# Booth lemniscate

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)

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How to Cite This Entry:
Booth lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Booth_lemniscate&oldid=53313
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article