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 
and 
(which can be considered a kind of foci
with respect to multiplication instead of addition) is a constant 
. This gives the Cartesian
equation
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(1)
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Squaring both sides gives
![[(x-a)^2+y^2][(x+a)^2+y^2]=a^4,](/images/equations/Lemniscate/NumberedEquation2.svg) |
(2)
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and simplifying results in the beautiful form
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(3)
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The half-width (distance from crossing point at the origin to a horizontal extremity) of a lemniscate is
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(4)
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while its half-height is
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(5)
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Switching to polar coordinates gives the equation
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(6)
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or
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(7)
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Note that this equation is defined only for angles 
and 
.
The parametric equations for the lemniscate with half-width 
are
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(8)
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(9)
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The two-center bipolar coordinates equation with origin at a focus is
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(10)
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and in pedal coordinates with the pedal point at the center, the equation is
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(11)
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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.
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 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
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(12)
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with radius from the center of the hole to the center of the torus 
and tube radius 
with the plane 
gives an intersection described by
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(13)
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illustrated above. While the curve of intersection is close to the equation of a lemniscate in the 
-plane
with parameter 
:
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(14)
|
it is not equivalent due to the difference in the 
term as illustrated below:
However, in the special case of a torus with 
, the toric section becomes exactly a lemniscate with
half-width
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(15)
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The area of the lemniscate is
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(16)
|
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(17)
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(18)
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The arc length as a function of 
is given by
 |  | ![sqrt(2)aint_0^t[3-cos(2t)]^(-1/2)dt](/images/equations/Lemniscate/Inline32.svg) |
(19)
|
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(20)
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where 
is an elliptic integral of the first
kind. The arc length of the entire curve is then
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(21)
|
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(22)
|
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(23)
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(24)
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(OEIS A064853), where 
is the lemniscate constant,
which plays a role for the lemniscate analogous to that of 
for the circle.
The curvature and tangential angle of the lemniscate are
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(25)
|
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(26)
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