The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most common form is the locus of points the product of whose distances from two fixed points (called the foci) a distance away is the constant . This gives the Cartesian equation

(1)

where both sides of the equation have been squared. Expanding and simplifying then gives

(2)

Jakob Bernoulli published an article in Acta Eruditorum in 1694 in which he called this curve the lemniscus (Latin for "a pendant ribbon"). Bernoulli was not aware that the curve he was describing was a special case of Cassini ovals which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by G. Fagnano in 1750 (MacTutor Archive). Gauss's and Euler's investigations of the arc length of the curve led to later work on elliptic functions.

The most general form of the lemniscate is a toric section of a torus

(3)

cut by a plane . Plugging in and rearranging gives the equation

(4)

In the special case (and rewriting as ), this becomes

(5)

which is the same form obtained in equation (1).

The two-center bipolar coordinates equation with origin at a focus is

(6)

Switching to polar coordinates gives the equation

(7)

usually simply written

(8)

where is a constant (differing from the torus radius by a factor of ). Note that this equation is only defined for angles and . The parametric equations for the lemniscate with width are

			(9)
			(10)

The bipolar equation of the lemniscate is

(11)

and in pedal coordinates with the pedal point at the center, the equation is

(12)

The lemniscate can also be generated as the envelope of circles centered on a rectangular hyperbola and passing through the center of the hyperbola (Wells 1991).

The lemniscate is the inverse curve of the hyperbola with respect to its center.

The area of the lemniscate is

			(13)
			(14)
			(15)

The arc length at a function of is given by

			(16)
			(17)

where is an elliptic integral of the first kind. The arc length of the entire curve is then

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

(Sloane's A064853), which is known as the lemniscate constant. If , then is related to Gauss's constant by

(24)

The quantity (or sometimes ) is called the lemniscate constant and plays a role for the lemniscate analogous to that of for the circle.

The curvature and tangential angle of the lemniscate are

			(25)
			(26)

Ayoub, R. "The Lemniscate and Fagnano's Contributions to Elliptic Integrals." Arch. Hist. Exact Sci. 29, 131-149, 1984.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 220, 1987.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Gray, A. "Lemniscates of Bernoulli." §3.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 52-53, 1997.

Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 143, 2002.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 120-124, 1972.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 37, 1983.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1967.

MacTutor History of Mathematics Archive. "Lemniscate of Bernoulli." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lemniscate.html.

Sloane, N. J. A. Sequence A064853 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 139-140, 1991.

Yates, R. C. "Lemniscate." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 143-147, 1952.

CITE THIS AS:

Weisstein, Eric W. "Lemniscate." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Lemniscate.html