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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} $).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017000a.gif" />

Figure: b017000a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017000b.gif" />

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|Persian curve]].

Named after J. Booth [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Booth,   "A treatise on some new geometrical methods" , '''1–2''' , London  pp. 1873–1877</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  pp. 144–146  (In Russian)</TD></TR></table>