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Lemniscate


Lemniscate

The lemniscate, also called the lemniscate of Bernoulli, is a polar curve defined as the locus of points such that the the product of distances from two fixed points (-a,0) and (a,0) (which can be considered a kind of foci with respect to multiplication instead of addition) is a constant a^2. This gives the Cartesian equation

 sqrt((x-a)^2+y^2)sqrt((x+a)^2+y^2)=a^2.
(1)

Squaring both sides gives

 [(x-a)^2+y^2][(x+a)^2+y^2]=a^4,
(2)

and simplifying results in the beautiful form

 (x^2+y^2)^2=2a^2(x^2-y^2).
(3)

The half-width (distance from crossing point at the origin to a horizontal extremity) of a lemniscate is

 x_(max)=a,
(4)

while its half-height is

 y_(max)=a/(2sqrt(2)).
(5)

Switching to polar coordinates gives the equation

 r^2=a^2cos(2theta),
(6)

or

 r=asqrt(cos(2theta)).
(7)

Note that this equation is defined only for angles -pi/4<theta<pi/4 and 3pi/4<theta<5pi/4.

The parametric equations for the lemniscate with half-width a are

x=(acost)/(1+sin^2t)
(8)
y=(asintcost)/(1+sin^2t).
(9)

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

 rr^'=1/2a^2,
(10)

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

 pa^2=r^3.
(11)

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

LemniscateEnvelope

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).

LemniscateToricSection

The lemniscate resembles certain toric sections when the cutting plane is tangent to the torus along the circumference of its central hole. For example, intersecting a torus

 (c^'-sqrt(x^2+y^2))^2+z^2=a^('2)
(12)

with radius from the center of the hole to the center of the torus c^'=1 and tube radius a^'=4/10 with the plane y=6/10 gives an intersection described by

 8/5x^2-x^4-(12)/5z^2-2x^2z^2-z^4,
(13)

illustrated above. While the curve of intersection is close to the equation of a lemniscate in the xz-plane with parameter a=2sqrt(2/5):

 8/5x^2-x^4-8/5z^2-2x^2z^2-z^4,
(14)

it is not equivalent due to the difference in the z^2 term as illustrated below:

LemniscateToricSectionComparison

However, in the special case of a torus with c^'=2a^', the toric section becomes exactly a lemniscate with half-width

 a=2sqrt(a^'c^').
(15)

The area of the lemniscate is

A=2(1/2intr^2dtheta)
(16)
=a^2int_(-pi/4)^(pi/4)cos(2theta)dtheta
(17)
=a^2.
(18)

The arc length as a function of t is given by

s(t)=sqrt(2)aint_0^t[3-cos(2t)]^(-1/2)dt
(19)
=aF(t,i),
(20)

where F(z,k) is an elliptic integral of the first kind. The arc length of the entire curve is then

s=4int_0^a(dr)/(sqrt(1-(r/a)^4))
(21)
=4aint_0^1(1-t^4)^(-1/2)dt
(22)
=2La
(23)
=5.2441151086...a
(24)

(OEIS A064853), where L is the lemniscate constant, which plays a role for the lemniscate analogous to that of pi for the circle.

The curvature and tangential angle of the lemniscate are

kappa(t)=(3sqrt(2)cost)/(asqrt(3-cos(2t)))
(25)
phi(t)=3tan^(-1)(sint).
(26)

See also

Cassini Ovals, Devil's Curve, Dumbbell Curve, Eight Curve, Figure Eight, Lemniscate Constant, Lemniscate Function, Lichtenfels Minimal Surface, Mandelbrot Set Lemniscate, Toric Section, Viviani's Curve

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References

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."Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 329, 1958.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. https://mathworld.wolfram.com/Lemniscate.html

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