# Booth lemniscate

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

If , the Booth lemniscate is called elliptic (it has singular point (Fig. a), where ). If , the Booth lemniscate is called hyperbolic (it has a nodal point at the coordinate origin, cf. Fig. b, where ).

Figure: b017000a

Figure: b017000b

The equation of an elliptic Booth lemniscate in polar coordinates is

If , the equation of a hyperbolic Booth lemniscate has the form

and if

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

while that bounded by a hyperbolic Booth lemniscate is

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. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Booth_lemniscate&oldid=18130