthogonal 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/b/b017/b017000/b0170001.png" /></td> </tr></table>
+
$$
 +
(x  ^ {2} + y  ^ {2} )  ^ {2} -
 +
(2m  ^ {2} + n) x  ^ {2} -
 +
(2m  ^ {2} - n) y  ^ {2}  = 0.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170002.png" />, the Booth lemniscate is called elliptic (it has singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170003.png" /> (Fig. a), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170004.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170005.png" />, the Booth lemniscate is called hyperbolic (it has a nodal point at the coordinate origin, cf. Fig. b, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170006.png" />).
+
If $  | n | < 2 m  ^ {2} $,  
 +
the Booth lemniscate is called elliptic (it has singular point $  O $(
 +
Fig. a), where $  0 < n < 2 m  ^ {2} $).  
 +
If $  | n | > 2 m  ^ {2} $,  
 +
the Booth lemniscate is called hyperbolic (it has a nodal point at the coordinate origin, cf. Fig. b, where $  n > 2 m  ^ {2} $).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017000a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017000a.gif" />
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The equation of an elliptic Booth lemniscate in polar coordinates is
 
The equation of an elliptic Booth lemniscate in polar 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/b/b017/b017000/b0170007.png" /></td> </tr></table>
+
$$
 +
\rho  ^ {2}  = a  ^ {2}  \cos  ^ {2}  \phi +
 +
b  ^ {2}  \sin  ^ {2}  \phi \ \
 +
\textrm{ or } \  \rho  \equiv  0.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b0170008.png" />, the equation of a hyperbolic Booth lemniscate has the form
+
If $  n > 2 m  ^ {2} $,  
 +
the equation of a hyperbolic Booth lemniscate 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/b/b017/b017000/b0170009.png" /></td> </tr></table>
+
$$
 +
\rho  ^ {2}  = \
 +
a  ^ {2}  \cos  ^ {2}  \phi - b  ^ {2}  \sin  ^ {2}  \phi ;
 +
$$
  
and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017000/b01700010.png" />
+
and if $  n < -2 m  ^ {2} $
  
<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/b/b017/b017000/b01700011.png" /></td> </tr></table>
+
$$
 +
\rho  ^ {2}  = \
 +
- a  ^ {2}  \cos  ^ {2}  \phi + b ^ {2}  \sin  ^ {2}  \phi
 +
$$
  
<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/b/b017/b017000/b01700012.png" /></td> </tr></table>
+
$$
 +
(a  ^ {2}  = | 2m  ^ {2} + n | ,\  b  ^ {2}  = | 2m  ^ {2} - n | ).
 +
$$
  
 
The arc length of a Booth lemniscate is expressed by elliptic integrals. The area bounded by an elliptic Booth lemniscate is
 
The arc length of a Booth lemniscate is expressed by elliptic integrals. The area bounded by an elliptic Booth lemniscate 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/b/b017/b017000/b01700013.png" /></td> </tr></table>
+
$$
 +
= {
 +
\frac \pi {2}
 +
}
 +
(a  ^ {2} + b ^ {2} ),
 +
$$
  
 
while that bounded by a hyperbolic Booth lemniscate is
 
while that bounded by a hyperbolic Booth lemniscate 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/b/b017/b017000/b01700014.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{a  ^ {2} - b ^ {2} }{2}
 +
\
 +
\mathop{\rm arctg}  {
 +
\frac{a}{b}
 +
} + {
 +
\frac{ab}{2}
 +
} .
 +
$$
  
 
The Booth lemniscate is a special case of a [[Persian curve|Persian curve]].
 
The Booth lemniscate is a special case of a [[Persian curve|Persian curve]].

Revision as of 06:28, 30 May 2020


A plane algebraic curve of order four whose equation in orthogonal Cartesian coordinates is

$$ (x ^ {2} + y ^ {2} ) ^ {2} - (2m ^ {2} + n) x ^ {2} - (2m ^ {2} - n) y ^ {2} = 0. $$

If $ | n | < 2 m ^ {2} $, the Booth lemniscate is called elliptic (it has singular point $ O $( Fig. a), where $ 0 < n < 2 m ^ {2} $). If $ | n | > 2 m ^ {2} $, the Booth lemniscate is called hyperbolic (it has a nodal point at the coordinate origin, cf. Fig. b, where $ n > 2 m ^ {2} $).

Figure: b017000a

Figure: b017000b

The equation of an elliptic Booth lemniscate in polar coordinates is

$$ \rho ^ {2} = a ^ {2} \cos ^ {2} \phi + b ^ {2} \sin ^ {2} \phi \ \ \textrm{ or } \ \rho \equiv 0. $$

If $ n > 2 m ^ {2} $, the equation of a hyperbolic Booth lemniscate has the form

$$ \rho ^ {2} = \ a ^ {2} \cos ^ {2} \phi - b ^ {2} \sin ^ {2} \phi ; $$

and if $ n < -2 m ^ {2} $

$$ \rho ^ {2} = \ - a ^ {2} \cos ^ {2} \phi + b ^ {2} \sin ^ {2} \phi $$

$$ (a ^ {2} = | 2m ^ {2} + n | ,\ b ^ {2} = | 2m ^ {2} - n | ). $$

The arc length of a Booth lemniscate is expressed by elliptic integrals. The area bounded by an elliptic Booth lemniscate is

$$ S = { \frac \pi {2} } (a ^ {2} + b ^ {2} ), $$

while that bounded by a hyperbolic Booth lemniscate is

$$ S = \frac{a ^ {2} - b ^ {2} }{2} \ \mathop{\rm arctg} { \frac{a}{b} } + { \frac{ab}{2} } . $$

The Booth lemniscate is a special case of a Persian curve.

Named after J. Booth [1].

References

[1] J. Booth, "A treatise on some new geometrical methods" , 1–2 , London pp. 1873–1877
[2] A.A. Savelov, "Planar curves" , Moscow (1960) pp. 144–146 (In Russian)
How to Cite This Entry:
Booth lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Booth_lemniscate&oldid=46118
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article